2. Section 8.5 Proving Triangles are Similar
Chapter 8 Similarity
Section 8.5
Proving Triangles are Similar
USING SIMILARITY THEOREMS
USING SIMILAR TRIANGLES IN REAL LIFE

3. C D E A D and C F  ABC ~ DEF F B A
USING SIMILARITY THEOREMS
Postulate A C B D F E
A D and C F 
ABC ~ DEF

4. AA Similarity Postulate
W  W
WVX  WZY
AA Similarity

5. AA Similarity Postulate WSU ~ VTU AA Similarity Postulate
USING SIMILARITY THEOREMS
AA Similarity Postulate
WSU ~ VTU
AA Similarity Postulate
CAB ~ TQR

6. USING SIMILARITY THEOREMS
THEOREM Side-Side-Side (SSS) Similarity Theorem
If the corresponding sides of two triangles are proportional, then the triangles are similar. P Q R A B C
If = =
A B PQ BC QR CA RP
then ABC ~ PQR.

7. Because all of the ratios are equal,  ABC ~  DEF
Using the SSS Similarity Theorem
Which of the following three triangles are similar? A C B 12 6 9 E F D 8 6 4 G J H 14 6 10
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

8. Using the SSS Similarity Theorem
Which of the following three triangles are similar? E F D 8 6 4 A C B 12 9 G J H 14 10 A 12 C E F D 8 6 4 G 14 J 6 9 6 10 B H
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
Since  ABC is similar to  DEF and  ABC is not similar to  GHJ,  DEF is not similar to  GHJ.
= = 1, 6 AB GH
Shortest sides
= = , 12 14 CA JG 6 7
Longest sides = 9 10 BC HJ
Remaining sides
Because all of the ratios are not equal,  ABC and  DEF are not similar.

9. USING SIMILARITY THEOREMS
THEOREM Side-Angle-Side (SAS) Similarity Theorem X Z Y M P N
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. ZX PM XY MN
If X M and =
then XYZ ~ MNP.

10. Use the given lengths to prove that  RST ~  PSQ.
Using the SAS Similarity Theorem
Use the given lengths to prove that  RST ~  PSQ.
SOLUTION
GIVEN
SP = 4, PR = 12, SQ = 5, QT = 15
PROVE
 RST ~  PSQ P Q S R T
Paragraph Proof Use the SAS Similarity Theorem. Find the ratios of the lengths of the corresponding sides. 12 4 5 15
= = = = 4 SR SP 16 4
SP + PR
= = = = 4 ST SQ 20 5
SQ + QT
The side lengths SR and ST are proportional to the corresponding side lengths of  PSQ.
Because S is the included angle in both triangles, use the SAS Similarity Theorem to conclude that  RST ~  PSQ.

11. USING SIMILARITY THEOREMS
Nothing is known about any corresponding congruent angles SSS ~ Theorem is the only choice 9 6
SSS ~ Theorem
ABC ~ XYZ

12. Only one Angle is Known Use SAS ~ Theorem
USING SIMILARITY THEOREMS
Nothing is known about any corresponding congruent angles SSS ~ Theorem is the only choice
Only one Angle is Known Use SAS ~ Theorem 9 6 6 3
SSS ~ Theorem
ABC ~ XYZ

13. Parallel lines give congruent angles Use AA ~ Postulate
USING SIMILARITY THEOREMS
Parallel lines give congruent angles Use AA ~ Postulate
Only one Angle is Known Use SAS ~ Theorem

14. No, Need to know the included angle.
USING SIMILARITY THEOREMS
No, Need to know the included angle.

15. No, Need to know the included angle. Yes, AA ~ Postulate DRM ~ XST
USING SIMILARITY THEOREMS 40
No, Need to know the included angle.
Yes, AA ~ Postulate
DRM ~ XST

16. USING SIMILARITY THEOREMS
SSS ~ Theorem
AA ~ Theorem
SAS ~ Theorem

17. HW Pg :6;9;11;13-17;19-25;27-29;32-34;39-47