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83
Similarity
3.1 Expanding Your MindDilations of Triangles ................................................... 85
3.2 Look-AlikesSimilar Triangles ...........................................................97
3.3 Prove It!AA, SAS, and SSS Similarity Theorems ........................ 105
The pupils
of a cat's eyes are shaped differently from ours. In brighter
light, they appear narrow, like a diamond.
But cat's eyes dilate just like
ours do.
84 • Chapter 3 Similarity
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3.1 Dilations of Triangles • 85
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Expanding Your MindDilations of Triangles
Key Terms dilation
center of dilation
scale factor
dilation factor
enlargement
reduction
Learning GoalsIn this lesson, you will:
Dilate triangles that result in an enlargement of the
original triangle.
Dilate triangles that result in a reduction of the
original triangle.
Dilate triangles in a coordinate plane.
What does it mean if someone says that the pupils of your eyes are dilated?
When a light source changes, the pupils of your eyes either shrink or enlarge to
control the passage of light. When it is very sunny outside, your pupils will shrink
to allow less light in. When it is very dark at night, your pupils will enlarge to allow
more light in.
A change in light isn’t the only thing that makes your pupils dilate. Your pupils can
also enlarge when your eyes look at something you like: a favorite show, a cute
animal, an interesting picture, or even a special someone.
Can you make your fellow pupils’ pupils dilate? Try it out in your groups.
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86 • Chapter 3 Similarity
1. How is the ratio distance of the image from the center of dilation : distance of the
original figure from the center of dilation represented? Is the scale factor less than 1,
equal to 1, or greater than 1? Explain your reasoning.
2. Measure each side of triangle ABC in millimeters.
m ___
AB 5
m ___
BC 5
m ___
AC 5
Problem 1 Maintaining Ratios—Enlargements
In mathematics, dilations are transformations that produce images that are the same
shape as the original image, but not the same size. Each point on the original figure is
moved along a straight line and the straight line is drawn from a fixed point known as the
center of dilation. The distance each point moves is determined by the scale factor used.
The scale factor or dilation factor is the ratio of the distance of the image from the
center of dilation to the distance of the original figure from the center of dilation.
When the scale factor is greater than one, the image is called an enlargement.
P
B' C'
A'
B
A
C
Triangle ABC was dilated to produce triangle A9B9C9 using point P as the center
of dilation. Triangle A9B9C9 is an enlargement of triangle ABC.
Therefore, the scale factor can be expressed as PA9 ____ PA
5 PB9 ____ PB
5 PC9 ____ PC
.
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3. Measure each side of triangle A9B9C9 in millimeters.
m_____
A9B9 5
m_____
B9C9 5
m_____
A9C9 5
4. Measure each line segment.
m____
A9P5 mm m___
AP5 mm
m____
B9P5 mm m___
BP5 mm
m____
C9P5 mm m___
CP5 mm
5. Determine each ratio.
A9P____AP
5 B9P____BP
5
C9P____CP
5 A9B9 _____AB
5
B9C9 _____BC
5 A9C9 _____AC
5
6. Measure each angle in triangle ABC.
m/A5 °
m/B5 °
m/C5 °
7. Measure each angle in triangle A9B9C9.
m/A95 °
m/B95 °
m/C95 °
8. Compare triangle A9B9C9 to triangle ABC. What do you notice?
You will need a ruler and a protractor.
Can I now add markers on
triangles ABC and A
,B
,C
,?
3.1 Dilations of Triangles • 87
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1. How is the ratio distance of the image from the center of dilation : distance of the
original figure from the center of dilation represented? Is the scale factor less than 1,
equal to 1, or greater than 1? Explain your reasoning.
2. Measure each side of triangle DEF in millimeters.
m ___
DE 5
m ___
EF 5
m ___
DF 5
3. Measure each side of triangle D9E9F9 in millimeters.
m _____
D9E9 5
m ____
E9F9 5
m _____
D9F9 5
Problem 2 Maintaining Ratios—Reductions
When the scale factor or dilation factor is less than one, the image is called a reduction.
Triangle DEF was dilated to produce triangle D9E9F9 using point P as the center
of dilation. Triangle D9E9F9 is a reduction of triangle DEF.
Therefore, the scale factor can be expressed as PD9 ____ PD
5 PE9 ____ PE
5 PF9 ____ PF
.
P
E F
D
E'F'
D'
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3.1 Dilations of Triangles • 89
4. Measure each line segment.
m____
D9P5 mm m___
DP5 mm
m____
E9P5 mm m___
EP5 mm
m____
F9P5 mm m___
FP5 mm
5. Determine each ratio.
D9P____DP
5 E9P____EP
5
F9P____FP
5 D9E9 _____DE
5
E9F9 ____EF
5 D9F9 ____DF
5
6. Measure each angle in triangle DEF.
m/D5 °
m/E5 °
m/F5 °
7. Measure each angle in triangle D9E9F9.
m/D95 °
m/E95 °
m/F95 °
8. Compare triangle D9E9F9 to triangle DEF. What do you notice?
How do these dilation ratios compare to the dilation ratios
from Problem 1?
Can I add markers on
triangles DEF and D
,E
,F
,?
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90 • Chapter 3 Similarity
Problem 3 Dilating Triangles on a Coordinate Plane
1. Enlarge triangle WXY with P as the center of dilation and a scale factor of 2.
Follow the steps given.
P
W
X Y
Step 1: Measure ___
PW , ___
PX , and ___
PY in millimeters.
m ___
PW 5
m ___
PX 5
m ___
PY 5
Step 2:
● Extend line segment PW to point W9 such
that m ____
PW9 5 2 3 m ___
PW .
● Extend line segment PX to point X9 such
that m ___
PX9 5 2 3 m ___
PX .
● Extend line segment PY to point Y9 such
that m ___
PY9 5 2 3 m ___
PY .
Step 3: Join points W9, X9, and Y9 to form triangle W9X9Y9.
How can you verify triangle
W,X
,Y
, was enlarged
correctly?
Take your time and use your
straightedge.
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3.1 Dilations of Triangles • 91
2. Analyze triangle ABC.
a. Dilate triangle ABC on the coordinate plane using the origin (0, 0) as the center of
dilation and a scale factor of 2 to form triangle A9B9C9.
x420
2
0
4
6
8
10
12
14
16
6 8 10 12 14 16 18
y
18
(3, 7)A
(7, 3)(3, 3)C B
b. What are the coordinates of points A9, B9, and C9?
3. Graph triangle ABC with the coordinates A(3, 7), B(7, 3), and C(3, 3) on the
grid provided.
x420
2
0
4
6
8
10
12
14
16
6 8 10 12 14 16 18
y
18
a. Dilate triangle ABC on the coordinate plane using point C as the center of dilation
and a scale factor of 3 to form triangle A9B9C.
b. What are the coordinates of points A9 and B9?
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92 • Chapter 3 Similarity
4. Reduce triangle HJK with P as the center of dilation and a scale factor of 1 __ 2
.
Follow the steps given.
H
JK
P
Step 1: Measure ___
PH , ___
PJ , and ___
PK in millimeters.
m ___
PH 5
m ___
PJ 5
m ___
PK 5
Step 2:
● Locate point H9 such that m ____
PH9 5 1 __ 2
3 m ___
PH .
● Locate point J9 such that m ___
PJ9 5 1 __ 2 3 m
___ PJ .
● Locate point K9 such that m ___
PK9 5 1 __ 2 3 m
___ PK .
Step 3: Join points H9, J9, and K9 to form triangle H9J9K9.
How can you verify that triangle H
,J
,K
, was reduced
correctly?
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3.1 Dilations of Triangles • 93
5. Analyze triangle ABC.
a. Dilate triangle ABC on the coordinate plane using the origin (0, 0) as the center of
dilation and a scale factor of 1 __ 2
to form triangle A9B9C9.
x420
2
0
4
6
8
10
12
14
16
6 8 10 12 14 16 18
y
18
(2, 14)A
(14, 6)B(6, 6)C
b. What are the coordinates of points A9, B9, and C9?
6. Graph triangle ABC with the coordinates A(3, 15), B(15, 3), and C(3, 3) on the
grid provided.
x420
2
0
4
6
8
10
12
14
16
6 8 10 12 14 16 18
y
18
a. Dilate triangle ABC on the coordinate plane using point C as the center of dilation
and a scale factor of 1 __ 2
to form triangle A9B9C.
b. What are the coordinates of points A9 and B9?
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94 • Chapter 3 Similarity
Talk the Talk
In this lesson, several triangles were dilated. Whether it was an enlargement or a
reduction, the same conclusions can be drawn about the relationship between
corresponding angles and the relationship between the corresponding sides of a triangle
and its image resulting from dilation.
1. Describe the relationship between the corresponding angles in an original triangle and
its image resulting from dilation.
2. Describe the relationship between the corresponding sides in an original figure and its
image resulting from dilation.
3. Does dilation result in an image that is the same shape as the original?
Why or why not?
4. Does dilation result in an image that is the same size as the original?
Why or why not?
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3.1 Dilations of Triangles • 95
5. If two triangles are congruent, what is the relationship between the
corresponding angles?
6. If two triangles are congruent, what is the relationship between the
corresponding sides?
7. Describe how a triangle is dilated when the ratio distance of the image from the
center of dilation : distance of the original figure from the center of dilation is:
● less than 1.
● equal to 1.
● greater than 1.
Be prepared to share your solutions and methods.
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96 • Chapter 3 Similarity
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3.2 Similar Triangles • 97
Turquoise, navy, cobalt, robin’s egg, cornflower, ultramarine, aquamarine,
cerulean, and periwinkle—all of these are names for different shades of the color
blue. There are an infinite number of possibilities for shades of blue, but all of
them are similar in one way: They are all blue.
What are some examples of similarity you have learned in mathematics?
Learning GoalsIn this lesson, you will:
Define similar triangles.
Identify the corresponding parts of similar triangles.
Write triangle similarity statements.
Determine the measure of corresponding parts of
similar triangles.
Look-AlikesSimilar Triangles
Key Term similar triangles
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Problem 1 Similar Triangles
Similar triangles are triangles that have the same shape.
In the previous lesson, you learned that when a triangle is dilated, the resulting image is an
enlarged or reduced triangle that maintains the same shape as the original triangle.
Dilations resulted in congruent corresponding angles, and proportional corresponding
sides based on the scale factor or dilation ratio.
P
B
B'
C'
A'
A
C
In the figure shown, triangle ABC is similar to triangle A9B9C9. This can be
expressed using symbols as △ABC , △A9B9C9.
1. Use the figure shown to answer each question.
a. Identify the congruent corresponding angles.
b. Write ratios to identify the proportional sides.
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3.2 Similar Triangles • 99
2. Given △TRP , △WMY:
a. Identify the congruent corresponding angles.
b. Write ratios to identify the proportional sides.
3. Suppose /K > /H, /P > /O, /E > /W, and KP ____ HO
5 PE ____ OW
5 KE ____ HW
.
Write a triangle similarity statement.
100 • Chapter 3 Similarity
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Problem 2 Unknown Measurements
1. Given: △ZAP , △EDP
PZ 5 5 cm, ZA 5 4 cm, and ED 5 12 cm
Z A
DE
P
5 cm
4 cm
12 cm
a. What other measurement(s) can you determine? Explain how
you know.
b. Determine the measurement(s).
Think about the similar triangles
given. What does this tell you?
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3.2 Similar Triangles • 101
2. Given: △ZAP , △EDP
m/E 5 47°
Z A
DE47 o
P
a. What other measurement(s) can you determine? Explain how you know.
b. Determine the measurement(s).
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102 • Chapter 3 Similarity
3. Given: △WRM , △WGQ
WQ 5 5 cm, WG 5 6 cm, and GR 5 8 cm
M
W
Q
G R
5 cm
6 cm 8 cm
a. What other measurement(s) can you determine? Explain how you know.
b. Determine the measurement(s).
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3.2 Similar Triangles • 103
4. Given: △DFH , △TKH
TK 5 5.5 ft, KH 5 6 ft, and FK 5 15 ft
D
F H
T
K
5.5 ft
15 ft 6 ft
a. What other measurement(s) can you determine? Explain how you know.
b. Determine the measurement(s).
104 • Chapter 3 Similarity
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5. Given: △WBE , △SEP
BP 5 EP, WE 5 58 mm
W
B P
S
E
a. What other measurement(s) can you determine? Explain how you know.
b. Determine the measurement(s).
Be prepared to share your solutions and methods.
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3.3 AA, SAS, and SSS Similarity Theorems • 105
Graphic artists often use knowledge about similarity to create realistic-looking
perspective drawings. Choose where the horizon should be and a vanishing
point—a point where all parallel lines in the drawing should appear to meet—and
you too can create a perspective drawing.
Can you see how similarity was used to create this drawing? Can you use
similarity to create your own perspective drawing?
vanishing point
horizon
Key Terms AA Similarity Theorem
SAS Similarity Theorem
SSS Similarity Theorem
Learning GoalsIn this lesson, you will:
Explore the AA Similarity Theorem.
Explore the SAS Similarity Theorem.
Explore the SSS Similarity Theorem.
Use the AA, SAS, and SSS Similarity Theorems
to identify similar triangles.
Prove It!AA, SAS, and SSS Similarity Theorems
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Problem 1 Two Angles
In the previous lesson, you determined that when two triangles are similar, the
corresponding angles are congruent and the corresponding sides are proportional. To
show that two triangles are similar, do you need to show that all of the corresponding
sides are proportional and all of the corresponding angles are congruent? In this lesson,
you will explore efficient methods for showing that two triangles are similar.
1. If the measures of two angles of a triangle are known, is that enough information to
draw a similar triangle? Let’s explore this possibility.
a. Use a straightedge to draw triangle ABC in the space provided.
b. Use a protractor to measure, /A and /B, of triangle ABC and record the
measurements.
m/A 5 m/B 5
c. Do you need a protractor to determine m/C? Why or why not?
d. Use the measurements in part (b) to draw triangle DEF in the space provided.
Do you remember what the sum of the
angle measures in a triangle is?
106 • Chapter 3 Similarity
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3.3 AA, SAS, and SSS Similarity Theorems • 107
e. Based on your knowledge from the previous lesson, what other information is needed
to determine if the two triangles are similar and how can you acquire that information?
f. Determine the measurements to get the additional information needed and decide if
the two triangles are similar.
You have just shown that given the measures of two pairs of congruent corresponding
angles of two triangles, it is possible to determine that two triangles are similar. In the
study of geometry, this is expressed as a theorem.
The Angle-Angle (AA) Similarity Theorem states that if two angles of one triangle are
congruent to the corresponding angles of another triangle, then the triangles are similar.
108 • Chapter 3 Similarity
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2. Analyze triangle ABC.
a. Dilate triangle ABC on the coordinate plane using the origin (0, 0) as the center of
dilation and a scale factor of 3 to form triangle A9B9C9.
42
2
4
6
8
10
12
14
16
6 8 10 12 14 16 18 20
y
18
20
x
(3, 7)A
(7, 3)(3, 3)C B
00
b. What are the coordinates of points A9, B9, and C9?
c. Use the AA Similarity Theorem and a protractor to
determine if the original triangle, △ABC, and the
image resulting from the dilation, △A9B9C9, are
similar triangles.
If the center of dilation is at the
origin, can that help you determine the coordinates
of A,, B
,, and C
,?
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3.3 AA, SAS, and SSS Similarity Theorems • 109
Problem 2 Two Sides and the Included Angle
If the lengths of two sides and the measure of the included angle of a triangle are known,
is that enough information to draw a similar triangle?
1. Let’s explore this possibility.
a. Use a straightedge to draw triangle ABC in the space provided.
b. Use a ruler to measure the lengths of ___
AB and ___
BC , of triangle ABC and record the
measurements.
m ___
AB 5 m ___
BC 5
c. Use a protractor to measure /B, the included angle in triangle ABC, and record
the measurement.
m/B 5
d. Use the measurements in parts (b) to draw two sides of a triangle that are
proportional to the corresponding sides of triangle ABC, and use the angle
measure in part (c) to draw an included angle that is congruent, in order to form
triangle DEF in the space provided.
Remember, you explored a similar
situation when analyzing congruent triangles.
110 • Chapter 3 Similarity
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e. Based on your knowledge from the previous lesson, what other information is
needed to determine if the two triangles are similar and how can you acquire
that information?
f. Determine the measurements to get the additional information needed and decide
if the two triangles are similar.
You have just shown that given the lengths of two sides of a triangle and the measure of
the included angle, it is possible to determine that two triangles are similar. In the study of
geometry, this is expressed as a theorem.
The Side-Angle-Side (SAS) Similarity Theorem states that if two pairs of corresponding
sides of two triangles are proportional and the included angles are congruent, then the
triangles are similar.
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3.3 AA, SAS, and SSS Similarity Theorems • 111
2. Use the SAS Similarity Theorem and a protractor to determine if the two triangles
drawn on the coordinate plane are similar. Use a protractor to verify the measure of
the included angle.
x86
2
4
8
10 12–2–2
42–4
–4
–6
–8
–8
–10–12
y
10
12
–10
–12
(7, 6)
(2, 6)
(3.5, 4)
(1, 28)
(8, 212)(22, 212)
P
N
M
A
Q
R6
–60
112 • Chapter 3 Similarity
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Remember, you explored
a similar situation with
congruent triangles.
Problem 3 Three Sides
If the lengths of three sides of a triangle are known, is that enough
information to draw a similar triangle?
1. Let’s explore this possibility.
a. Use a straightedge to draw triangle ABC in the space provided.
b. Use a ruler to measure the length of each side, AB, BC, and
AC, of triangle ABC and record the measurements.
m ___
AB 5 m ___
BC 5 m ___
AC 5
c. Use the measurements in parts (b) to draw three sides of a triangle that are
proportional to these measurements to form triangle DEF in the space provided.
d. Michael says that based on what he’s learned so far, he needs to find the
measures of the three corresponding angles of the triangles to determine if
they are similar. Is he correct? Why or why not?
e. Determine the measurements to get the additional information needed and decide
if the two triangles are similar.
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3.3 AA, SAS, and SSS Similarity Theorems • 113
You have just shown that given the length of three sides of a triangle, it is possible to
determine that two triangles are similar. In the study of geometry, this method is expressed
as a theorem.
The Side-Side-Side (SSS) Similarity Theorem states that if three pairs of corresponding
sides of two triangles are proportional, then the triangles are similar.
2. Use the SSS Similarity Theorem to determine if the two triangles drawn on the
coordinate plane are similar.
x86
2
4
8
10 12–2–2
42–4
–4
–6
–8
–8
–10–12
y
10
12
–10
–12
(7, 1)(4, 1)
(5.5, 7)
(29, 22) (23, 22)
(26, 10)
P
Q
M
A
R N
6
–60
114 • Chapter 3 Similarity
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Talk the Talk
Determine if each pair of triangles are similar by AA, SAS, or SSS.
1.
30 cm
35 cm
6 cm
7 cmE
KJ M
TR
2.
31o31o
Y
V Z
W
X
3.
29o
29o
T
SN
M R
4.
4 mm
5 mm
5 mm
6 mm
6.25 mm
13.5 mmF G
P
Q R
Be prepared to share your solutions and methods.
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Chapter 3 Summary • 115
Chapter 3 Summary
Key Terms dilation (3.1)
center of dilation (3.1)
scale factor (3.1)
dilation factor (3.1)
enlargement (3.1)
reduction (3.1)
similar triangles (3.2)
AA Similarity Theorem (3.3)
SAS Similarity Theorem (3.3)
SSS Similarity Theorem (3.3)
Dilating Triangles
Dilations are transformations that produce images that are the same shape as the original
image, but not the same size. Each point on the original figure is moved along a straight
line and the straight line is drawn from a fixed point known as the center of dilation.
The scale factor is the ratio formed when comparing the distance of the image from the
center of dilation to the distance of the original figure from the center of dilation.
Example
Enlarge triangle ABC with P as the center of dilation and a scale factor of 2.
A
B
C
A’
B’
C’
P
First, measure PA. Then, extend the line PA to the point A9 such that PA9 5 2PA.
Next, measure PB. Then, extend the line PB to the point B9 such that PB9 5 2PB.
Finally, measure PC. Then, extend the line PC to the point C9 such that PC9 5 2PC.
Because the scale factor is greater than one, the image A9B9C9 is called an enlargement.
If the scale factor had been less than one, the image would be called a reduction.
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Properties of Similar Triangles
Similar triangles are triangles that have the same shape.
Example
In the figure shown, triangle DEF is similar to triangle D9E9F9. This can be expressed using
symbols as nDEF | nD9E9F9.
D
E
F
D’
E’
F’
P
1. Identify the congruent corresponding angles.
/D > /D9
/E > /E9
/F > /F9
2. Write ratios to identify the proportional sides.
D9E9 _____ DE
5 E9F9 ____ EF
5 F9D9 _____ FD
I am going to be an architect and they
use these skills all the time. Understanding
this already will be a big help when it comes time to go to college.
116 • Chapter 3 Similarity
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Chapter 3 Summary • 117
Using Similar Triangles to Find Unknown Measures
The properties of similar triangles can be used to determine unknown measures of the
triangles.
Example
In the figure shown, triangle ACE is similar to triangle BCD. Find the length of side AC.
A
B
C
D
E
6 cm
12 cm
9 cm
Using the proportional relationship between corresponding sides of similar triangles,
you know BC ___ AC
5 BD ___ AE
.
Substitute the known values for the sides.
BC ___ AC
5 BD ___ AE
9 ___ AC
5 6 ___ 12
6AC 5 (9)(12)
6AC 5108
AC 518
The length of side AC is 18 centimeters.
118 • Chapter 3 Similarity
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AA, SAS, and SSS Similarity Theorems
The Angle-Angle (AA) Similarity Theorem states: “If two angles of one triangle are
congruent to the corresponding angles of another triangle, then the triangles are similar.”
The Side-Angle-Side (SAS) Similarity Theorem states: “If two pairs of corresponding sides
of two triangles are proportional and the included angles are congruent, then the triangles
are similar.”
The Side-Side-Side (SSS) Similarity Theorem states: “If three pairs of corresponding sides
of two triangles are proportional, then the triangles are similar.”
Example
Determine if each pair of triangles are similar by AA, SAS, or SSS.
1.
36°
36°
72°
72°
A
B
C
D
E F
/A 5 /F
/B 5 /D
nABC | nFDE
The triangles are similar by AA.
2.
A
B
C
DE
6 cm
6 cm
3 cm
3 cm
AB ___ AC
5 AE ___ AD
6 __ 9 5 6 __
9
/A 5 /A
nABE | nACD
The triangles are similar by SAS.
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Chapter 3 Summary • 119
3.
A
B
C
D
E
F
12 cm
14 cm
14 cm6 cm
7 cm
7 cm
AB ___ DE
5 14 ___ 7
BC ___ EF
5 12 ___ 6
5 2 __ 1 5 2 __ 1
CA ___ FD
5 14 ___ 7
5 2 __ 1
AB ___ DE
5 BC ___ EF
5 CA ___ FD
2 __ 1
5 2 __ 1
5 2 __ 1
nABC | nDEF
The triangles are similar by SSS.
120 • Chapter 3 Similarity
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