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continued Your answer is 9. Your answer does not depend on the number you choose. Your answer will always be 15 if you change Step 3 to “Add 30 to your answer.” You have to add twice the value you want as an answer. Then when you divide your answer in Step 3 by 2 and subtract your original answer, you will always get the value you want as an answer. 3. C; The following statement is the result of inductive reasoning: On Wednesdays, the math teacher can expect about 73 students to attend her classes. 4. F; If a person needs a library card to check out books at the public library, and Kate checked out a book at the public library, then Kate has a library card. 5. 1, 2, 5, 10, 17, 4. Answers will vary. 2.5 Guided Practice (pp. 106–108) 26, . . . 11 13 15 17 19 111 Add 1 to get the second number, then add 3 to get the third number, then add 5 to get the fourth number, then add 7 to get the fifth number, then add 9 to get the sixth number. To find the seventh number, add the next consecutive odd integer, which is 11. So, the next number is 37. 4x 1 9 5 23x 1 2 1. 7x 1 9 5 2 Subtraction Property of Equality 7x 5 27 Simplify. x 5 21 Division Property of Equality 2. 14x 1 3(7 2 x) 5 21 Explore: Step 2 Given 14x 1 21 2 3x 5 21 Sample answer: Distributive Property 11x 1 21 5 21 a. 16 a. 42 a. 63 b. 16 + 2 b. 42 + 2 b. 63 + 2 c. 32 1 4 c. 84 1 4 c. 126 1 4 d. 36 + 5 d. 88 + 5 d. 130 + 5 e. 180 1 12 e. 440 1 12 e. 650 1 12 f. 192 + 10 f. 452 + 10 f. 662 + 10 h. 4200 h. 6300 Simplify. 11x 1 21 2 21 5 21 2 21 11x 5 222 x 5 22 3. Subtraction Property of Equality Simplify. Division Property of Equality 1 A 5 }2 bh 2A 5 bh 2A h }5b g. 1920 2 320 g. 4520 2 320 g. 6620 2 320 h. 1600 Simplify. 7x 1 9 2 9 5 2 2 9 Lesson 2.5 Investigating Geometry Activity 2.5 (p. 104) Given 4x 1 9 1 3x 5 23x 1 2 1 3x Addition Property of Equality 4. Symmetric Property of Equality 5. Transitive Property of Equality Your answer is always the number you picked in part (a). 1. a. x d. (2x 1 4) + 5 2.5 Exercises (pp. 108–111) e. (10x 1 20) 1 12 f. (10x 1 32) + 10 Skill Practice g. (100x 1 320) 2 320 h. 100x 4 100 c. 2x 1 4 2. x is chosen, 2x doubles x, 2x 1 4 is four more than 2x, 5(2x 1 4) is five times the previous number, 5(2x 1 4) 1 12 is 12 more than the previous number, 10[5(2x 1 4) 1 12] multiplies the previous number by 10, 10[5(2x 1 4) 1 12] 2 320 reduces the previous number by 320, crossing out the zeros (dividing by 100) leaves x. 3. Sample answer: 5 5+2 10 1 18 28 4 2 14 2 5 38 6. Reflexive Property of Equality b. 2x Geometry Worked-Out Solution Key 1. Reflexive Property of Equality for angle measure r 2 154 2. To check your answer, substitute a 5 } into the 20.70 original equation. Simply to see if the result is a true statement. 3. 3x 2 12 5 7x 1 8 24x 2 12 5 8 24x 5 20 x 5 25 4. 5(x 2 1) 5 4x 1 13 5x 2 5 5 4x 1 13 x 2 5 5 13 x 5 18 Given Subtraction Property of Equality Addition Property of Equality Division Property of Equality Given Distributive Property Subtraction Property of Equality Addition Property of Equality Copyright © by McDougal Littell, a division of Houghton Mifflin Company. Chapter 2, Chapter 2, continued 5. D; The statement illustrates the Transitive Property of Equality for segment length. 6. 5x 2 10 5 240 Given 5x 5 230 Addition Property of Equality x 5 26 Division Property of Equality 7. 4x 1 9 5 16 2 3x 7x 1 9 5 16 y 5 210x 1 4 18. 3x 1 9y 5 27 x51 7 1 y 5 2}3 x 2 }9 15x 2 100 5 210 Addition Property of Equality Division Property of Equality 9. 3 (2x 1 11) 5 9 6x 1 33 5 9 Given Distributive Property 6x 5 224 x 5 24 (2x 10. 2 2 5) 5 12 22x 2 10 5 12 22x 5 22 x 5 211 1 8 y 5 }3 x 1 }3 Subtraction Property of Equality Division Property of Equality Given 22. If m1 5 m 2, then m2 5 m1. Distributive Property 23. If AB 5 CD, then AB 1 EF 5 CD 1 EF. Addition Property of Equality 24. If 5(x 1 8) 5 2, then 5x 1 40 5 2. Division Property of Equality 25. If m1 5 m2 and m2 5 m3, then m1 5 m3. 36 5 212x 23 5 x 12. 4 (5x 2 9) 5 22 (x 1 7) Given Distributive Property Simplify. Addition Property of Equality Division Property of Equality Given 20x 2 36 5 22x 2 14 Distributive Property 22x 2 36 5 214 Addition Property of Equality 22x 5 22 Addition Property of Equality x51 Division Property of Equality 13. 2x 2 15 2 x 5 21 1 10x Given x 2 15 5 21 1 10x Simplify. 29x 2 15 5 21 29x 5 36 x 5 24 Subtraction Property of Equality Addition Property of Equality Division Property of Equality 14. 3 (7x 2 9) 2 19x 5 215 Given 21x 2 27 2 19x 5 215 Distributive Property 2x 2 27 5 215 Simplify. 2x 5 12 y 5 25x 1 18 Given 2}4 y 5 2}2 x 2 2 2 Division Property of Equality 21. If AB 5 20, then AB 1 CD 5 20 1 CD. 26x 1 36 5 218x 16. 24x 1 2y 5 8 3 Subtraction Property of Equality Division Property of Equality 44 2 6x 2 8 5 218x x56 Given 3 1 20. } x 2 } y 5 22 4 2 Subtraction Property of Equality 11. 44 2 2 (3x 1 4) 5 218x 15. 5x 1 y 5 18 Division Property of Equality y 5 20.25x 1 8 Distributive Property x56 Subtraction Property of Equality 2y 5 20.5x 1 16 Given 15x 5 90 Division Property of Equality 19. 2y 1 0.5x 5 16 Division Property of Equality 8. 5 (3x 2 20) 5 210 Subtraction Property of Equality Given 9y 5 23x 2 7 Given Subtraction Property of Equality Given 23y 5 30x 2 12 Addition Property of Equality 7x 5 7 Copyright © by McDougal Littell, a division of Houghton Mifflin Company. 17. 12 2 3y 5 30x Addition Property of Equality 26. In step 2, the Subtraction Property of Equality should have been used instead of the Addition Property of Equality. 7x 5 x 1 24 6x 5 24 x54 Given Subtraction Property of Equality Division Property of Equality 27. Answers will vary. 28. Equation (Reason) AD 5 AB, DC 5 BC (Given) AC 5 AC (Reflexive Property of Equality) AD 1 DC 5 AB 1 DC (Addition Property of Equality) AD 1 DC 5 AB 1 BC (Substitution) AD 1 DC 1 AC 5 AB 1 BC 1 AC (Addition Property of Equality) 29. Equation (Reason) AD 5 CB, DC 5 BA (Given) AC 5 AC (Reflexive Property of Equality) AD 1 DC 5 CB 1 DC (Addition Property of Equality) AD 1 DC 5 CB 1 BA (Substitution) AD 1 DC 1 AC 5 CB 1 BA 1 AC (Addition Property of Equality) Division Property of Equality Given Subtraction Property of Equality Given 2y 5 4x 1 8 Addition Property of Equality y 5 2x 1 4 Division Property of Equality Geometry Worked-Out Solution Key 39 Chapter 2, continued 30. ZY 1 YX 5 ZX l ZY 5 ZX 2 YX 34. a. YX 1 XW 5 YW l XW 5 YW 2 YX A ZY 5 (5x 1 17) 2 3 5 5x 1 14 B C D b. AB 1 BC 5 AC XW 5 (10 2 2x) 2 3 5 7 2 2x } } Because ZY > XW, ZY 5 XW. BC 1 CD 5 BD Because AC 5 BD, AB 1 BC 5 BC 1 CD, so, AB must equal CD. 5x 1 14 5 7 2 2x 7x 1 14 5 7 c. 7x 5 27 x 5 21 So, ZY 5 5(21) 1 14 5 9 and XW 5 7 2 2(21) 5 9. AC 5 BD Given BC 5 BC Reflexive Property of Equality AC 5 AB 1 BC Segment Addition Postulate BD 5 BC 1 CD Segment Addition Postulate Problem Solving P 5 2* 1 2w Given P 2 2w 5 2* Subtraction Property of Equality P }2w5* 2 AB 1 BC 5 BC 1 CD Substitution Property of Equality AB 5 CD Subtraction Property of Equality Division Property of Equality When P 5 55 and w 5 11: 55 *5} 2 11 5 16.5 2 35. m1 1 m2 1 m3 5 1808 m1 1 m2 1 m1 5 1808 The length is 16.5 meters. 32. 1 A 5 }2 bh m2 5 1808 2 2m1 Given 2A 5 bh 2A b }5h m1 1 m2 5 1488 m2 5 1488 2 m1 Multiplication Property of Equality 1808 2 2m1 5 1488 2 m1 Division Property of Equality When A 5 1768 and b 5 52: 328 5 m1 So, m2 5 1808 2 2(328) 5 1168. 2(1768) h5} 5 68 52 5 C 5 }9 (F 2 32) 36. a. The height is 68 inches. 40 5 Equation Explanation Reason m1 5 m4, mEHF 5 908, mGHF 5 908 Marked in diagram. Given mEHF 5 GHF Substitute mGHF for 908. Substitution Property of Equality mEHF 5 m1 1 m2 mGHF 5 m3 1 m4 Add measures of adjacent angles. Angle Addition Postulate m1 1 m2 5 m3 1 m4 Write expressions equal to the angle measures. Substitution Property of Equality m1 1 m2 5 m3 1 m1 Substitute m1 for m4. Substitution Property of Equality m2 5 m3 Subtract m1 from each side. Subtraction Property of Equality Geometry Worked-Out Solution Key 160 C 5 }9 F 2 } 9 160 5 5 }9 F C1} 9 9 5 1 160 9 2 Mult. Property of Equality 9 5 } C 1 32 5 F b. Distributive Property Addition Property of Equality } C1} 5F Distributive Property Temperature (8C) 0 20 Temperature (8F) 32 68 c. Temperature (8C) 33. Given 32 41 89.6 105.8 y 60 40 20 0 0 20 40 60 80 Temperature (8F) 100 x This is a linear function. 37. The relationship is symmetric. “Yen worked the same hours as Jim” is the same as “Jim worked the same hours as Yen.” Copyright © by McDougal Littell, a division of Houghton Mifflin Company. 31. Chapter 2, continued 2.6 Exercises (pp. 116–119) 38. The relationship is transitive. One example of an explanation is that 24 is less than 22 and 22 is less than 21, so 24 is less than 21. Skill Practice 1. A theorem is a statement that can be proven. A postulate Mixed Review for TAKS is a rule that is accepted without proof. 39. A; 2. Sample answer: You can use postulates such as the Angle Addition Postulate. You can use properties such as the Reflexive Property of Equality. You can also use definitions such as the definition of a right angle. Cost per Number Cost per Number Total + a block of + of blocks 1 apple of apples cost cheese of cheese 4.5 + 1 c 0.5 + a a 26 3. Quiz 2.4–2.5 (p. 111) 1. True 2. False 4. x 1 20 5 35 3. True Given x 5 15 Subtraction Property of Equality 5. 5x 2 14 5 16 1 3x 2x 2 14 5 16 Statements Reasons 1. AB 5 5, BC 5 6 1. Given Given 2. AC 5 AB 1 BC 2. Segment Addition Postulate 3. AC 5 5 1 6 3. Substitution Property of Equality 4. AC 5 11 4. Simplify. Subtraction Property of Equality 2x 5 30 Addition Property of Equality 4. A; Transitive Property of Equality x 5 15 Division Property of Equality 5. SE > SE } } 6. If AB 5 CD, then AB 2 EF 5 CD 2 EF. 6. If JKL > RST, then RST > JKL. 7. If a 5 b and b 5 c, then a 5 c. 7. If F > J and J > L, then F > L. 8. Symmetric Property of Congruence Lesson 2.6 9. Reflexive Property of Congruence 2.6 Guided Practice (pp. 112–115) Copyright © by McDougal Littell, a division of Houghton Mifflin Company. 1. Statements Reasons 1. AC 5 AB 1 AB 1. Given 2. AB 1 BC 5 AC 2. Segment Addition Postulate 10. Transitive Property of Congruence 11. Reflexive Property of Equality 12. C; Symmetric Property of Congruence } Property of Segment Congruence, not the Reflexive Property of Segment Congruence. } } } } } } Because MN > LQ and LQ > PN, then MN > PN by the Transitive Property of Congruence. 3. AB 1 AB 5 AB 1 BC 3. Substitution Property 4. AB 5 BC 4. Subtraction Property of Equality } 13. The reason that MN > PN should be the Transitive 14. Six planes that intersect at right angles: 2. Reflexive Property of Segment Congruence 3. Symmetric Property of Angle Congruence 4. Step 5 would be MB 1 MB 5 AB. Step 6 would be 2 MB 5 AB. 15. Shops along the boardwalk: Cottage Snack Bike Arcade Shop Rentals 1 Step 7 would be MB 5 }2 AB. 5. It does not matter what the actual distances are in order 6. 16. Statements Kite Shop Reasons to prove the relationship between AB and CD. What matters are the positions of the stores relative to each other. 1. RT 5 5, RS 5 5, 1. Given } } RT > TS A 2. RS 5 RT 2. Transitive Property of Equality 3. RT 5 TS 3. Definition of congruent segments 4. RS 5 TS } } 5. RS > TS 4. Transitive Property of Equality B E C D From Example 4, you know that AB 5 CD. You also } } know that BE > EC by the definition of a midpoint. By the definition of congruent segments, BE 5 EC. So, AB 1 BE 5 EC 1 CD, or AE 5 ED. The food court and the bookstore are also the same distance from the clothing store. 5. Definition of congruent segments Geometry Worked-Out Solution Key 41