Name
Class
Date
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?
Resource
Locker
G.8.A Prove theorems about similar triangles, including the Triangle Proportionality theorem,
and apply these theorems to solve problems. Also G.2.B.
Constructing Similar Triangles
Explore
In the following activity you will see one way to construct a triangle similar to a given triangle.

Do your work for Steps A–C in the space provided. Draw a triangle.
Label it ABC as shown.
A
B
B
C
_
Select a point on AB. Label it E.
A
E
B
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
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.
A
E
B


F
C
_
‹ ›
−
Why are EF and BC parallel?
_ _ _
_
Graphics\02192014\From Ramesh\Econ 2016\W
Use a ruler to measure AE, EB\\192.168.9.251\07Macdata\07Vol1Data\From
, AF, and FC. Then compare the
AE
AF
___
___
ratios EB and FC .
Module 12
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Lesson 1
Reflect
1.
Discussion How can you show that △AEF ∼ △ABC? Explain.
2.
AE
AF
and ___
? Explain.
What do you know about the ratios ___
AB
AC
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
AE = _
AF
_
EB FC
A
E
B
F
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If a line parallel to a side
of a triangle intersects
the other two sides, then
it divides those sides
proportionally.
Conclusion
C
EF ∥ BC
Example 1

Prove the Triangle Proportionality Theorem
‹ › _
−
Given: EF ∥ BC
A
AE
AF
= ___
Prove: ___
EB
FC
E 1
Step 1 Show that △AEF ∼ △ABC.
‹ › _
−
Because EF ∥ BC, you can conclude that ∠1 ≅ ∠2 and
by ∠3 ≅ ∠4 the
4 C
2
B
Theorem.
So, △AEF ∼ △ABC by the
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.
688
Lesson 1
Step 2 Use the fact that corresponding sides of similar triangles are proportional to
AE
AF
prove that ___
= ___
.
EB
FC
AB =
_
AE
Corresponding sides are proportional.
AE + EB
_
=
AE
Segment Addition Postulate
a+b
EB =
1+_
AE
a
b
__
__
Use the property that ____
c = c + 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
Example 2
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
Applying the Triangle Proportionality
Theorem
Find the length of each segment.
_
CY
B
4 X
9
C
Y
10
A
_ _ AX ___
AY by the Triangle Proportionality Theorem.
= YC
It is given that XY ∥ BC so ___
XB
Substitute 9 for AX, 4 for XB, and 10 for AY. Then solve for CY.
9=_
10
_
4
CY
Take the reciprocal of both sides.
4=_
CY
_
9
10
Next, multiply both sides by 10.
() ( )
CY 10
4 = _
10 _
9
10
40 = CY, or 4_
4 = CY
_
9
9
Module 12
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Lesson 1
BFind PN.
_ _
NQ
It is given that PQ​ ​   ∥ ​ LM​ 
, so ___
​ QM  ​ = by the
L
Triangle Proportionality Theorem.
3
M
Q 2
P
5
Substitute for NQ, for QM, and 3 for .
N
5  ​ = _
​ _
​  NP ​
 
2
3
Multiply both sides by .
()
(  )
​_
​  5 ​  ​ =
2
​_
​  NP ​   ​
3
= NP
Reflect
5.
Justify Reasoning Suppose you are given △PQR. Explain how you can use
coordinates to determine whether the Triangle Proportionality Theorem applies to
△PQR. Justify your reasoning.
y
6
P
N
4
M
-4
Q
-2
2
x
0
-2
2
4
R
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Your Turn
Find the length of each segment.
_
E
6. DG​
​   
32
24
F
Module 12
C
7.
40
_
RN​
​   
Q
8
D
M
G
690
10
5 P
R
N
Lesson 1
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
A
If a line divides two sides of
a triangle proportionally,
then it is parallel to the
third side.
Example 3

Conclusion
AE
AF
=
EB
FC
‹ › _
−
EF || BC
F
E
B
C
Prove the Converse of the Triangle Proportionality Theorem
AE = _
AF
Given: _
EB
FC
‹ › _
−
Prove: EF ∥ BC
A
F
Step 1 Show that △AEF ∼ △ABC.
E
AE = _
AF , and taking the reciprocal
It is given that _
EB
FC
of both sides shows that
C
B
. Now add 1 to
AF to the right side.
AE to the left side and _
both sides by adding _
AE
AF
This gives
.
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Adding and using the Segment Addition Postulate gives
.
Since ∠A ≅ ∠A, △AEF ∼ △ABC by the
Criterion.
‹ › _
−
Step 2 Use corresponding angles of similar triangles to show that EF ∥ BC.
‹ › _
−
∠AEF ≅ ∠
and are corresponding angles. So, EF ∥ BC by the
Theorem.
Reflect
8.
_
Critique Reasoning
A
student
states
that
UV
must
_
be parallel to ST. Do you agree? Why or why not?
R
U
S
Module 12
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V
T
Lesson 1
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.
_
_
MN and KL
JN
JM
30 = 2
42 = 2
_
_
=_
=_
21
15
MK
NL
JM
JN _ _
Since _ = _, MN || KL by the Converse of the
MK
NL
42
21 M
K
L
15
J
30
N
Triangle Proportionality Theorem.
B
_
_
DE and AB (Given that AC = 36 cm, and BC = 27 cm)
A
D
20 cm
AD = AC - DC = 36 - 20 = 16
B
BE = BC =
-
CD = _ = _
_
DA
E
15 cm
C
CE = _ =
_
EB
=
_ _
CD = _, DE
Since _
|| AB by the
DA
Theorem.
Reflect
9.
Module 12
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Communicate Mathematical Ideas In △ABC, in the example, what is the value
AB
of ___
? Explain how you know.
DE
Lesson 1
Your Turn
Verify that the given segments are parallel.
_
_
A
10. AB​
​   
and CD​
​   
_
_
11. TU​
​   
and RS​
​   
B
1.5
R
72
1.5
C
T
90
D
1.5
1.5
V
67.5
U
54
S
E
Elaborate
_ _
12. In △ABC, XY​ ​  | | ​ BC​. Use what you know about similarity
and proportionality to identify as many different proportions
as possible.
A
X
Y
B
C
13. 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?
―
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14. Essential Question Check-In Suppose a line parallel to side BC​
​   of
―
― at points X and Y, respectively, and ___
▵ABC intersects sides AB​
​   and AC​
​   
​ AX
  ​ = 1. What
XB
do you know about X and Y? Explain.
Module 12
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Lesson 1
Evaluate: Homework and Practice
1.
2.
_ −
‹ ›
XN
XM
ZY || MN . Write a paragraph proof to show that ___
= ___
.
NY
MZ
X
N
Y
Z
M
Find the length of each segment.
_
3. KL
H
G
6
4.
_
XZ
X
m
8
J
n
K
4
30
I
15
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6.
9
30
U
_
BH
B
Z
Y
18
L
5.
• Online Homework
• Hints and Help
• Extra Practice
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.
V
_
VM
D
M
8
U
H
9
V
C
Module 12
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694
N 14
T
Lesson 1
Verify that the given segments are parallel.
_
_
7. AB​
​   
and CD​
​   
9.
_
_
​ MN​ 
and QR​
​   
A
C 4
12
E
8.
P
9
D 42
3
14
_
_
and NM​
​   
​ PQ​ 
_
_
10. ​ WX​ 
and DE​
​   
100
12 Q
L
9P
F
1.5
W
M
75
10
N
3
R
2.7 M
Q
B
2.1
X
3.5
2.5
N
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D
11. Use the Converse of the Triangle Proportionality Theorem to
identify parallel lines in the figure.
E
M
16
A
8
B
15
20
N
L
16
12
C
Module 12
695
Lesson 1
12. On the map, 1st Street and 2nd Street are parallel. What is the distance from City Hall
to 2nd Street along Cedar Road?
2.8
City
Hall
m
2.1
As p
mi
Rd.
Library
i
2.4 mi
1st St.
en
Cedar Rd.
2nd St.
13. 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?
5th
Ave.
6th
Ave.
Main
St.
g
rin
Sp
0.3
7th
Ave.
0.4 km
St.
0.5
km
ge07se_c07l04007a
km
G
14. Multi-Step The storage unit has horizontal siding that is parallel to
the base.
a. Find LM.
11.3 ft
L
b. Find GM.
c.
M
N
10.4 ft
H
J 2.2 ft
K
D 2.6 ft
E
F
Find MN to the nearest tenth of a foot.
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HJ
LM
and __
as decimals
d. Make a Conjecture Write the ratios ___
MN
JK
to the nearest hundredth and compare them. Make a conjecture
‹ › −
‹ ›
‹ ›
−
−
about the relationship between the parallel lines LD , ME , and NF
‹ ›
‹ ›
−
−
and the transversals GN and GK.
Module 12
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Lesson 1
15. 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 
   
 ​
B
A
AC
AC
BX ___
BX
BD ___
BD
___
___
___
___
Prove: ​  CE  ​= ​  XE  ​, ​  XE  ​= ​  DF  ​, ​  CE  ​= ​  DF  ​
X
D
C
E
F
Statements
‹ › −
‹ › −
‹ › −
‹ ›
−
1.​  AB ​
   
 ∥​   CD ​
 
 , ​  CD ​
  ∥
 
  ​   AF ​
 
 
‹ ›
‹ ›
−
−
2.Draw​  EB ​
   
  intersecting​  CD ​
   
 at X.
Reasons
1.Given
2.Two points AC
3.​ ___
 ​ = __
​  BX
 ​ 
XE
CE
3. BX
4.​ __
 ​ = ___
​ BD
 ​ 
XE
DF
4. AC
5.​ ___
 ​ = ___
​ BD
  ​
DF
CE
5. Property of Equality
― ―― ―
16. Given that DE​
​   ∥ BC​
​  , XY​
​   
∥ AD​
​  . Find EC to the nearest tenth.
A
17
X
15
D
7.5
18 Y 16
B
E
C
― ―
∥ ST​
​  ? Select all
17. Which of the given measures allow you to conclude that UV​
​   
that apply.
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A. SR = 12, TR = 9
U
20
B. SR = 16, TR = 20
S
T
V
16
C. SR = 35, TR = 28
D. SR = 50, TR = 48
R
E. SR = 25, TR = 20
18. Suppose that LM = 24. Use the Triangle Proportionality Theorem
to find PM.
L
P
K
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15
M
Lesson 1
H.O.T. Focus on Higher Order Thinking
― ―
19. Algebra For what value of x is GF​ ​   
∥ ​ HJ​? 
G
40
H
4x + 4
F
45
J 5x + 1
E
20. Communicate Mathematical Ideas John used
△ABC to write a proof of the Centroid Theorem. He
― and CL​
―
began by drawing medians AK​
​   
​  , intersecting at
―
―
Z. Next he drew midsegments LM​
​   
and NP​
​  , both
―
parallel to median AK​
​  . 
―
―
Given: △ABC with medians AK​
​  , and
― and​ NP​
― and CL​
​   
midsegments LM​
​   
Prove: Z is located _​ 23 ​of the distance from each vertex
of △ABC to the midpoint of the opposite site.
a. Complete each statement to justify the first part of
John’s proof.
A
L
B
Z
M
L
M
K
N
P
C
Z
K
P
C
​ K. By the definition
By the definition of , MK = ​ _12 B
​ KC   ​= 2.
of , BK = KC. So, by , MK = _​ 12 ​KC, or ___
MK
― ―
​  ∥
 AK​
​  , so ___
​ ZC
  ​= ___
​  KC  ​ by the Consider △LMC. LM​
MK
LZ
  ​= ___
​ 2LZ
  ​= _
​ 23 ​, and Z is located _​ 32 ​
Theorem, and ZC = 2LZ. Because LC = 3LZ, ___
​  ZC
LC
3LZ
of the distance from vertex C of △ABC to the midpoint of the opposite side.
b. Explain how John can complete his proof.
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Module 12
698
Lesson 1
21. 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.
© Houghton Mifflin Harcourt Publishing Company
y
9
B (5.5, y)
8
7
F (7, 6)
6
5
4
3
2
C (10, 2)
1 A (1, 2) E (4, 2)
x
0
1 2 3 4 5 6 7 8 9
Module 12
699
Lesson 1
Lesson Performance Task
S hown 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
A
2.5 ft
B
2.CD
2.25 ft
3. HG
C
4. GF
5. the perimeter of △AEF
J
1.8 ft
I
1.2 ft
H
D
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
G
3.5 ft
6 ft
E
6.5 ft
F
© Houghton Mifflin Harcourt Publishing Company
Module 12
700
Lesson 1