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Similarity Lesson 12-3 15. XZ ≈ 3.34 (pp. 731–737) Mental Math 3 a. 12 ft 16. 93 in. 17. a. 200 mi 1 b. _______ 2,534,400 b. 3 18. about 17.14 ft c. 16 Guided Example 1 a. similarity; a size transformation; a reflection b. m∠H c. 1.8; 6; 1.8; 10.8 19. Yes, the rectangles must be similar because corresponding angles are congruent (all angles are right angles) and corresponding lengths are 3 cm 2 cm = ____ proportional _____ . ( 4 ft ) 20. B'C' = 32 21. x = –5 Questions 1. a. The letters are similar because the d is the image of the b under a reflection and size transformation. 22. x - __3 ( 2 ) + ( y - __72 ) 2 2 29 = __ 2 23. 96.28% air, 3.72% leather 24. 8; rectangle, rhombus, isosceles trapezoid, square, parallelogram . b. __3 4 c. __4 3 2. Two figures are similar if and only if there is a composite of size transformations and reflections mapping one onto the other. 3. a. false b. true 4. Yes, the figures are similar because LMNK = S2.3 rm(ABED). 5. GF = 6.8 in.; LK = 15.64 in. 6. HJ = 100 mm; DE = 100 mm 7. ∠ABE and ∠LKN 8. CM = 9.2 9. yes 10. yes 11. No; Figure I and Figure III in Question 4 are similar, but AB ≠ LK. 12. No; Figure I and Figure II in Question 4 are similar, but oppositely oriented. 13. PA = 4.8 14. m∠O = 80; m∠A = 119 A221 6 ft Geometry 25. true 26. There are comparable properties for similarity. Proof of the Reflexive Property of Similarity: Let F be any figure. We need to show that F is similar to itself. To do that, reflect F over the line m, apply size transformation S with center C and magnitude k, then apply size transformation S' with center C and magnitude __1 , and reflect that image k back over m. There is a composite of reflections and size transformations mapping F onto itself. Thus by the definition of similarity, F ∼ F. Proof of the Symmetric Property of Similarity: If F ∼ G then there is a composite of reflections and size transformations that maps F onto G. To map G back onto F, reflect G back over the same lines and apply size transformations with reciprocal magnitudes in reverse order. This means F is the image of G under a composite of reflections and size transformations, so G ∼ F. Proof of the Transitive Property of Similarity: If F ∼ G then there is a composite of reflections and size transformations that maps F onto G. If G ∼ H then there is a composite of reflections and size transformations that maps G onto H. The composite of these composites is another composite of reflections and size transformations which maps F onto H, so F ∼ H. A222 Geometry