1. Bell Ringer

2. Proving Triangles are Similar by AA,SS, & SAS

3. Use the AA Similarity Postulate Example 1 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° Triangle Sum Theorem m  F + 151° = 180° Add. m  F = 29° Subtract 151° from each side.

4. Use the AA Similarity Postulate Example 2 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.

5. Now You Try Use the AA Similarity Postulate Determine whether the triangles are similar. If they are similar, write a similarity statement. 1. 2. ANSWER yes;  RST ~  MNL ANSWER yes;  GLH ~  GKJ

6. Use Similar Triangles Example 3 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? AB DE = BC EC Write a proportion. 25 x = 40 28 Substitute x for DE, 25 for AB, 28 for EC, and 40 for BC. x · 40 = 25 · 28 Cross product property

7. Use Similar Triangles Example 3 40x = 700 Multiply. 40 40x = 40 700 Divide each side by 40. x = 17.5 Simplify. The teammate will pick up the pass 17.5 feet from the wall. ANSWER

8. Checkpoint Use Similar Triangles Write a similarity statement for the triangles. Then find the value of the variable. 3. 4. ANSWER  ABC ~  DEF ; 9  ABD ~  EBC ; 3 ANSWER Now You Try

9. Example 1 Use the SSS Similarity Theorem Find the ratios of the corresponding sides. SOLUTION 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. All three ratios are equal. So, the corresponding sides of the triangles are proportional. PR SU 12 6 12 ÷ 6 6 ÷ 6 = = 2 1 = RQ UT 10 5 10 ÷ 5 5 ÷ 5 = = 2 1 = QP TS 8 4 8 ÷ 4 4 ÷ 4 = = 2 1 =

10. Example 1 Use the SSS Similarity Theorem ANSWER The scale factor of Triangle B to Triangle A is. 2 1 By the SSS Similarity Theorem,  PQR ~  STU.

11. Example 2 Use the SSS Similarity Theorem Is either  DEF or  GHJ similar to  ABC ? SOLUTION Look at the ratios of corresponding sides in  ABC and  DEF. 1. Shortest sides = 6 4 AB DE 3 2 = Longest sides = 12 8 CA FD 3 2 = Remaining sides = 9 6 BC EF 3 2 = ANSWER Because all of the ratios are equal,  ABC ~  DEF.

12. Example 2 Use the SSS Similarity Theorem Look at the ratios of corresponding sides in  ABC and  GHJ. 2. Shortest sides = 6 6 AB GH 1 1 = Longest sides = 12 14 CA JG 6 7 = Remaining sides BC HJ 9 10 = ANSWER Because the ratios are not equal,  ABC and  GHJ are not similar. Is either  DEF or  GHJ similar to  ABC ?

13. Checkpoint Use the SSS Similarity Theorem ANSWER yes;  ABC ~  DFE ANSWER no Determine whether the triangles are similar. If they are similar, write a similarity statement. 1. 2. Now You Try

14. 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. = AC DF 3 5 Shorter sides = 6 10 CB FE 3 5 = Longer sides The lengths of the sides that include  C and  F are proportional. ANSWER By the SAS Similarity Theorem,  ABC ~  DEF.

15. Example 4 Similarity in Overlapping Triangles Show that  VYZ ~  VWX. Separate the triangles,  VYZ and  VWX, and label the side lengths. SOLUTION  V   V by the Reflexive Property of Congruence. Shorter sides 4 + 8 4 = VY VW 12 4 = 3 1 = Longer sides 5 + 10 5 = ZV XV 15 5 = 3 1 =

16. Example 4 Similarity in Overlapping Triangles ANSWER By the SAS Similarity Theorem,  VYZ ~  VWX. The lengths of the sides that include  V are proportional.

17. Checkpoint 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 6 8 8 12 ≠. Now You Try

18. Checkpoint Use the SAS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. 4. ANSWER Yes;  P   P, and the lengths of the sides that include  P are proportional, so  PQR ~  PST by the SAS Similarity Theorem., 6 3 PS PQ == 2 1 10 5 PT PR == 2 1 ; Now You Try

19. Page 375 #s 8-26 & Page 382 #s 6-18 even only