7-3 Triangle Similarity

1. 7-3 Triangle Similarity • I CAN • -Use the triangle similarity theorems to • determine if two triangles are similar. • Use proportions in similar triangles to solve • for missing sides.

2. Recall in 7-2, to prove that two polygons are similar you had to: Show all corresponding angles are congruent AND Show all corresponding sides are proportional Triangle Similarity Theorems are “shortcuts” for showing two triangles are similar.

4. Example D 9 E B 12 10 5 C A 6 18 F Similarity Statement Reason ABC~ EFD by AA Because A E and C D justification

5. Example Explain why the triangles are similar and write a similarity statement. By the Triangle Sum Theorem, mC = 47°, so C F. B E by the Right Angle Congruence Theorem. Therefore, ∆ABC ~ ∆DEF by AA ~.

7. Example Verify that the triangles are similar. ∆PQR and ∆STU Therefore ∆PQR ~ ∆STU by SSS ~.

9. Example Verify that the triangles are similar. ∆DEF and ∆HJK D  H by the Definition of Congruent Angles. Therefore ∆DEF ~ ∆HJK by SAS ~.

10. Example Verify that ∆TXU ~ ∆VXW. TXU VXW by the Vertical Angles Theorem. Therefore ∆TXU ~ ∆VXW by SAS ~.

11. Your Turn

12. Your Turn

13. Your Turn

14. Find the value of x such that ∆ACE ~ ∆BCD C Why is ∆ACE ~ ∆BCD? C 3 3 12 D B D B x 12 28 E C A 3 = 12 x + 3 28 x + 3 E A 12(x + 3) = 84 28 12x + 36 = 84 – 36 – 36 12x = 48 x = 4

15. Example Explain why ∆ABE ~ ∆ACD, and then find CD. Step 1 Prove triangles are similar. A A by Reflexive Property of , and B C since they are both right angles. Therefore ∆ABE ~ ∆ACD by AA ~. Step 2 Find CD. x(9) = 5(3 + 9) 9x = 60

16. Example Explain why ∆RSV ~ ∆RTU and then find RT. Step 1 Prove triangles are similar. It is given that S T. R R by Reflexive Property of . Therefore ∆RSV ~ ∆RTU by AA ~. Step 2 Find RT. RT(8) = 10(12) 8RT = 120 RT = 15

17. Your Turn

18. Properties of Similar Triangles7-4 • I can use the triangle proportionality theorem and its converse. • I can set up and solve problems using properties of similar triangles.

19. Example: AF = 3, FC = 5, AE = 2. Find BE OR

21. It is given that , so by the Triangle Proportionality Theorem. Example Find US. US(10) = 56

22. Example Find PN. 2PN = 15 PN = 7.5

23. Your Turn Solve for x.

24. Example Solve for x.

25. Example

26. Example I Solve for x, y, and w. x 4 N E w y 5 L A 3 2 D S 12

27. Example I Solve for x, y, and w. x = 6 4 N E w y 5 L A 3 2 D S 12 OR OR

28. Example I Solve for x, y, and w. x = 6 4 N E w 5 L A 3 2 D S 12

30. Example BD = 8; DF = 6; CE = 16. EG = ________

31. Example BD = 2x – 2; DF = 4; CE = x + 2; EG = 8 Find BD and CE

32. Your Turn

33. Example: SHOW EF BC if BE = 21, AE = 42, CF = 15, and AF = 30 ? So, EF BC

34. Verify that . Since , by the Converse of the Triangle Proportionality Theorem. Example

35. AC = 36 cm, and BC = 27 cm. Verify that . Since , by the Converse of the Triangle Proportionality Theorem. Your Turn

36. The previous theorems and corollary lead to the following conclusion.

37. by the ∆ Bisector Theorem. Example Find PS and SR. PS = x – 2 40(x – 2) = 32(x + 5) = 30 – 2 = 28 40x – 80 = 32x + 160 SR = x + 5 40x – 80 = 32x + 160 = 30 + 5 = 35 8x = 240 x = 30

38. by the ∆ Bisector Theorem. Example Find AC and DC. 4y = 4.5y – 9 –0.5y = –9 y = 18 So DC = 9 and AC = 16.

39. Your Turn