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Bell Ringer

Bell Ringer. Proving Triangles are Similar by AA,SS, & SAS. Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. m  F + 90° + 61° = 180°. Triangle Sum Theorem. m  F + 151° = 180°. Add. Subtract 151° from each side.

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Bell Ringer

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  1. Bell Ringer

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

  3. Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. mF+ 90°+ 61°= 180° Triangle Sum Theorem mF + 151° = 180° Add. Subtract 151° from each side. mF = 29° SOLUTION If two pairs of angles are congruent, then the triangles are similar. G  Lbecause they are both marked as right angles. 1. Example 1 Use the AA Similarity Postulate

  4. Are you given enough information to show that RST is similar to RUV? Explain your reasoning. SOLUTION Redraw the diagram as two triangles: RUVand RST. Example 2 Use the AA Similarity Postulate From the diagram, you know that both RSTand RUVmeasure 48°, so RSTRUV. Also, RRby the Reflexive Property of Congruence. By the AA Similarity Postulate, RST~RUV.

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

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

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

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

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

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

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

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

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

  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 measure61°, soC  F. Compare the ratios of the side lengths that include CandF. FE 10 5 5 DF Shorter sides Longer sides = CB 6 = 3 = AC 3 The lengths of the sides that include CandFare proportional. By the SAS Similarity Theorem, ABC ~ DEF. ANSWER

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

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

  17. Checkpoint . Now You Try  Use the SAS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. 3. 12 8 No; H M but 6 8 ANSWER ≠

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

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

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