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Transcript
Name
LESSON
7-3
Date
Class
Practice B
Triangle Similarity: AA, SSS, SAS
For Exercises 1 and 2, explain why the triangles are similar and write
a similarity statement.
1.
#
4
2.
$
!
7
8
—
—
%
"
5
6
9
Possible answer: ACB and
Possible answer: Every equilateral
ECD are congruent vertical
triangle is also equiangular, so
angles. mB mD 100°,
each angle in both triangles
so B D. Thus, ABC measures 60°. Thus, TUV EDC by AA .
WXY by AA .
For Exercises 3 and 4, verify that the triangles are similar. Explain why.
3. JLK and JMN
4. PQR and UTS
+
2.1
0
16
*
24
1
1.8
2
3.6
3
4
3
8
12
3.5
5
6
,
.
Possible answer: It is given that
JL __
KL ___
4.
JMN L. ___
MN JM 3
Possible answer:
Thus, JLK JMN by SAS .
Thus, PQR UTS by SSS .
PQ ___
QR ___
3.
PR __
___
UT
TS
US
5
%
For Exercise 5, explain why the triangles are
similar and find the stated length.
5
3.25
#
5. DE
$
1.25
'
&
Possible answer: C C by the Reflexive Property. CGD and F are
right angles, so they are congruent. Thus, CDG CEF by AA .
DE 9.75
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
20
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
�
Copyright © by Holt, Rinehart and Winston.
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