Chapter 8 Similarity

1. USING SIMILARITY THEOREMS USING SIMILAR TRIANGLESIN REAL LIFE Chapter 8Similarity Section 8.5 Proving Triangles are Similar

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

3. USING SIMILARITY THEOREMS THEOREMS P A AB PQ BC QR CA RP Q R If = = B C THEOREM 8.2 Side-Side-Side (SSS) Similarity Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. then ABC ~ PQR.

4. Proof of Theorem 8.2 M S ST MN RS LM RS PS ST SQ TR NL TR QP GIVEN = = = = RST ~ LMN PROVE L N R T Locate P on RS so that PS = LM. Draw PQ so that PQRT. Then RST ~PSQ, by the AA Similarity Postulate, and . P Q SOLUTION Paragraph Proof Because PS = LM, you can substitute in the given proportion and find that SQ = MN and QP = NL. By the SSS Congruence Theorem, it follows that PSQ LMN. Use the definition of congruent triangles and the AA Similarity Postulate to conclude that RST ~ LMN.

5. USING SIMILARITY THEOREMS Determine if the triangles are similar Compare Side Lengths of LKM and NOP Ratios Different, triangles not similar

6. USING SIMILARITY THEOREMS Determine if the triangles are similar Compare Side Lengths of LKM and NOP Ratios Same, triangles are similar RQS ~ LKM

7. USING SIMILARITY THEOREMS THEOREMS X M P N Z Y XY MN ZX PM If XM and= THEOREM 8.3 Side-Angle-Side (SAS) Similarity Theorem 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. then XYZ ~ MNP.

8. USING SIMILARITY THEOREMS CED 44° 68° 20

9. USING SIMILARITY THEOREMS Statements Reasons

10. USING SIMILARITY THEOREMS Statements Reasons ~

11. Finding Distance Indirectly D B 5 ft E 6.5 ft 85 ft A C Similar triangles can be used to find distances that are difficult to measure directly. ROCK CLIMBING You are at an indoor climbing wall. To estimate the height of the wall, you place a mirror on the floor 85 feet from the base of the wall. Then you walk backward until you can see the top of the wall centered in the mirror. You are 6.5 feet from the mirror and your eyes are 5 feet above the ground. Use similar triangles to estimate the height of the wall. Not drawn to scale

12. Finding Distance Indirectly Due to the reflective property of mirrors, you can reason that ACB  ECD. D B 5 ft E 6.5 ft 85 ft A C Use similar triangles to estimate the height of the wall. SOLUTION Using the fact that ABC and EDCare right triangles, you can apply the AA Similarity Postulate to conclude that these two triangles are similar.

13. Finding Distance Indirectly DE BA EC AC = D DE 5 85 6.5 =  DE 65.38 B 5 ft E 6.5 ft 85 ft A C Use similar triangles to estimate the height of the wall. SOLUTION Ratios of lengths of corresponding sides are equal. So, the height of the wall is about 65 feet. Substitute. Multiply each side by 5 and simplify.

14. Finding Distance Indirectly The Tree is 72 feet tall

15. Finding Distance Indirectly 72 The Tree is 72 feet tall 4 x The mirror would need to be placed 36 feet from the tree

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