### 31_ Similar Polygons AA_SSS_ and SAS

```Bell Ringer
Proving Triangles are Similar by
AA,SS, & SAS
Example 1
Use the AA Similarity Postulate
Determine whether the triangles
are similar. If they are similar, write
a similarity statement. Explain your
reasoning.
SOLUTION
If two pairs of angles are congruent, then the triangles are
similar.
1. G  L because they are both marked as right angles.
mF + 90° + 61° = 180°
mF + 151° = 180°
mF = 29°
Triangle Sum Theorem
Add.
Subtract 151° from each side.
Example 2
Use the AA Similarity Postulate
Are you given enough information to show
that RST is similar to
RUV? Explain your reasoning.
SOLUTION
Redraw the diagram as two triangles: RUV and RST.
From the diagram, you know that both RST and RUV
measure 48°, so RST  RUV. Also, R  R by the Reflexive
Property of Congruence. By the AA Similarity Postulate, RST ~
RUV.
Now You Try 
Use the AA Similarity Postulate
Determine whether the triangles are similar. If they are similar, write a similarity
statement.
1.
ANSWER
yes; RST ~ MNL
ANSWER
yes; GLH ~ GKJ
2.
Example 3
Use Similar Triangles
A hockey player passes the puck to a teammate by bouncing the puck off
the wall of the rink, as shown below. According to the laws of physics, the
angles that the path of the puck makes with the wall are
congruent. How far from the wall will the teammate pick up the pass?
DE
AB
x
25
=
=
EC
BC
28
40
x · 40 = 25 · 28
Write a proportion.
Substitute x for DE, 25 for AB,
28 for EC, and 40 for BC.
Cross product property
Example 3
Use Similar Triangles
40x = 700
Multiply.
40x 700
=
40
40
Divide each side by 40.
x = 17.5
ANSWER
Simplify.
The teammate will pick up the pass 17.5 feet
from the wall.
Now Checkpoint
You Try 
Use Similar Triangles
Write a similarity statement for the triangles. Then find the value of the variable.
3.
ANSWER
ABC ~ DEF; 9
4.
ANSWER
ABD ~ EBC; 3
Example 1
Use the SSS Similarity Theorem
Determine whether the triangles are
similar. If they are similar, write a
similarity statement and find the
scale factor of Triangle B to Triangle A.
SOLUTION
Find the ratios of the corresponding sides.
SU
6
6÷6
1
=
PR
12 = 12 ÷ 6 = 2
UT
5
5÷5
1
=
=
=
RQ
10 ÷ 5
2
10
TS
4
4÷4
1
=
=
=
QP
8÷4
2
8
All three ratios are
equal.
So, the corresponding
sides of the triangles
are proportional.
Example 1
Use the SSS Similarity Theorem
ANSWER
By the SSS Similarity Theorem, PQR ~ STU.
The scale factor of Triangle B to Triangle A is
.
1
2
Example 2
Use the SSS Similarity Theorem
Is either DEF or GHJ similar to ABC?
SOLUTION
1. Look at the ratios of corresponding sides in ABC and DEF.
Shortest sides
Longest sides
DE 4 2
= =
AB
6 3
FD 8 2
=
CA 12 = 3
ANSWER
Remaining sides
Because all of the ratios are equal,
ABC ~ DEF.
EF 6 2
= =
BC
9 3
Example 2
Use the SSS Similarity Theorem
2. Look at the ratios of corresponding sides in ABC and GHJ.
Shortest sides
GH
AB
ANSWER
=
6
1
=
1
6
Longest sides
JG
CA
=
14
7
=
6
12
Remaining sides
HJ
10
=
BC
9
Because the ratios are not equal, ABC and GHJ are not
similar.
Is either DEF or GHJ similar to ABC?
NowCheckpoint
You Try 
Use the SSS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a
similarity statement.
1.
ANSWER
yes; ABC ~ DFE
ANSWER
no
2.
Example 3
Use the SAS Similarity Theorem
Determine whether the triangles
are similar. If they are similar,
write a similarity statement.
SOLUTION
C and F both measure 61°, so C  F.
Compare the ratios of the side lengths that include C and F.
DF
5
=
Shorter sides AC
3
FE
10
5
=
Longer sides
=
CB
6
3
The lengths of the sides that include C and F are proportional.
ANSWER
By the SAS Similarity Theorem,
ABC ~ DEF.
Example 4
Similarity in Overlapping Triangles
Show that VYZ ~ VWX.
SOLUTION
Separate the triangles, VYZ and VWX, and label the side lengths.
V  V by the Reflexive Property of Congruence.
Shorter sides
Longer sides
VW
4
1
4
=
=
=
VY
4+8
12 3
XV
5
1
5
=
=
=
ZV
5 + 10 15 3
Example 4
Similarity in Overlapping Triangles
The lengths of the sides that include V are proportional.
ANSWER
By the SAS Similarity Theorem,
VYZ ~ VWX.
NowCheckpoint
You Try 
Use the SAS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a
similarity statement. Explain your reasoning.
3.
ANSWER
No; H  M but
12
8
.
≠
6
8
NowCheckpoint
You Try 
Use the SAS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a
similarity statement. Explain your reasoning.
4.
ANSWER
PQ 3
PR
1
5
1
,
Yes; P  P,
and
the
=
=
=
= ;
PS
2
6
PT 10 2
lengths of the sides that include P are proportional,
so PQR ~ PST by the SAS Similarity Theorem.
Page 375 #s 8-26
&
Page 382 #s 6-18 even only
```