© Houghton Mifflin Harcourt Publishing Company
Name Class Date
Explore
Constructing Similar Triangles
In the following activity you will see one way to construct a triangle similar to a given triangle.
A
Do your work for Steps A–C in the space provided. Draw a triangle.
Label it ABC as shown.
B
Select a point on
_
AB . Label it E.
C
Construct an angle with vertex E that is congruent to ∠B. Label the point where the
side of the angle you constructed intersects
_
AC as F.
D
Why are
‹
−
›
EF and
_
BC parallel?
E
Use a ruler to measure
_
AE ,
_
EB ,
_
AF , and
_
FC . Then compare the
ratios
AE
___
EB
and
AF
___
FC
.
Resource
Locker
B
A
C
B
E
A
C
\\192.168.9.251\07Macdata\07Vol1Data\From Graphics\02192014\From Ramesh\Econ 2016\Worksheet
B
E
A
C
F
Module 12
631
Lesson 1
12.1 Triangle Proportionality
Theorem
Essential Question: When a line parallel to one side of a triangle intersects the other two
sides, how does it divide those sides?
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=NL-D;CA-D
© Houghton Mifflin Harcourt Publishing Company
Reflect
1. Discussion How can you show that △AEF ∼ △ABC? Explain.
2. What do you know about the ratios
AE
___
AB
and
AF
___
AC
? Explain.
3. Make a Conjecture Use your answer to Step E to make a conjecture about the line
segments produced when a line parallel to one side of a triangle intersects the other
two sides.
Explain 1
Proving the Triangle Proportionality
Theorem
As you saw in the Explore, when a line parallel to one side of a triangle intersects the other
two sides of the triangle, the lengths of the segments are proportional.
Triangle Proportionality Theorem
Theorem Hypothesis Conclusion
If a line parallel to a side
of a triangle intersects
the other two sides, then
it divides those sides
proportionally.
AE
_
EB
=
AF
_
FC
Example 1 Prove the Triangle Proportionality Theorem
A
Given:
‹
−
›
EF ∥
_
BC
Prove:
AE
___
EB
=
AF
___
FC
Step 1 Show that △AEF ∼ △ABC.
Because
‹
−
›
EF ∥
_
BC , you can conclude that ∠1 ≅ ∠2 and
∠3 ≅ ∠4 by the Theorem.
So, △AEF ∼ △ABC by the .
B
E
F
A
C
EF ∥ BC
B
E
1
2
3
4
F
A
C
Module 12
632
Lesson 1
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=NL-D;CA-D
© Houghton Mifflin Harcourt Publishing Company
Step 2 Use the fact that corresponding sides of similar triangles are proportional to
prove that
AE
___
EB
=
AF
___
FC
.
AB
_
AE
=
Corresponding sides are proportional.
AE + EB
_
AE
=
Segment Addition Postulate
1 +
EB
_
AB
=
Use the property that
a + b
____
c
=
a
__
c
+
b
__
c
.
EB
_
AE
=
Subtract 1 from both sides.
AE
_
EB
=
Take the reciprocal of both sides.
Reflect
4. Explain how you conclude that ▵AEF ∼▵ABC without using ∠3 and ∠4.
Explain 2
Applying the Triangle Proportionality
Theorem
Example 2 Find the length of each segment.
A
_
CY
It is given that
_
XY ∥
_
BC so
AX
___
XB
=
AY
___
YC
by the Triangle Proportionality
Theorem.
Substitute 9 for AX, 4 for XB, and 10 for AY.
Then solve for CY.
9
_
4
=
10
_
CY
Take the reciprocal of both sides.
4
_
9
=
CY
_
10
Next, multiply both sides by 10.
10
(
4
_
9
)
=
(
CY
_
10
)
10 →
40
_
9
= CY, or 4
4
_
9
= CY
B
Find PN.
It is given that
_
PQ ∥
_
LM , so
NQ
___
QM
= by the
Triangle Proportionality Theorem.
Substitute for NQ, for QM, and 3 for .
5
_
2
=
NP
_
3
Multiply both sides by :
(
5
_
2
)
=
(
NP
_
3
)
→ = NP
B
X
Y
C
10
4
9
A
L
P Q
N
M
3
5
2
Module 12
633
Lesson 1
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=NL-D;CA-D
© Houghton Mifflin Harcourt Publishing Company
Your Turn
Find the length of each segment.
5.
_
DG 6.
_
RN
Explain 3
Proving the Converse of the Triangle
Proportionality Theorem
The converse of the Triangle Proportionality Theorem is also true.
Converse of the Triangle Proportionality Theorem
Theorem Hypothesis Conclusion
If a line divides two sides of
a triangle proportionally,
then it is parallel to the
third side.
‹
−
›
EF ||
_
BC
Example 3 Prove the Converse of the Triangle Proportionality Theorem
A
Given:
AE
_
EB
=
AF
_
FC
Prove:
‹
−
›
EF ∥
_
BC
Step 1 Show that △AEF ∼ △ABC.
It is given that
AE
_
EB
=
AF
_
FC
, and taking the reciprocal
of both sides shows that . Now add 1 to
both sides by adding
AE
_
AE
to the left side and
AF
_
AF
to the right side.
This gives .
Adding and using the Segment Addition Postulate gives .
Since ∠A
≅ ∠A, △AEF ∼ △ABC by the Theorem.
Step 2 Use corresponding angles of similar triangles to show that
‹
−
›
EF ∥
_
BC .
∠AEF ≅ ∠ and are corresponding angles.
So,
‹
−
›
EF ∥
_
BC by the Theorem.
E
F
C D
G
24
32 40
P
M N
Q
R
10
8
5
A
B
E
C
F
AE
EB
AF
FC
=
A
B
E
C
F
Module 12
634
Lesson 1
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=NL-D;CA-D
© Houghton Mifflin Harcourt Publishing Company
Reflect
7. Critique Reasoning A student states that
_
UV must
be parallel to
_
ST . Do you agree? Why or why not?
Explain 4
Applying the Converse of the Triangle
Proportionality Theorem
You can use the Converse of the Triangle Proportionality Theorem to verify that a line is
parallel to a side of a triangle.
Example 4 Verify that the line segments are parallel.
A
_
MN and
_
KL
JM
_
MK
=
42
_
21
= 2
JN
_
N L
=
30
_
15
= 2
Since
JM
_
MK
=
JN
_
N L
,
_
MN ||
_
KL by the Converse of the
Triangle Proportionality Theorem.
B
_
DE and
_
AB
(
Given that AC = 36 cm, and BC = 27 cm
)
AD = AC - DC = 36 - 20 = 16
BE = BC -
= - =
CD
_
DA
=
_
=
_
CE
_
EB
=
_
=
Since
CD
_
DA
=
_
,
―
DE ||
―
AB by the Theorem.
Reflect
8. Communicate Mathematical Ideas In △ABC, in the example, what is the value
of
AB
___
DE
? Explain how you know.
Your Turn
9. Verify that
_
TU and
_
RS are parallel.
R
S
T
U
V
90
67.5 54
72
K
L
M
42
30
15
21
N
J
A
B
D
E
C
15 cm
20 cm
R
S T
U
V
Module 12
635
Lesson 1
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=NL-D;CA-D
© Houghton Mifflin Harcourt Publishing Company
Elaborate
10. In △ABC,
_
XY ||
_
BC . Use what you know about similarity
and proportionality to identify as many different proportions
as possible.
11. Discussion What theorems, properties, or strategies are common to the proof of
the Triangle Proportionality Theorem and the proof of Converse of the Triangle
Proportionality Theorem?
12. Essential Question Check-In Suppose a line parallel to side
―
BC of
▵ABC intersects sides
―
AB and
―
AC at points X and Y, respectively, and
AX
___
XB
= 1. What
do you know about X and Y? Explain.
1. Copy the triangle ABC that you drew for the Explore activity. Construct a line
‹
−
›
FG parallel to
_
AB using the same method you used in the Explore activity.
2.
_
ZY ||
‹
−
›
MN . Write a paragraph proof to show that
XM
___
MZ
=
XN
___
NY
.
• Online Homework
• Hints and Help
• Extra Practice
Evaluate: Homework and Practice
M
N
X
Y
Z
A
B C
X Y
Module 12
636
Lesson 1
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=NL-D;CA-D
© Houghton Mifflin Harcourt Publishing Company
Find the length of each segment.
3.
_
KL 4.
_
XZ 5.
_
VM
Verify that the given segments are parallel.
6.
_
AB and
_
CD 7.
_
MN and
_
QR 8.
_
WX and
_
DE
9. Use the Converse of the Triangle Proportionality Theorem to
identify parallel lines in the figure.
10. On the map, 1st Street and 2nd Street are parallel. What is the distance from
City Hall to 2nd Street along Cedar Road?
G
6
4
8
H
m
n
J
K
L
U
V T
M
N
14
8
49
X
Y
U V
30
18
30
Z
A
B
C
D
E
12
14
4
4
2
3
P
Q R
N
M
3
2.7
9
10
D
W X
E
F
1.5
2.5
3.5
2.1
A B
C
L
M
N
20
16
12
15
816
2.4 mi
2.1 mi
2.8 mi
Cedar Rd.
Aspen Rd.
1st St. 2nd
St.
City
Hall
Library
ge07se_c07l04007a
Module 12
637
Lesson 1
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=NL-D;CA-D
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©Stuart Walker/Alamy
11. On the map, 5th Avenue, 6th Avenue, and 7th Avenue are parallel. What is the length
of Main Street between 5th Avenue and 6th Avenue?
12.
Multi-Step The storage unit has horizontal siding that is parallel to the base.
a. Find LM.
b. Find GM.
c. Find MN to the nearest tenth of a foot.
d.
Make a Conjecture Write the ratios
LM
___
MN
and
HJ
__
JK
as
decimals to the nearest hundredth and compare them.
Make a conjecture about the relationship between parallel
lines
‹
−
›
LD ,
‹
−
›
ME , and
‹
−
›
NF and transversals
‹
−
›
GN and
‹
−
›
GK.
13. A corollary to the Converse of the Triangle Proportionality Theorem states that
if three or more parallel lines intersect two transversals, then they divide the
transversals proportionally. Complete the proof of the corollary.
Given: Parallel lines
‹
−
›
AB ∥
‹
−
›
CD,
‹
−
›
CD ∥
‹
−
›
EF
Prove:
AC
___
CE
=
BX
___
XE
,
BX
___
XE
=
BD
___
DF
,
AC
___
CE
=
BD
___
DF
Statements Reasons
1.
‹
−
›
AB ∥
‹
−
›
CD ,
‹
−
›
CD ∥
‹
−
›
AF 1. Given
2. Draw
‹
−
›
EB intersecting
‹
−
›
CD at X. 2. Two points
3.
AC
___
CE
=
BX
__
XE
3.
4.
BX
__
XE
=
BD
___
DF
4.
5.
AC
___
CE
=
BD
___
DF
5. Property of Equality
0.3 km
0.5 km
Spring St.
0.4 km
Main
St.
7th
Ave.
6th
Ave.
5th
Ave.
The storage unit has horizontal siding that is parallel to the base.
G
L
K
D
H
11.3 ft
10.4 ft
2.6 ft
2.2 ft
J
M E
N F
A
B
C
D
E
F
X
Module 12
638
Lesson 1
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=NL-D;CA-D
© Houghton Mifflin Harcourt Publishing Company
14. Suppose that LM = 24. Use the Triangle Proportionality Theorem
to find PM.
15. Which of the given measures allow you to conclude that
―
UV ∥
―
ST ? Select
all that apply.
A. SR = 12, TR = 9
B. SR = 16, TR = 20
C. SR = 35, TR = 28
D. SR = 50, TR = 48
E. SR = 25, TR = 20
H.O.T. Focus on Higher Order Thinking
16. Algebra For what value of x is
_
GF ∥
―
HJ ?
17.
Communicate Mathematical Ideas John used △ABC to write a proof
of the Centroid Theorem. He began by drawing medians
―
AK and
―
CL ,
intersecting at Z. Next he drew midsegments
―
LM and
―
NP , both
parallel to median
―
AK .
Given: △ABC with medians
―
AK and
―
CL , and midsegments
―
LM
and
―
NP
Prove: Z is located
2
_
3
of the distance from each vertex of △ABC to
the midpoint of the opposite side.
a. Complete each statement to justify the first part of John’s proof.
By the definition of , MK =
1
_
2
BK. By the definition
of , BK = KC. So, by , MK =
1
_
2
KC, or
KC
___
MK
= 2.
Consider
△LMC.
―
LM ∥
―
AK (and therefore
―
LM ∥
―
ZK ), so
ZC
___
LZ
=
KC
___
MK
by the
Theorem, and ZC = 2LZ. Because
LC = 3LZ,
ZC
___
LC
=
2LZ
___
3LZ
=
2
_
3
, and Z is located
2
_
3
of the distance from vertex C
of △ABC to the midpoint of the opposite side.
b. Explain how John can complete his proof.
K
N
10 15
M
L
P
U
S
T
V
R
20
16
A
B C
M
L
N
Z
K P
45
40
5x + 1
4x + 4
G
F
J
E
H
L
M
K
P
C
Z
Module 12
639
Lesson 1
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=NL-D;CA-D
© Houghton Mifflin Harcourt Publishing Company
18. Persevere in Problem Solving Given △ABC with FC = 5, you want to find BF.
First, find the value that y must have for the Triangle Proportionality Theorem to
apply. Then describe more than one way to find BF, and find BF.
Shown here is a triangular striped sail, together with some of its dimensions.
In the diagram, segments BJ, CI, and DH are all parallel to segment EG. Find
each of the following:
1. AJ
2. CD
3. HG
4. GF
5. the perimeter of △AEF
6. the area of △AEF
7. the number of sails you could make for $10,000 if the sail
material costs $30 per square yard
Lesson Performance Task
0
1
2
3
4
5
6
7
8
9
1 2 63 4 5 7 8 9
x
y
A (1, 2) E (4, 2) C (10, 2)
F (7, 6)
B (5.5, y)
E
A
F
G
D
C
B
J
I
H
3.5 ft
6.5 ft
6 ft
2.25 ft
2.5 ft
1.8 ft
1.2 ft
Module 12
640
Lesson 1
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=NL-D;CA-D
