1. There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.

2. Example 1: Using the AA Similarity Postulate Prove:

3. AA Angle Congruence Thm Prove: 4. 4. 3. 3. . 2. 1. 1. Justifications Conclusions

6. Example 2A: Verifying Triangle Similarity Verify that the triangles are similar. ∆ PQR and ∆ STU

7. Example 2A: Verifying Triangle Similarity Verify that the triangles are similar. ∆ PQR and ∆ STU

8. Example 2A: Verifying Triangle Similarity Verify that the triangles are similar. ∆ PQR and ∆ STU

9. Example 2A: Verifying Triangle Similarity Verify that the triangles are similar. ∆ PQR and ∆ STU

10. Example 2A: Verifying Triangle Similarity Verify that the triangles are similar. ∆ PQR and ∆ STU Therefore ∆ PQR ~ ∆ STU by SSS ~ .

11. Example 2B: Verifying Triangle Similarity ∆ DEF and ∆ HJK Verify that the triangles are similar.

12. Example 2B: Verifying Triangle Similarity ∆ DEF and ∆ HJK Verify that the triangles are similar.  D   H by the Definition of Congruent Angles.

13. Example 2B: Verifying Triangle Similarity ∆ DEF and ∆ HJK Verify that the triangles are similar.  D   H by the Definition of Congruent Angles.

14. Example 2B: Verifying Triangle Similarity ∆ DEF and ∆ HJK Verify that the triangles are similar.  D   H by the Definition of Congruent Angles. Therefore ∆ DEF ~ ∆ HJK by SAS ~.

15. Example 3: Finding Lengths in Similar Triangles Explain why ∆ ABE ~ ∆ ACD , and then find CD .

16.  A   A by Reflexive Property of  , and  B   C since they are both right angles. Example 3: Finding Lengths in Similar Triangles Explain why ∆ ABE ~ ∆ ACD , and then find CD . Step 1 Prove triangles are similar.

17.  A   A by Reflexive Property of  , and  B   C since they are both right angles. Example 3: Finding Lengths in Similar Triangles Explain why ∆ ABE ~ ∆ ACD , and then find CD . Step 1 Prove triangles are similar. Therefore ∆ ABE ~ ∆ ACD by AA ~.

18. Example 3 Continued Step 2 Find CD .

19. Example 3 Continued Step 2 Find CD .

20. Example 3 Continued Step 2 Find CD . Corr. sides are proportional. Seg. Add. Postulate. Substitute x for CD, 5 for BE, 3 for CB, and 9 for BA.

21. Example 3 Continued Step 2 Find CD . Corr. sides are proportional. Seg. Add. Postulate. Substitute x for CD, 5 for BE, 3 for CB, and 9 for BA. Cross Products Prop. x (9) = 5(3 + 9)

22. Example 3 Continued Step 2 Find CD . Corr. sides are proportional. Seg. Add. Postulate. Substitute x for CD, 5 for BE, 3 for CB, and 9 for BA. Cross Products Prop. x (9) = 5(3 + 9) Simplify. 9 x = 60 Divide both sides by 9.

23. You have learned that the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. These properties also hold true for similarity of triangles.

24. You have learned that the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. These properties also hold true for similarity of triangles.

25. HW for 13-1: Read Lesson 13.1 Questions: CTR, ATM, Review 13-2: Read Lesson 13.2 Questions: CTR, ATM, Review