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Chapter 2, continued 3. A; 6. If-then: If an angle measures 348, then it is an acute angle. You know that 1 and 2 are a pair of vertical angles because m1 5 m2, while m3 5 3m1. m3 5 m4 because 3 and 4 are the other pair of vertical angles. 1 and 3 are linear pair, so m1 1 m3 5 1808. Let m1 5 x, then m3 5 3x and x 1 3x 5 180, or x 5 45. So, m1 5 458, m2 5 458, m3 5 3(458) 5 1358, and m4 5 1358. 4. J; Converse: If an angle is an acute angle then it measures 348. Inverse: If an angle does not measure 348, then it is not an acute angle. Contrapositive: If an angle is not an acute angle, then it does not measure 348. 7. This is a valid definition because it can be written as a T 5 c(1 1 s) true biconditional statement. T 5 c 1 cs 8. All the interior angles of a polygon are congruent if and T 2 c 5 cs only if the polygon is an equiangular polygon. T2c }5s c 9. Because B is a right angle it satisfies the hypothesis, so c T }2}5s c c 10. The conclusion of the second statement is the hypothesis the conclusion is also true. So, B measures 908. of the first statement, so you can write the following new statement if 4x 5 12, then 2x 5 6. T }215s c 11. Look for a pattern: 5. C; 1 1 3 5 4, 5 1 7 5 12, 9 1 3 5 12 Conjecture: Odd integer 1 odd integer 5 even integer Let 2n 1 1 and 2m 1 1 be any two odd integers (2n 1 1) 1 (2m 1 1) 5 2n 1 2m 1 2 5 2(n 1 m 1 1) 2(n 1 m 1 1) is the product of 2 and an integer (n 1 m 1 1). So, 2(m 1 n 1 1) is an even integer. The sum of any two odd integers is an even integer. 12. K 6. m1 1 m2 1 m3 1 m4 5 3608 m1 1 m1 1 808 1 808 5 3608 2 + m1 5 2008 m1 5 1008 Chapter 2 Review (pp. 134–137) 1. A statement that can be proven is called a theorem. 2. The inverse negates the hypothesis and conclusion of a conditional statement. The converse exchanges the hypothesis and conclusion of a conditional statement. 3. When m A 5 m B and m B 5 mC, then m A 5 mB. 4. 220,480, 25120, 21280, 2320, 44 44 44 ... 44 Each number in the pattern is the previous number divided by 4. The next three numbers are 280, 220, 25. 224 5. Counterexample: } 5 3 28 Because a counterexample exists, the conjecture is false. C D E 13. B; With no right angle marked, you cannot assume } CD > plane P. 14. 29x 2 21 5 220x 2 87 Addition Property of Equality 11x 5 266 Addition Property of Equality x 5 26 Division Property of Equality 15. 15x 1 22 5 7x 1 62 Geometry Worked-Out Solution Key Given 8x 1 22 5 62 Subtraction Property of Equality 8x 5 40 Subtraction Property of Equality x55 16. 3(2x 1 9) 5 30 6x 1 27 5 30 6x 5 3 1 x 5 }2 50 Given 11x 2 21 5 287 Division Property of Equality Given Distributive Property Subtraction Property of Equality Division Property of Equality Copyright © by McDougal Littell, a division of Houghton Mifflin Company. AB and ###$ AF BAC and CAF are a linear pair because ###$ are opposite rays. mBAC 1 mCAF 5 1808 by the Linear Pair Postulate. mCAF 5 mCAE 1 mEAF and mCAE 5 mCAD 1 mDAE by the Angle Addition Postulate. mCAD 1 mDAE 5 908 because CAD and DAE are complements. So, mCAE 5 908, and mBAC 1 mCAF 5 mBAC 1 (mCAE 1 mEAF) 5 mBAC 1 (908 1 mEAF) 5 (mBAC 1 mEAF) 1 9085 1808. So, mBAC 1 mEAF 5 908. Chapter 2, continued 17. 5x 1 2(2x 2 23) 5 2154 3. 26, 21, Given 5x 1 4x 2 46 5 2154 Distributive Property 9x 2 46 5 2154 Simplify. 9x 5 2108 Addition Property of Equality x 5 212 Division Property of Equality ... 15 15 15 15 Each number in the pattern is five more than the previous number. The next number is 14. 4. 100, 250, 1 25, 1 212.5, 1 ... 1 3 2}2 3 2}2 3 2}2 3 2}2 1 19. Reflexive Property of Congruence number. The next number is 6.25. 20. Transitive Property of Equality 5. If-then form: If two angles are right angles, then they 21. Given: 1 > 2 and 2 > 3 are congruent. Prove: 1 > 3 Converse: If two angles are congruent, then they are right angles. Statements Reasons 1. Given 2. m1 5 m2 2. Definition of congruent angles m2 5 m3 Inverse: If two angles are not right angles, then they are not congruent. Contrapositive: If two angles are not congruent, then they are not right angles. 6. If-then form: If an animal is a frog, then it is 3. m1 5 m3 3. Transitive Property of Equality an amphibian. 4. 1 > 3 Converse: If an animal is an amphibian, then it is a frog. 4. Definition of congruent angles 22. Given: m1 5 1148 m3 5 m1 5 1148 23. Given: m4 5 578 m2 5 m4 5 578 m1 1 m2 5 1808 m4 1 m3 5 1808 1148 1 m2 5 1808 578 1 m3 5 1808 m2 5 668 m4 5 m2 5 668 Copyright © by McDougal Littell, a division of Houghton Mifflin Company. 9, Each number in the pattern is 2}2 times the previous 18. Symmetric Property of Congruence 1. 1 > 2 2 > 3 4, m3 5 1238 m1 5 m3 5 1238 Inverse: If an animal is not a frog, then it is not an amphibian. Contrapositive: If an animal is not an amphibian, then it is not a frog. 7. If-then form: If x 5 22, then 5x 1 4 5 26. Converse: If 5x 1 4 5 26, then x 5 22. Inverse: If x Þ 22, then 5x 1 4 Þ26. Contrapositive: If 5x 1 4 Þ 26, then x Þ 22. 9. Because you are going to the football game satisfies the 24. Statements Reasons 1. 12 and 11 are complementary. m10 1 m11 5 908 1. Given 2. 10 and 11 are complementary. 2. Definition of complementary angles 3. 12 > 10 3. Congruent Complements Theorem hypothesis, the conclusion is true. So, you will miss band practice. 10. The conclusion of the first statement is the hypothesis of the second statement, so you write the following new statement. Chapter 2 Test (p. 128) 1. The figure is rotated counterclockwise. The next figure is: If Margot goes to college, then she will need to buy a lab manual. 11. Sample answers: Line @##$ TQ contains points N, Q, and T. 12. Sample answer: Plane Y contains points Q, R, and S. 13. Planes X and Y intersect at @##$ TQ. 14. 9x 1 31 5 223 9x 5 254 x 5 26 15. 27(2x 1 2) 5 42 2. Two pieces are added to the figure alternating between unshaded and shaded pattern. The next figure is: 7x 2 14 5 42 Given Subtraction Property of Equality Division Property of Equality Given Distributive Property 7x 5 56 Addition Property of Equality x58 Division Property of Equality Geometry Worked-Out Solution Key 51 Chapter 2, continued 16. 26 1 2(3x 1 11) 5 218x 26 1 6x 1 22 5 218x Given Distributive Property 6x 1 48 5 218x Simplify. 48 5 224x 22 5 x Subtraction Property of Equality Division Property of Equality 17. B; Symmetric Property of Congruence 18. A; Reflexive Property of Congruence 19. C; Transitive Property of Congruence 20. 7y 5 5y 1 36 } } } } } } 11. 2Ï 180 5 2Ï 36 + Ï 5 5 26Ï 5 12. 6Ï 128 5 6Ï 64 + Ï 2 5 68Ï 2 } } } } } } } } 13. Ï 2 2 Ï 18 1 Ï 6 5 Ï 2 2 3Ï 2 1 Ï 6 5 Ï 6 2 2Ï 2 } } } } } } 14. Ï 28 2 Ï 63 2 Ï 35 5 2Ï 7 2 3Ï 7 2 Ï 35 } } } 5 2Ï7 2 Ï 35 } } } } 15. 4Ï 8 1 3Ï 32 5 8Ï 2 1 12Ï 2 5 20Ï 2 } } } 16. (6Ï 5 )(2Ï 2 ) 5 12Ï 10 } } } } } } The measures of the angles are: 7(18) 5 1268 5(18) 1 36 5 1268 3(25) 2 21 5 548 Statements } Reasons 1. AX > DX, } } XB > XC 1. Given 2. AX 5 DX, XB 5 XC 2. Definition of congruent segments 3. AX 1 XC 5 AC 3. Segment Addition Postulate DX 1 XB 5 BD 4. DX 1 XC 5 AC 4. Substitution Property of DX 1 XC 5 BD Equality 5. AC 5 BD } 6. AC > BD 5. Transitive Property of Equality 6. Definition of congruent segments Chapter 2 Algebra Review (p. 139) 2 5+x+x+x+x x 5x4 1. }2 5 }} 5 } 5+4+x+x 4 20x 24 + 3 + a + b + b + b 24b2 212ab3 } 2. } 5 }} 5 2 3a 3+3+a+a+b 9a b 5(m 1 7) 5m 1 35 3. } 5 } 5 m 1 7 5 5 212m 36m 2 48m 4. } 5 } 5 22 6m 6m k13 5. } ; cannot be simplified 22k 1 3 m14 1 m14 6. } 5} 5} m m(m 1 4) m2 1 4m 2(6x 1 8) 12x 1 16 7. } 5 } 5 2 8 1 6x 8 1 6x 3+x+x+x 3x2 3x3 } 8. } 5} 5 2 5 1 8x x(5 1 8x) 5x 1 8x Geometry Worked-Out Solution Key 19. Ï(25)2 5 25 20. Ïx2 5 x 21. Ï(2a) 2 5 Ïa2 5 a } } Ï(3y) 2 5 Ï9y 2 5 3y 22. 2(25) 1 4 5 548 } } } } } } } 23. Ï32 1 22 5 Ï9 1 4 5 Ï13 24. Ïh2 1 k 2 ; cannot be simplified } TAKS Practice (pp. 142–143) 1. C; The student section has 40 rows, with 36 seats in each row, and student admission is $4 per person. Because student tickets sell out, to calculate the total amount of student ticket sales, multiply 40 by 36 and then multiply that product by $4. 2. J; n2 1 n represents the pattern shown in the table. 3. C; The other information that is needed to determine the amount of discount per person is the number of students on the trip. 4. J; The charge for labor was $38 per hour, so the charge for h hours of labor was 38h. This number plus the charge for parts, $126, gives the total charge to repair Andy’s car, $278. So, the equation 38h 1 126 5 278 can be used to find the number of hours it took to repair his car. 5. B; A scatter plot would be most helpful in determining whether there is a correlation between the amount of TV watched and the number of extracurricular activities. 6. F; d 5 rt r 5 3 mi/h 2 t 5 40 min 5 }3 h d 5 31 }3 2 5 2 2 Jane can walk 2 miles in 40 minutes. Copyright © by McDougal Littell, a division of Houghton Mifflin Company. x 5 25 52 } } 18. (2Ï 6 ) 2 5 (2) 2 + (Ï 6 ) 2 5 4 + 6 5 24 3x 2 21 5 2x 1 4 } } } } y 5 18 } } } 10. Ï 75 5 Ï 25 + Ï 3 5 5Ï 3 17. (24Ï 10 )(25Ï 5 ) 5 20Ï 50 5 20 + 5Ï 2 5 100Ï 2 2y 5 36 21. 3x(x 2 2) 3x 2 2 6x x22 9. } 5} 5} 3x(2x 2 1) 2x 2 1 6x 2 2 3x Chapter 2, continued 7. C; The maximum value of the parabola shown is at (1, 3). 8. G; Number Number Total Price Price of desk of wall dollar per wall 5 calendars + per desk 1 calendars + amount calendar calendar sold sold sold 9 + d 1 3 + w 5 70.5 4 + d 1 6 + w 5 71 9. C; Cost per Number of Cost per + 1 notebook notebooks pencil + 2.5 1 n 0.5 Number Maximum + of a total cost pencils + p a 10 10. F; (23, 2), (0, 8) 822 6 5 }3 5 2 m5} 0 2 (23) Because the line passes through the point (0, 8), the y-intercept, b, is 8. So, the equation y 5 2x 1 8 represents the line that passes through the points (23, 2) and (0, 8). 11. A; 24(x 1 5) 1 2(2x 1 1) 5 24x 2 20 1 4x 1 2 5 218 12. J; (3, 0), (0, 1) 120 1 Copyright © by McDougal Littell, a division of Houghton Mifflin Company. 5 2}3 m5} 023 1 The y-intercept, b, is 1. So, the equation y 5 2}3 x 1 1 best represents the graph. 13. Of the 750 students in Hannah’s school, 72%, or 0.72(750) 5 540 students have at least one pet. Of these 540 students with pets, 30%, or 0.3(540) 5 162 students own dogs and 5%, or 0.05(540) 5 27 students own two or more dogs. So, 162 2 27 5 135 students own exactly one dog. Geometry Worked-Out Solution Key 53