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Name
Date
Class
Reteach
LESSON
7-3
Triangle Similarity: AA, SSS, and SAS
$
"
78° #
If two angles of one triangle
are congruent to two angles
of another triangle, then the
triangles are similar.
Angle-Angle (AA)
Similarity
57°
57°
&
!
78°
%
ABC DEF
Side-Side-Side (SSS)
Similarity
#
15
18
12
10
&
!
14.4
%
ABC DEF
If two sides of one triangle
are proportional to two sides
of another triangle and
their included angles are
congruent, then the triangles
are similar.
Side-Angle-Side (SAS)
Similarity
$
12
"
If the three sides of one
triangle are proportional
to the three corresponding
sides of another triangle,
then the triangles are similar.
$
"
#
15
57°
10
57°
12
!
&
18
%
ABC DEF
Explain how you know the triangles are similar, and write a similarity statement.
1.
2
2.
5
49°
+
(
24
18
*
1 92°
3
4
92°
39°
16
'
6
Q T by the Def. of . By
27
,
the Sum Thm., mS 39 and
HJG LJK by the Vert. HJ ___
GJ __
2. GHJ Thm. ___
LJ
KJ 3
mU 49, so S V and
KLJ by SAS .
R U. QRS TUV
by AA .
3. Verify that ABC MNP.
BC ____
CA __
4 ; ABC AB ____
____
MN
NP
PM
!
12
12
5
#
MNP by SSS .
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All rights reserved.
0
22
8
"
15
10
.
15
Holt Geometry
Name
Date
Class
Reteach
LESSON
7-3
Triangle Similarity: AA, SSS, and SAS
continued
You can use AA Similarity, SSS Similarity, and SAS Similarity to solve problems.
First, prove that the triangles are similar. Then use the properties of similarity to
find missing measures.
Explain why ADE ABC and then find BC.
Step 1
#
2
Prove that the triangles are similar.
A A by the Reflexive Property of .
"
3 __
AD __
1
___
AB
6
3.5
3
$
3
%
2
!
2
AE __
2 __
1
___
AC
4
2
Therefore, ADE ABC by SAS .
Step 2
Find BC.
AD ___
DE
___
AB
BC
3 ___
3.5
__
6
BC
Corresponding sides are proportional.
Substitute 3 for AD, 6 for AB, and 3.5 for DE.
3(BC) 6(3.5)
Cross Products Property
3(BC) 21
Simplify.
BC 7
Divide both sides by 3.
Explain why the triangles are similar and then find each length.
4. GK
5. US
*
4
+
8
7
'
11
42
28
12
3
&
_
(
_
6
26
5
JK FH , so J H, and
It is given that S WVU.
K F by the Alt. Int. Thm.
U U by the Reflex. Prop.
JKG HFG by AA .
1
GK 7 __
3
of . UVW UST by AA .
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
US 39
23
Holt Geometry
Name
LESSON
7-3
Date
Class
Name
Practice A
LESSON
7-3
Triangle Similarity: AA, SSS, SAS
Fill in the blanks to complete each postulate or theorem.
proportional
1. If the three sides of one triangle are
another triangle, then the triangles are similar.
Date
Practice B
Triangle Similarity: AA, SSS, SAS
For Exercises 1 and 2, explain why the triangles are similar and write
a similarity statement.
to the three sides of
1.
similar
3. If two angles of one triangle are
triangle, then the triangles are similar.
congruent
�
�
to two angles of another
5.
�
37°
�
�
�
�
.
37°
�
�
�
����
����
Name two pairs of congruent angles in Exercises 4 and 5 to show that
the triangles are similar by the Angle-Angle (AA) Similarity Postulate.
4.
2.
�
�
2. If two sides of one triangle are proportional to two sides of another triangle and their
included angles are congruent, then the triangles are
Class
�
�
�
Possible answer: �ACB and
Possible answer: Every equilateral
�ECD are congruent vertical
triangle is also equiangular, so
angles. m�B � m�D � 100°,
each angle in both triangles
so �B � �D. Thus, �ABC �
measures 60°. Thus, �TUV �
�EDC by AA �.
�WXY by AA �.
�
�
�
�
�
�B � �Y; �C � �Z
�
�
�
For Exercises 3 and 4, verify that the triangles are similar. Explain why.
�F � �V; �D � �T
4. �PQR and �UTS
3. �JLK and �JMN
�
Substitute side lengths into the ratios in Exercise 6. If the ratios are equal,
the triangles are similar by the Side-Side-Side (SSS) Similarity Theorem.
6.
12
� 6
12 �
9
�
�
�
�
6
___
�
�
12
JK
�
�
_1_
9
___
HI �
___
2
20
8
�
_1_
�
18
KL
12
___
GI �
___
2
_1_
�
24
JL
2
�
congruent angles:
�Q � �T
8
___
10
PQ �
___
ST
_4_
5
�
19
16
___
20
QR �
___
TU
Date
�
Class
Holt Geometry
Thus, �PQR � �UTS by SSS �.
PQ � ___
QR � ___
PR � _3_.
___
UT
3.25
�
20
Name
TS
7-3
Possible answer: �ABC and �ADB share �A. They
3. Use the similarity ratios you found in Exercise 1 and the answer
to Exercise 2 to find the perimeters of �ADB and �BDC.
�
4
�
2
�
�
3
Date
Class
���
���
�
�
�
�
78°
�
�ABC � �DEF
Side-Side-Side (SSS)
Similarity
3 � � 3 ; 3 � 3� 3
�
�
15
18
12
10
�
�
14.4
�
�ABC � �DEF
If two sides of one triangle
are proportional to two sides
of another triangle and
their included angles are
congruent, then the triangles
are similar.
Side-Angle-Side (SAS)
Similarity
�
12
�
If the three sides of one
triangle are proportional
to the three corresponding
sides of another triangle,
then the triangles are similar.
�
�
�
15
57°
12
10
57°
18
�
�
�
�ABC � �DEF
Explain how you know the triangles are similar, and write a similarity statement.
_
_
and QT. �G and �P are right angles, so they are
��
�
�
��
��
�
�
��
�
�
�
�
�
1.
�
� ��
�
2.
�
49°
��
�
�
�
PT
PQ 2
IJ � _3_, so �HIJ � �QRS by SAS�. Because �GHK
HI � ___
�I � �R. ___
QR RS 2
HJ � _3_
HK � _3_ and �GHK � �PQT. Because �HIJ � �QRS, ___
� �PQT, ___
QT 2
QS 2
and �IHJ � �RQS. It is given that �H � �Q. So by the Angle Addition
�
92°
39°
�Q � �T by the Def. of � �. By
�HJG � �LJK by the Vert. �
m�U � 49�, so �S � �V and
�KLJ by SAS �.
BC � ____
CA � _4_; �ABC �
AB � ____
____
MN
NP
PM
�
12
5
�
�MNP by SSS �.
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
55
�
HJ � ___
GJ � _2_. �GHJ �
Thm. ___
LJ
KJ 3
3. Verify that �ABC � �MNP.
PQRST by the definition of similar polygons.
27
the � Sum Thm., m�S � 39� and
�R � �U. �QRS � �TUV
by AA �.
HJ � _3_, so �KHJ � �TQS by SAS�.
HK � ___
Postulate, �KHJ � �TQS. ___
QT QS 2
JK � ___
HK � _3_. All the corresponding angles are
Because �KHJ � �TQS, ___
ST QT 2
congruent; all the corresponding sides are proportional. Thus, GHIJK �
�
16
�
�
24
18
�
� 92°
GK � ___
GH � _3_, so �GHK � �PQT by SAS�. It is given that
congruent. ___
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
57°
57°
�
Possible answer: Draw diagonals HK, HJ, QS,
21
78° �
If two angles of one triangle
are congruent to two angles
of another triangle, then the
triangles are similar.
Angle-Angle (AA)
Similarity
�
Holt Geometry
�
�
�
5. Use triangle similarity to prove that GHIJK � PQRST.
_ _
�
Triangle Similarity: AA, SSS, and SAS
�
7
5
�
1.25
�
�
�
5
US
Reteach
LESSON
also each have a right angle, so �ABC � �ADB by AA �. They have a
similarity ratio of _2_. �ABC and �BDC share �C. They also each have
1
a right angle, so �ABC � �BDC by AA �. They have a similarity ratio
�
2�3 to 1. By the Transitive Property of Similarity, �ADB � �BDC.
of ____
�
3
�3 to 1.
They have a similarity ratio of ___
3
�
6 � 2�3
AD � 1 and DC � 3. Find the perimeter of �ABC.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Thus, �JLK � �JMN by SAS �.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
1. Prove similarity relationships between triangles in the figure.
Give a similarity ratio for each relationship you find.
3
Possible answer:
right angles, so they are congruent. Thus, �CDG � �CEF by AA �.
Triangle Similarity: AA, SSS, SAS
13
___
Possible answer: It is given that
JL � _4_.
KL � ___
�JMN � �L. ___
MN JM 3
Possible answer: �C � �C by the Reflexive Property. �CGD and �F are
_4_
5
Practice C
4. Find ST.
�
DE � 9.75
Use the figure for Exercises 1–3.
2.
3.5
6
�
5. DE
�
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All rights reserved.
7-3
�
3
�
For Exercise 5, explain why the triangles are
similar and find the stated length.
�
Name
LESSON
�
�
10
16
1.8
3.6
�
Name one pair of congruent angles and substitute side lengths into
the ratios in Exercise 7. If the ratios are equal and the congruent
angles are in between the proportional sides, the triangles are similar
by the Side-Angle-Side (SAS) Similarity Theorem.
7.
8
12
24
18
GH �
___
24
�
�
2.1
�
16
22
�
12
8
�
15
10
�
�
15
Holt Geometry
Holt Geometry
Name
Date
Class
Name
Reteach
LESSON
7-3
7-3
continued
You can use AA Similarity, SSS Similarity, and SAS Similarity to solve problems.
First, prove that the triangles are similar. Then use the properties of similarity to
find missing measures.
Explain why �ADE � �ABC and then find BC.
2
�
AD � _3_ � _1_
___
6
AB
�
3.5
�A � �A by the Reflexive Property of �.
3
�
Similar Triangles Within a Right Triangle
Take a rectangular sheet of paper and draw the segments
shown at right. Cut out the three triangles and align them
as shown. If your work is accurate, the triangles should
appear similar. In fact, you have demonstrated the following
important theorem about right triangles.
�
Prove that the triangles are similar.
RIGHT TRIANGLE ALTITUDE THEOREM
2
�
3
In a right triangle, the altitude from the vertex of the right angle to the
hypotenuse forms two triangles that are similar to the given triangle
and to each other.
2
AE � _2_ � _1_
___
4
AC
2
Therefore, �ADE � �ABC by SAS �.
Find BC.
AD � ___
DE
___
AB
BC
Step 2
Substitute 3 for AD, 6 for AB, and 3.5 for DE.
BC
3(BC) � 6(3.5)
Cross Products Property
3(BC) � 21
Simplify.
BC � 7
STATEMENTS
_
_
QS � PR
Explain why the triangles are similar and then find each length.
5. US
�
�
�
8
�
11
42
28
12
�
�
_
�
26
�
�
_
�
1.
Given
2.
Definition of altitude
�PSQ and �QSR are right angles.
3.
Definition of perpendicular
m�PSQ � m�QSR � m�PQR � 90�
4.
Definition of right angle
�PSQ � �PQR ; �QSR � �PQR
5.
Definition of congruent angles
�P � �P; �R � �R
6.
Reflexive Property of Congruence
�PSQ � �PQR ; �QSR � �PQR
7.
AA Similarity Postulate
�PSQ � �QSR
8.
Transitive Property
�
�ABC; �ACD � �CBD
�K � �F by the Alt. Int. � Thm.
�U � �U by the Reflex. Prop.
of �. �UVW � �UST by AA �.
�
�
�
�ACD � �ABC; �CBD �
�JKG � �HFG by AA �.
GK � 7 _1_
3
�
�
9. a. Name all pairs of similar triangles in the figure at the right.
It is given that �S � �WVU.
JK � FH , so �J � �H, and
�
REASONS
�PQR
is a right angle.
_
QS is the altitude of �PQR
drawn from the right angle.
Divide both sides by 3.
4. GK
�
Complete the following proof of the Right Triangle
Altitude Theorem.
Given: _
�PQR is a right angle.
QS is the altitude of �PQR drawn from the right angle.
Prove: �PSQ � �PQR; �QSR � �PQR; �PSQ � �QSR
Corresponding sides are proportional.
3.5
_3_ � ___
6
Class
Challenge
LESSON
Triangle Similarity: AA, SSS, and SAS
Step 1
Date
�
�
�
�
�
�
_
_c_
b. Complete each proportion: _a_ � _c_ and _b
e �b
f
a
c. On a separate sheet of paper, write a proof of the Pythagorean Theorem that
utilizes the figure above and your results from parts a and b. (Hint: Apply the
Cross-Multiplication Property to the proportions and add the resulting equations.)
US � 39
Proofs will vary.
23
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All rights reserved.
Name
Date
Class
Holt Geometry
Problem Solving
LESSON
7-3
7-3
Two triangles
_
In the diagram of the tandem bike, AE � BD.
1. Explain why �CBD ~ �CAE.
������
�
and �C � �C by the Reflex. Prop.
�
�
�
�CBD � �CAE by Corr. � Thm.
�����
�
Are two angles of one triangle congruent
to two angles of the other triangle?
������
Are three sides of one triangle proportional
to three sides of the other triangle?
Are two sides of one triangle proportional
to two sides of the other triangle and the
included angles congruent?
46.7 in.
3. Is �WXZ ~ �XYZ ? Explain.
4. Find RQ. Explain how you found it.
�
�
13
16.3
44
�
40
�
�
19.3
25
24
13.8
27.5
�
1.
�
Corr. sides of ~ � are
�
3.
��4
�
�
�
7. To measure the distance EF across the
lake, a surveyor at S locates points E, F,
G, and H as shown. What is EF ?
8
�
�
� � 3.5
A 8
B 9
The triangles are similar by
Side-Angle-Side (SAS)
Similarity.
�
4
�
2 �
�
C 12
D 16
�
10
�
�
4
12
6
SSS �
4.
�
�
�
�
5.
����
�
�
3 �
�
2 4
�
�
�
����
�
�
�
AA �
6.
9
�
12
�
6
12
�
�
SSS �
�
A 25 m
B 36 m
�
5
�
�
����
����
�
2.
�
3
�
�
6. Triangle STU has vertices at S(0, 0),
T(2, 6), and U(8, 2). If �STU ~ �WXY
and the coordinates of W are (0, 0), what
are possible coordinates of X and Y?
F X(1, 3) and Y(4, 1)
G X(1, 3) and Y(2, 0)
H X(3, 1) and Y(2, 4)
yes
no conclusion
20
�
The triangles are similar by
Side-Side-Side (SSS) Similarity.
SAS �
Choose the best answer.
�
3
�
4
proportional.
5. Find the value of x that makes
�FGH ~ �JKL.
yes
Use the flowchart to determine, if possible, whether the following pairs of
triangles are similar. If similar, write AA �, SSS �, or SAS �—the postulate
or theorem you used to conclude that they are similar. If it is not possible
to conclude that they are similar, write no conclusion.
�
15; �MNP ~ �RQP by SAS ~.
WX � ___
XZ
No; ____
XY
YZ
The triangles are similar by
Angle-Angle (AA) Similarity.
no
2. Find CE to the nearest tenth.
�
yes
no
of �. So �CBD ~ �CAE by AA ~.
�
Holt Geometry
Class
Use a Graphic Aid
Use the diagram for Exercises 1 and 2.
11
Date
Reading Strategies
LESSON
Triangle Similarity: AA, SSS, and SAS
_
24
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All rights reserved.
Name
������
7.
C 45 m
D 90 m
�
22
8.
�
�
1
�
�
3
�
�
�
�
6
�
no conclusion
�
�
�
11
�
12
4
�
�
SAS �
AA �
J X(0, 3) and Y(4, 0)
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All rights reserved.
25
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All rights reserved.
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
56
26
Holt Geometry
Holt Geometry