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Proving Triangles are Similar8.5
Proving Triangles are Similar8.5
Chapter 8SimilaritySection 8.5
Proving Triangles are Similar
USING SIMILARITY THEOREMS
USING SIMILAR TRIANGLES IN REAL LIFE
Proving Triangles are Similar8.5
Postulate
A
C
B
D
F
E
A D and C F
ABC ~ DEF
USING SIMILARITY THEOREMS
Proving Triangles are Similar8.5
AA Similarity Postulate
W W
WVX WZY
AA Similarity
Proving Triangles are Similar8.5
AA Similarity Postulate
CAB ~ TQR
USING SIMILARITY THEOREMS
AA Similarity Postulate
WSU ~ VTU
Proving Triangles are Similar8.5
USING SIMILARITY THEOREMS
THEOREM S
THEOREM 8.2 Side-Side-Side (SSS) Similarity Theorem
If the corresponding sides of two triangles are proportional, then the triangles are similar.
If = =A BPQ
BCQR
CARP
then ABC ~ PQR.
A
B C
P
Q R
Proving Triangles are Similar8.5
E
F D8
6 4A C
B
12
6 9
G J
H
14
6 10
Using the SSS Similarity Theorem
Which of the following three triangles are similar?
SOLUTION To decide which of the triangles are similar, consider the
ratios of the lengths of corresponding sides.
Ratios of Side Lengths of ABC and DEF
= = , 6 4
AB DE
3 2
Shortest sides
= = , 12 8
CA FD
3 2
Longest sides
= = 9 6
BC EF
3 2
Remaining sides
Because all of the ratios are equal, ABC ~ DEF
Proving Triangles are Similar8.5
= = ,1214
CA JG
67
Longest sides
E
F D8
6 4
Using the SSS Similarity Theorem
A C
B
G J
H
Which of the following three triangles are similar?
SOLUTION To decide which of the triangles are similar, consider the
ratios of the lengths of corresponding sides.
Ratios of Side Lengths of ABC and GHJ
12 14
6 6 109
= = 1, 6 6
AB GH
Shortest sides
= 910
BC HJ
Remaining sides
Because all of the ratios are not equal, ABC and DEF are not similar.
E
F D8
6 4A C
B
12
6 9
G J
H
14
6 10
Since ABC is similar to DEF and ABC is not similar to GHJ, DEF is not similar to GHJ.
Proving Triangles are Similar8.5
USING SIMILARITY THEOREMS
THEOREM S
THEOREM 8.3 Side-Angle-Side (SAS) Similarity Theorem
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.
then XYZ ~ MNP.
ZXPM
XYMN
If X M and =
X
Z Y
M
P N
Proving Triangles are Similar8.5
Using the SAS Similarity Theorem
Use the given lengths to prove that RST ~ PSQ.
SOLUTION PROVE RST ~ PSQ
GIVEN SP = 4, PR = 12, SQ = 5, QT = 15
Paragraph Proof Use the SAS Similarity Theorem. Find the ratios of the lengths of the corresponding sides.
= = = = 4 SR SP
16 4
SP + PR SP
4 + 12 4
= = = = 4 ST SQ
20 5
SQ + QT SQ
5 + 15 5
Because S is the included angle in both triangles, use the SAS Similarity Theorem to conclude that RST ~ PSQ.
The side lengths SR and ST are proportional to the corresponding side lengths of PSQ.
12
4 5
15
P Q
S
R T
Proving Triangles are Similar8.5
USING SIMILARITY THEOREMS
Nothing is known about any corresponding congruent angles
SSS ~ Theorem is the only choice9 6
69
46
69
23
SSS ~ Theorem
ABC ~ XYZ
Proving Triangles are Similar8.5
USING SIMILARITY THEOREMS
Nothing is known about any corresponding congruent angles
SSS ~ Theorem is the only choice9 6
69
46
69
23
SSS ~ Theorem
ABC ~ XYZ
Only one Angle is Known Use SAS ~ Theorem
6 3
Proving Triangles are Similar8.5
USING SIMILARITY THEOREMS
Only one Angle is Known Use SAS ~ Theorem
Parallel lines give congruent angles Use
AA ~ Postulate
Proving Triangles are Similar8.5
USING SIMILARITY THEOREMS
No, Need to know the included angle.
Proving Triangles are Similar8.5
USING SIMILARITY THEOREMS
No, Need to know the included angle.
40
Yes, AA ~ Postulate
DRM ~ XST
Proving Triangles are Similar8.5
USING SIMILARITY THEOREMS
SSS ~ Theorem
AA ~ Theorem
SAS ~ Theorem
Proving Triangles are Similar8.5
HW Pg :6;9;11;13-17;19-25;27-
29;32-34;39-47