Determine whether the triangles are similar. If so, write a similarity

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage1

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage2

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage3

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage4

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage5

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage6

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage7

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage8

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage9

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage10

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage11

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage12

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage13

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage14

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage15

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage16

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage17

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage18

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage19

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage20

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage21

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

eSolutionsManual-PoweredbyCogneroPage22

9-3 Similar Triangles

Determine whether the triangles are similar. If

so, write a similarity statement. Explain your

reasoning.

1.

SOLUTION:

We can prove byAASimilarity.



1) We can prove that because

theyarealternateinterioranglesand

.(AlternateInteriorAnglesTheorem)



2)Wecanprovethat because

they are vertical angles. ( Vertical angles Theorem)

2.

SOLUTION:

We can prove bySASSimilarity.



1) We can prove that because they are

both right angles.(All right angles are congruent.)



2) Since these are right triangles, we can use the

PythagoreanTheoremtofindthemissingsides









Now, since we are using SAS Similarity to prove this

relationship, we can set up ratios of corresponding

sides to see if they are equal. We will match short

sidetoshortsideandmiddlesidetomiddleside.



So, bySASSimilarity.

3.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles can be

proven similar by using the SSS Similarity

Theorem.This requires that we determine if each pair

ofcorrespondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are not

allthesame,thesetrianglesarenotsimilar.

4.

SOLUTION:

Since no angles measures are provided in these

triangles, we can determine if these triangles are

similar by using the SSS Similarity Theorem.This

requires that we determine if each pair of

correspondingsideshaveanequalratio.



We know the following correspondences exist

because we are matching longest side to longest side,

middle to middle, and shortest to shortest:



Since the ratios of the corresponding sides are

equal, by SSS Similarity.



5.

MULTIPLE CHOICE

In the figure,

intersects atpointC. Which additional

information would be enough to prove that

DAC

and

ECB

are congruent.

 and arecongruent.

and areparallel.

CBE

is a right angle.

SOLUTION:

Since , by the Vertical Angle

Theorem,optionCisthebestchoice.Ifweknow

that , then we know that the alternate

interior angles formed by these segments and sides

and are congruent. This would allow us to

useAASimilaritytoprovethetrianglesaresimilar.



CCSSSTRUCTUREIdentifythesimilar

triangles. Find each measure.

6.KL

SOLUTION:

By AA Similarity,



Use the corresponding side lengths to write a

proportion.



Solve for x.

7.VS

SOLUTION:

Wecanseethat because all right

trianglesarecongruent.

Additionally, ,byReflexiveProperty.



Therefore, by AA Similarity, .



Use the corresponding side lengths to write a

proportion.



Solve for x.

8.

COMMUNICATION

A cell phone tower casts a

100-foot shadow. At the same time, a 4-foot 6-inch

post near the tower casts a shadow of 3 feet 4

inches. Find the height of the tower.

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the tower’sheight.(Note that 1 foot = 12

inches and covert all the dimensions to inches)



100 ft = 1200 inches

4 feet 6 inches = 54 inches

3 feet 4 inches = 40 inches.



Substitute these corresponding values in the

proportion.



So, the cell phone tower is 1620 inches or 135 feet

tall.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

9.

SOLUTION:



Matching up short to short, middle to middle, and long

to long sides, we get the following ratios:



Since, then

bySSSSimilarity

10.

SOLUTION:

No; needstobeparallelto for

byAASimilarity.Additionally,there

are no given side lengths to compare to use SAS or

SSSSimilaritytheorems.

11.

SOLUTION:

We know that , because their

measures are equal. We also can match up the

adjacent sides that include this angle and determine if

they have the same ratio. We will match short to

shortandmiddletomiddlelengths.



Yes; since and , we

know that bySASSimilarity.

Determine whether the triangles are similar. If

so, write a similarity statement. If not, what

would be sufficient to prove the triangles

similar? Explain your reasoning.

12.

SOLUTION:

We know that due to the Reflexive

property. Additionally, we can prove that

, because they are corresponding

anglesformedbyparallellines.Therefore,

byAASimilarity.

13.

SOLUTION:

The known information for relates to a SAS

relationship, whereas the known information for

is a SSA relationship. Since they are no the

same relationship, there is not enough information to

determineifthetrianglesaresimilar.



If JH = 3 or WY = 24, then all the sides would have

the same ratio and we could prove

bySSSSimilarity.

14.

SOLUTION:

No; the angles of are 59, 47, and 74 degrees

and the angles of are 47, 68, and 65 degrees.

Since the angles of these triangles won't ever be

congruent, so the triangles can never be similar.

15.

CCSS MODELING

When we look at an object, it

is projected on the retina through the pupil. The

distances from the pupil to the top and bottom of the

object are congruent and the distances from the pupil

to the top and bottom of the image on the retina are

congruent. Are the triangles formed between the

object and the pupil and the object and the image

similar? Explain your reasoning.



SOLUTION:

Yes; sample answer: and

therefore, we can state that their ratios

areproportional,or We also know that

becauseverticalanglesare

congruent. Therefore, bySAS

Similarity.

ALGEBRA Identify the similar triangles. Then

find each measure.

16.JK

SOLUTION:

We know that vertical angles are congruent. So,

Additionally,wearegiventhat .

.

Therefore, by AA Similarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.

17.ST

SOLUTION:

By the Reflexive Property, we know that

.

Also, since , we know that

 ( Corresponding Angle Postulate).



Therefore,byAASimilarity,



Use the corresponding side lengths to write a

proportion.







Solve for x.



18.WZ, UZ

SOLUTION:

Wearegiventhat and we also

knowthat ( All right angles are

congruent.)



Therefore, by AA Similarity,



Use the Pythagorean Theorem to find WU.



Since the length must be positive, WU = 24.



Use the corresponding side lengths to write a

proportion.







Solve for x.



Substitute x = 12 in WZ and UZ.

= 3x – 6

=3(12)– 6

=30

= x + 6

=12+6

=18

19.HJ, HK

SOLUTION:

Since we are given two pairs of congruent angles,

we know that , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in HJ and HK.

= 4(2) + 7

=15

= 6(2) – 2

= 10

20.DB, CB

SOLUTION:

We know that ( All right angles

are congruent.) and we are given that

.

Therefore, , by AA Similarity.



Use the corresponding side lengths to write a

proportion.



Solve for x.



Substitute x = 2 in DB and CB.

= 2(2) + 1

=5

= 2 (2) – 1 + 12

=15

21.GD, DH

SOLUTION:

We know that ( Reflexive Property) and

are given .

Therefore, by AA Similarity.



Use the corresponding side lengths to write a

proportion:







Solve for x.



Substitute x=8inGD and DH.

= 2 (8) – 2

=14

= 2 (8) + 4

=20

22.

STATUES

Mei is standing next to a statue in the

park. If Mei is 5 feet tall, her shadow is 3 feet long,

and the statue’s shadow is feet long, how tall is

the statue?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Since two pairs of angles are congruent, the right

trianglesaresimilarbytheAASimilarityPostulate.



So, the following proportion can be written:



Let x be the statue’s height and substitute given

values into the proportion:



So, the statue's height is 17.5 feet tall.

23.

SPORTS

When Alonzo, who is tall,stands

next to a basketball goal, his shadow is long,and

the basketball goal’s shadow is long.About

how tall is the basketball goal?

SOLUTION:

Make a sketch of the situation. 4 feet 6 inches is

equivalent to 4.5 feet.

In shadow problems, you can assume that the angles

formed by the Sun’s rays with any two objects are

congruent and that the two objects form the sides of

two right triangles.



Sincetwopairsofanglesarecongruent,theright

triangles are similar by the AA Similarity Postulate.

So, the following proportion can be written:



Let x be the basketball goal’s height. We know that 1

ft=12in..Convertthegivenvaluestoinches.



Substitute.

24.

FORESTRY

A hypsometer, as shown, can be used

to estimate the height of a tree. Bartolo looks through

the straw to the top of the tree and obtains the

readings given. Find the height of the tree.

SOLUTION:

Triangle EFD in the hypsometer is similar to triangle

GHF.

Therefore, the height of the tree is (9 + 1.75) or

10.75 meters.

PROOF Write a two-column proof.

25.Theorem 9.3

SOLUTION:

A good way to approach this proof is to consider

how you can get by AA Similarity.

You already have one pair of congruent angles (

) , so you just need one more pair. This

can be accomplished by proving that

and choosing a pair of

corresponding angles as your CPCTC. To get those

triangles congruent, you will need to have proven that

but you have enough information in

the given statements to do this. Pay close attention to

how the parallel line statement can help. Once these

triangles are similar, you can create a proportion

statement and combine it with the given statements

to create the

relationship that .



Given: ,

Prove:

Proof:

Statements (Reasons)

2. , (Corr. 's

Post.)

3. (Trans.Prop.)

4. (AASimilarity)

6. AB·QP = AQ·BC; AB·EF = DE·BC (Cross

products)

7. QP = EF (Def. of segs.)

8. AB·EF = AQ·BC (Subst.)

9. AQ·BC = DE·BC (Subst.)

10. AQ = DE (Div. Prop.)

11.

12.

13. (CPCTC)

14. (Trans.Prop.)

15. (AASimilarity)

26.Theorem 9.4

SOLUTION:

This is a three-part proof, as you need to prove three

different relationships - that Reflexive, Symmetric,

and Transitive properties are true for similar

triangles. For each part of this proof, the key is to

find a way to get two pairs of congruent angles

which will allow you to use AA Similarity

Postulate.As you try these, remember that you

already know that these three properties already hold

for congruent triangles and can use these

relationshipsinyourproofs.



Reflexive Property of Similarity

Given:

Prove:

Proof:

Statements (Reasons)

2. , (Refl.Propof .)

3. (AASimilarity)



Transitive Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of

polygons)

3. (Trans.Prop.)

4. (AASimilarity)



Symmetric Property of Similarity

Given:

Prove:

Statements (Reasons)

2. (Def.of polygons)

3. (Symm.Propof.)

4. (AASimilarity)

PROOF Write a two-column proof.

27.

Given:

and arerighttriangles;

Prove:

SOLUTION:

The given information in this proof is almost all you

need to prove by SAS Similarity

theorem. You already have two pairs of proportional

corresponding sides. You just need to think about

how to get the included angles congruent to each

other.



Proof:

Statements (Reasons)

1. and arerighttriangles.(Given)

2. and arerightangles.(Def.ofrt.

)

3. (Allrt.anglesare .)

4. (Given)

5. (SASSimilarity)

28.

Given:

ABCD is a trapezoid.

Prove:

SOLUTION:

Think backwards when attempting this proof. In

order to prove that , we need to show that

. To prove triangles are similar, you

need to prove two pairs of corresponding angles are

congruent. Think about what you know about

trapezoids and how that can help you get

.



Proof:

Statements (Reasons)

1. ABCD is a trapezoid. (Given)

2. (Def.oftrap.)

3. (Alt.Int.angle

Thm.)

4. (AASimilarity)

5. (Corr.sidesof s are proportional.)

29.

CCSS MODELING

When Luis’s dad threw a

bounce pass to him, the angles formed by the

basketball’s path were congruent. The ball landed

ofthewaybetweenthembeforeitbouncedback

up. If Luis’s dad released the ball 40 inches above

the floor, at what height did Luis catch the ball?

SOLUTION:

Since the ball landed ofthewaybetweenthem,

the horizontal line is in the ratio of 2:1.

By AA Similarity, the given two triangles are similar.



Form a proportion and solve for x. Assume that Luis

will catch the ball at a height of x inches.



So, Luis will catch the ball 20 inches above the floor.

COORDINATE GEOMETRY and

haveverticesX(–1, –9), Y(5, 3), Z(–1,

6), W(1, –5), and V(1, 5).

30.Graph the triangles, and prove that

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now, find the ratios of the corresponding sides:



 



Since by

SSS Similarity.

31.Find the ratio of the perimeters of the two triangles.

SOLUTION:

We can prove that by using the

distance formula to determine the lengths of each

side of the triangles. Then, we can set up ratios to

determine if the ratios of corresponding sides are

equal and use SSS Similarity theorem to prove the

trianglesaresimilar.



The lengths of the sides of are:



YZ =



ZX = 6 – (–9)=15;



The lengths of the sides of are:



VW = 5 – (–5)=10;



WY = 



YV =



Now,findtheperimeterofeachtriangle:



32.

BILLIARDS

When a ball is deflected off a smooth

surface, the angles formed by the path are

congruent. Booker hit the orange ball and it followed

the path from A to B to C as shown below. What

was the total distance traveled by the ball from the

time Booker hit it until it came to rest at the end of

the table?

SOLUTION:

By AA Similarity, the given triangles are similar.

FormaproportionandsolveforBC.Convertthe

fractions to decimals.



So, the total distance traveled by the ball is about 61

in..

33.

PROOF

Use similar triangles to show that the slope

of the line through any two points on that line is

constant. That is, if points A, B, and areon

line usesimilartrianglestoshowthattheslopeof

the line from A to B is equal to the slope of the line

from to

SOLUTION:

In this proof, it is important to recognize that and

are both vertical lines and are, therefore,

parallel to each other. Using this relationship, along

with the fact that line is a transversal of these

segments, we can prove that

Once this is proven, you can use a proportion

statementtocompletetheproof.



since all rt. angles are . Line isa

transversalof||segments and so

sincecorrespondinganglesof||

lines are . Therefore, by AA Similarity,

So theslopeofline

throughpointsA and B, is equal to the

slope of line throughpoints and .

34.

CHANGING DIMENSIONS

Assume that

If the lengths of the sides of arehalfthe

length of the sides of andtheareaof

is40squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

If the lengths of the sides of arethree

times the length of the sides of andthearea

of is63squareinches,whatistheareaof

Howisthearearelatedtothescalefactor

of to

SOLUTION:

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 10 square

inches.

The ratio of the areas is the square of the scale

factor.

Let b and h be the base and height of the triangle

ABC respectively.

Thus, the area of the triangle JKL is 7 square inches.

The ratio of the areas is the cube of the scale factor.

35.

MEDICINE

Certain medical treatments involve

laser beams that contact and penetrate the skin,

forming similar triangles. Refer to the diagram. How

far apart should the laser sources be placed to ensure

that the areas treated by each source do not overlap?

SOLUTION:

For 100 cm, it covers an area that has a radius of 15

cm. It penetrates and go inside the skin for 5 cm. so,

the total height is 105 cm. Assume that for 105 cm,

laser source covers an area that has a radius of x

cm.

Form a proportion.

So, the laser beam covers 15.75 + 15.75 or 31.5 cm.

36.

MULTIPLE REPRESENTATIONS

In this

problem, you will explore proportional parts of

triangles.

a. GEOMETRIC

Draw a with

parallel

asshown.

b. TABULAR

Measure and record the lengths AD,

DB, CD, and EB and the ratios

and ina

table.

c. VERBAL

Make a conjecture about the segments

created by a line parallel to one side of a triangle and

intersecting the other two sides.

SOLUTION:

The triangle you draw doesn't have to be

congruent to the one in the text. However, measure

carefully so that is parallel to side

.

Sampleanswer:

When measuring the side lengths, it may be

easiest to use centimeters. Fill in the table with the

correspondingmeasures.



Sample answer:



Observe patterns you notice in the table that are

formed by the ratios of sides of a triangle cut by a

parallelline.



Sample answer: The segments created by a line || to

onesideofatriangleandintersectingtheothertwo

sides are proportional.

37.

WRITING IN MATH

Compare and contrast the

AA Similarity Postulate, the SSS Similarity Theorem,

and the SAS similarity theorem.

SOLUTION:

Sample answer: The AA Similarity Postulate, SSS

Similarity Theorem, and SAS Similarity Theorem are

all tests that can be used to determine whether two

trianglesaresimilar.



The AA Similarity Postulate is used when two pairs

ofcongruentanglesoftwotrianglesaregiven.



The SSS Similarity Theorem is used when the

corresponding side lengths of two triangles are

given.



The SAS Similarity Theorem is used when two

proportional side lengths and the included angle of

two triangles are given.



38.

CHALLENGE

isanaltitudeof Find

YW.

SOLUTION:

Both

are isosceles right

triangles, so by AA Similarity postulate, we know

thattheyaresimilar.Thisallowsustosetupa

proportionofcorrespondingsidelengthstofindYW:



First,weneedtofindthelengthofXZ.



Now, substitute the side lengths you know into the

proportion .



Therefore, YW= .



39.

REASONING

A pair of similar triangles has angle

measuresof45°,50°,and85°.Thesidesofone

triangle measure 3, 3.25, and 4.23 units, and the sides

of the second triangle measure x

–

0.46, x, and x +

1.81 units. Find the value of x

.



SOLUTION:

Using the given information, sketch two triangles and

label the corresponding sides and angles. Make sure

you use the Angle- Sides relationships of triangles to

place the shortest sides across from the smallest

angles,etc.



Form a proportion and solve for x.



40.

OPEN ENDED

Draw a triangle that is similar to

shown.Explainhowyouknowthatitis

similar.

SOLUTION:

When making a triangle similar to , keep in

mind the relationships that exist between the angles

of similar triangles, as well as the sides. We know

that the corresponding sides of similar triangles are

proportional and the corresponding angles are

congruent.



Sample answer:

becausethemeasuresofeach

side are half the measure of the corresponding side

and the measures of corresponding angles are equal.

41.

WRITINGINMATH

How can you choose an

appropriate scale?

SOLUTION:

Sample answer: You could consider the amount of

space that the actual object occupies and compare it

to the amount of space that is

available for the scale model or drawing. Then, you

could determine the amount of detail that you want

the scale model or drawing to have, and you could

use these factors to choose an appropriate scale.

42.

PROBABILITY

3.0

0.33



–

3x + 2



–

+ 2x

SOLUTION:

So, the correct option is D.

43.

EXTENDED RESPONSE

In the figure below,

Write a proportion that could be used to find x.

Find the value of x and the measure of

SOLUTION:



Since we know ,

and



because they are corresponding

anglesformedbyparallellines.Therefore,

and corresponding sides are

proportional.



Solve for x.



Substitute x = 9.5 in AB.

AB = x

–

= 9.5

–

= 7.5

44.

ALGEBRA

Which polynomial represents the area

of the shaded region?



+ r

π

–

SOLUTION:

The area of the circle is

.

The area of one white triangle is , because the

radius of the circle is both the height and the base of

thetriangle.



The area of two white triangles would be

.



To find the area of the shaded region, you can

subtract the area of the two white triangles from the

circle'sarea.

So, the correct option is J.

45.

SAT/ACT

The volume of a certain rectangular solid

is 16x cubic units. If the dimensions of the solid are

integers x, y, and z units, what is the greatest possible

value of z?

E 2

SOLUTION:

The volume of a rectangular solid with dimensions x,

y, and z is given by xyz. So xyz = 16. Since all

dimensions are integers, and since lengths must be

positive, the least possible value of x

and

y is 1. In

that case, z = 16. So the correct answer is B.

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

46.

SOLUTION:

The order of vertices in a similarity statement

identifies the corresponding angles and sides. Since

we know that , we can take the

corresponding angles of this statement and set them

congruent to each other. Then, since the

corresponding sides of similar triangles are

proportional to each other, we can write a proportion

thatrelatesthecorrespondingsidestoeachother.



47.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding a

this statement and set them congruent to each other.

the corresponding sides of similar triangles are propor

each other, we can write a proportion that relates the

correspondingsidestoeachother.



48.

SOLUTION:

The order of vertices in a similarity statement identifi

corresponding angles and sides. Since we know that

, we can take the corresponding

statement and set them congruent to each other. The

corresponding sides of similar polygons are proportio

other, we can write a proportion that relates the corr

sidestoeachother.



Solve each proportion.

49.

SOLUTION:



Cross multiply.



Solve for x.



50.

SOLUTION:



Cross multiply.



Solve for x.



51.

SOLUTION:



Cross multiply.



Solve for x.



52.

SOLUTION:



Cross multiply.



Solve for x.



53.

TANGRAMS

A tangram set consists of seven

pieces: a small square, two small congruent right

triangles, two large congruent right triangles, a

medium-sized right triangle, and a quadrilateral. How

can you determine the shape of the quadrilateral?

Explain.

SOLUTION:

Consider the properties of different quadrilaterals

when answering this question. The shape appears to

beaparallelogram,thereforeyoucantestthe

conditions of a parallelogram to see if they are true

forthisshape.



Sample answer: If one pair of opposite sides are

congruent and parallel, the quadrilateral is a

parallelogram.

Determine which postulate can be used to

prove that the triangles are congruent. If it is

not possible to prove congruence, write not

possible.

54.

SOLUTION:

We are given two pairs of congruent sides and a pair

of congruent angles. However, the congruent angles

are not the include angle between the two sides.

Therefore, it is a SSA relationship, not a SAS

relationship and it is not possible to prove these

trianglescongruent.

55.

SOLUTION:

We are given one pair of congruent angles and one

pair of congruent sides ( by Reflexive property).

However, it not possible to prove these triangles

congruent because we can't prove any other pair of

sidesoranglescongruent.

56.

SOLUTION:

Wearegiventhattwopairsofsidesarecongruent

and can prove the third pair of sides is, as well, by

using Reflexive property. Therefore, these triangles

can be proven congruent with SSS.

Write a two-column proof.

57.

Given:

Prove:

SOLUTION:

There are many angles in this diagram, so it is easy

to get confused by which ones to use. Notice how

the given statement guides you to using and .

Because they have different transversals, they are

not related to the same set of parallel lines. However,

they are both related to . If you can think about

how to get supplementary to , then you can

prove that

.



Given:

Prove:

Proof:

Statements (Reasons)

(Given)

(Consecutive Interior Angle

Theorem)

(Definitionofsupplementary

angles)

4. (Definition of congruent angles)

(Substitution)

(Definition of supplementary)

(Ifcons.int.

s are suppl., then lines are

||.)

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9-3 Similar Triangles