State the assumption you would make to start an indirect proof of each statement.

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1 1. State the assumption you would make to start an indirect proof of each statement. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. 2. is a scalene triangle. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. is an isosceles or equilateral triangle. 3. If 4x < 24, then x < 6. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. 4. A is not a right angle. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. A is a right angle. esolutions Manual - Powered by Cognero Page 1

2 Write an indirect proof of each statement. 5. If 2x + 3 < 7, then x < 2. For this problem, assume that x > 2 or x = 2 is true. Given: 2x + 3 < 7 Prove: x < 2 Indirect Proof: Step 1 Assume that x > 2 or x = 2 is true. Step 2 When x > 2, 2x + 3 > 7 and when x = 2, 2x + 3 = 7. Step 3 In both cases, the assumption leads to the contradiction of the given information that 2x + 3 < 7. Therefore, the assumption that must be false, so the original conclusion that x < 2 must be true. 6. If 3x 4 > 8, then x > 4. For this problem, assume that x < 4 or x = 4 is true. Given: 3x 4 > 8 Prove: x > 4 Indirect Proof: Step 1 Assume that x < 4 or x = 4 is true. Step 2 When x < 4, 3x 4 < 8 and when x = 4, 3x 4 = 8. Step 3 In both cases, the assumption leads to the contradiction of the given information that 3x 4 > 8. Therefore, the assumption that must be false, so the original conclusion that x > 4 must be true. esolutions Manual - Powered by Cognero Page 2

3 7. LACROSSE Christina scored 13 points for her high school lacrosse team during the last six games. Prove that her average points per game was less than 3. For this problem, assume that Christina's average points per game was greater than or equal to 3. Use. Indirect Proof: Step 1 Assume that Christina s average points per game was greater than or equal to 3,. Step 2 Step 3 The conclusions are false, so the assumption must be false. Therefore, Christina s average point per game was less than Write an indirect proof to show that if 5x 2 is an odd integer, then x is an odd integer. For this problem, assume that x is an even integer. Given: 5x 2 is an odd integer. Prove: x is an odd integer. Indirect Proof: Step 1 Assume that x is not an odd integer. That is, assume that x is an even integer. Step 2 Let x = 2k for some integer k. Since k is an integer, 5k 1 is also an integer. Let p represent the integer 5k 1. So 5x 2 can be represented by 2p, where p is an integer. This means that 5x 2 is an even integer, but this contradicts the given that 5x 2 is an odd integer. Step 3 Since the assumption that x is an even integer leads to a contradiction of the given, the original conclusion that x is an odd integer must be true. esolutions Manual - Powered by Cognero Page 3

4 Write an indirect proof of each statement. 9. The hypotenuse of a right triangle is the longest side. For this problem, assume that the hypotenuse of a right triangle is not the longest side. Given: is a right triangle; C is a right angle. Prove: AB > BC and AB > AC Indirect Proof: Step 1 Assume that the hypotenuse of a right triangle is not the longest side. That is, AB < BC and AB < AC. Step 2 If AB < BC, then m C < m A. Since m C = 90, m A > 90. So, m C + m A > 180. By the same reasoning, m C + m B > 180. Step 3 Both relationships contradict the fact that the sum of the measures of the angles of a triangle equals 180. Therefore, the hypotenuse must be the longest side of a right triangle. 10. If two angles are supplementary, then they both cannot be obtuse angles. For this problem, assume that A and B are both obtuse angles. Given: A and B are supplementary. Prove: A and B cannot both be obtuse angles. Indirect Proof: Step 1 Assume that A and B are both obtuse angles. Step 2 By the definition of obtuse angles, m A > 90 and m B > 90. So, m A + m B > 180. Step 3 This contradicts the given information, m A + m B = 180. Therefore, the original conclusion that A and B cannot both be obtuse angles must be true. State the assumption you would make to start an indirect proof of each statement. 11. If 2x > 16, then x > 8. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. esolutions Manual - Powered by Cognero Page 4

5 12. 1 and 2 are not supplementary angles. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. 1 and 2 are supplementary angles. 13. If two lines have the same slope, the lines are parallel. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. The lines are not parallel. 14. If the consecutive interior angles formed by two lines and a transversal are supplementary, the lines are parallel. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. The lines are not parallel. 15. If a triangle is not equilateral, the triangle is not equiangular. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. The triangle is equiangular. 16. An odd number is not divisible by 2. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. An odd number is divisible by 2. esolutions Manual - Powered by Cognero Page 5

6 Write an indirect proof of each statement. 17. If 2x 7 > 11, then x > 2. For this problem, assume that is true. Given: 2x 7 > 11 Prove: x > 2 Indirect Proof: Step 1 Assume that is true. Step 2 When x < 2, 2x 7 < 11 and when x = 2, 2x 7 = 11. Step 3 In both cases, the assumption leads to the contradiction of the given information that 2x 7 > 11. Therefore, the assumption that must be false, so the original conclusion that x > 2 must be true. 18. If 5x + 12 < 33, then x < 9. For this problem, assume that is true. Given: 5x + 12 < 33 Prove: x < 9 Indirect Proof: Step 1 Assume that is true. Step 2 When x > 9, 5x + 12 > 33 and when x = 9, 5x + 12 = 33. Step 3 In both cases, the assumption leads to the contradiction of the given information that 5x + 12 < 33. Therefore, the assumption that must be false, so the original conclusion that x < 9 must be true. esolutions Manual - Powered by Cognero Page 6

7 19. If 3x + 4 < 7, then x > 1. For this problem, assume that is true. Given: 3x + 4 < 7 Prove: x > 1 Indirect Proof: Step 1 Assume that is true. Step 2 When x < 1, 3x + 4 > 7 and when x = 1, 3x + 4 = 7. Step 3 In both cases, the assumption leads to the contradiction of the given information that 3x + 4 < 7. Therefore, the assumption that must be false, so the original conclusion that x > 1 must be true. 20. If 2x 6 > 12, then x < 9. For this problem, assume that is true. Given: 2x 6 > 12 Prove: x < 9 Indirect Proof: Step 1 Assume that is true. Step 2 When x > 9, 2x 6 < 12 and when x = 9, 2x 6 = 12. Step 3 In both cases, the assumption leads to the contradiction of the given information that 2x 6 > 12. Therefore, the assumption that must be false, so the original conclusion that x < 9 must be true. esolutions Manual - Powered by Cognero Page 7

8 21. COMPUTER GAMES Kwan-Yong bought two computer games for just over $80 before tax. A few weeks later, his friend asked how much each game cost. Kwan-Yong could not remember the individual prices. Use indirect reasoning to show that at least one of the games cost more than $40. For this problem, assume that and is true. Let the cost of one game be x and the other be y. Step 1 Given: x + y > 80 Prove: x > 40 or y > 40 Indirect Proof: Assume that and. Step 2 If and, then or. This is a contradiction because we know that. Step 3 Since the assumption that and leads to a contradiction of a known fact, the assumption must be false. Therefore, the conclusion that x > 40 or y > 40 must be true. Thus, at least one of the games had to cost more than $ FUNDRAISING Jamila s school is having a Fall Carnival to raise money for a local charity. The cost of an adult ticket to the carnival is $6 and the cost of a child s ticket is $2.50. If 375 total tickets were sold and the profit was more than $1460, prove that at least 150 adult tickets were sold. For this problem, assume that that fewer than 150 adult tickets were sold. Step 1 Assume that fewer than 150 adult tickets were sold. Step 2 If 149 adult tickets were sold, , or 226 child tickets were sold. The total profit for 149 adult tickets and 226 child tickets is (149)($6) + (226)($2.50) or $1459. Step 3 The conclusion is false, so the assumption must be false. Therefore, the number of adult tickets sold is greater than or equal to 150. esolutions Manual - Powered by Cognero Page 8

9 Write an indirect proof of each statement. 23. Given: xy is an odd integer. Prove: x and y are both odd integers. For this problem, assume that x or y are even integers. Given: xy is an odd integer. Prove: x and y are odd integers. Indirect Proof: Step 1 Assume that x and y are not both odd integers. That is, assume that either x or y is an even integer. Step 2 You only need to show that the assumption that x is an even integer leads to a contradiction, since the argument for y is an even integer follows the same reasoning. So, assume that x is an even integer and y is an odd integer. This means that x = 2k for some integer k and y = 2m + 1 for some integer m. Since k and m are integers, km + k is also an integer. Let p represent the integer km + k. So xy can be represented by 2p, where p is an integer. This means that xy is an even integer, but this contradicts the given that xy is an odd integer. Step 3 Since the assumption that x is an even integer and y is an odd integer leads to a contradiction of the given, the original conclusion that x and y are both odd integers must be true. 24. Given: n 2 is even. Prove: n 2 is divisible by 4. For this problem, assume that n is not divisible by 4. Given: n 2 is even. Prove: n 2 is divisible by 4. Indirect Proof: Step 1 Assume n 2 is not divisible by 4. In other words, 4 is not a factor of n 2. Step 2 If the square of a number is even, then the number is also even. So, if n 2 is even, n must be even. Let n = 2a. n = 2a n 2 = (2a) 2 or 4a 2 4 is a factor of n 2, which contradicts the assumption. Step 3 Since the assumption that n 2 is not divisible by 4 leads to a contradiction of the assumption, the original conclusion that n 2 is divisible by 4 must be true. esolutions Manual - Powered by Cognero Page 9

10 25. Given: x is an odd number. Prove: x is not divisible by 4. For this problem, assume that x is divisible by 4. Given: x is an odd number. Prove: x is not divisible by 4. Indirect Proof: Step 1 Assume x is divisible by 4. In other words, 4 is a factor of x. Step 2 Let x = 4n, for some integer n. x = 2(2n) So, 2 is a factor of x which means x is an even number, but this contradicts the given information. Step 3 Since the assumption that x is divisible by 4 leads to a contradiction of the given, the original conclusion x is not divisible by 4 must be true. 26. Given: xy is an even integer. Prove: x or y is an even integer. For this problem, assume that both x and y are odd integers. Given: xy is an even integer. Prove: x or y is an even integer. Indirect Proof: Step 1 Assume x and y are odd integers. Step 2 Let x = 2n + 1 and y = 2k +1, for some integer n and k. Since k and n are integers, 2nk + n + k is also an integer. Let p represent the integer 2nk + n + k. So xy can be represented by 2p + 1, where p is an integer. This means that xy is an odd integer, but this contradicts the given that xy is an even integer. Step 3 Since the assumption that x and y are odd integers leads to a contradiction of the given, the original conclusion x or y is an even integer must be true. esolutions Manual - Powered by Cognero Page 10

11 27. Given: XZ > YZ Prove: For this problem, assume that is true. Given: XZ > YZ Prove: is not congruent to Indirect Proof: Step 1 Assume that. Step 2 by the converse of the Isosceles Triangle Theorem. Step 3 This contradicts the given information that XZ > YZ. Therefore, the assumption original conclusion is not congruent to must be true. must be false, so the esolutions Manual - Powered by Cognero Page 11

12 28. Given: is equilateral. Prove: is equiangular. For this problem, assume that is not equiangular. Given: Prove: is equilateral. is equiangular. Indirect Proof: Step 1 Assume that is not equiangular. Step 2 Then m B > m C. Then by Angle Side Relationships in Triangles Theorem. Step 3 This contradicts the given information that is equilateral. Therefore, the assumption that is not equiangular must be false, so the original conclusion is equiangular must be true. esolutions Manual - Powered by Cognero Page 12

13 29. In an isosceles triangle neither of the base angles can be a right angle. For this problem, assume that is a right angle. Given: is isosceles. Prove: Neither of the base angles is a right angle. Indirect Proof: Step 1 Assume that B is a right angle. Step 2 By the Isosceles Triangle Theorem, C is also a right angle. Step 3 This contradicts the fact that a triangle can have no more than one right angle. Therefore, the assumption that B is a right angle must be false, so the original conclusion neither of the base angles is a right angle must be true. 30. A triangle can have only one right angle. For this problem, assume that has more than one right angle. Given: Prove: has no more than one right angle. Indirect Proof: Step 1 Assume that has more than one right angle. Step 2 If B and C are right angles, then m B + m C = 180. The m A + m B + m C = 180 because the angles of a triangle add to 180. By substitution, m A = 180, so m A = 0. Step 3 This contradicts the given information,. Therefore, the assumption that has more than one right angle must be false, so the original conclusion, has no more than one right angle, must be true. esolutions Manual - Powered by Cognero Page 13

14 31. Write an indirect proof for Theorem For this problem, assume that. Given: m A > m ABC Prove: BC > AC Proof: Assume. By the Comparison Property, BC = AC or BC < AC. Case 1: If BC = AC, then by the Isosceles Triangle Theorem. (If two sides of a triangle are congruent, then the angles opposite those sides are congruent.) But, contradicts the given statement that m A > m ABC. So,. Case 2: If BC < AC, then there must be a point D between A and C so that. Draw the auxiliary segment. Since DC = BC, by the Isosceles Triangle Theorem. Now BDC is an exterior angle of and by the Exterior Angles Inequality Theorem (the measure of an exterior angle of a triangle is greater than the measure of either corresponding remote interior angle) m BDC > m A. By the Angle Addition Postulate, m ABC = m ABD + m DBC. Then by the definition of inequality, m ABC > m DBC. By Substitution and the Transitive Property of Inequality, m ABC > m A. But this contradicts the given statement that m A > m ABC. In both cases, a contradiction was found, and hence our assumption must have been false. Therefore, BC > AC. esolutions Manual - Powered by Cognero Page 14

15 32. Write an indirect proof to show that if, then b is negative. For this problem, assume that b is positive or, in other words, b > 0. Given: Prove: b is negative. Indirect Proof: Step 1 Assume that b > 0. since that would make undefined. Step 2 b > 0. ; A positive number divided by a positive number is positive. Step 3 contradicts the given, so the assumption must be false. Thus, b must be negative. 33. BASKETBALL In basketball, there are three possible ways to score three points in a single possession. A player can make a basket from behind the three-point line, a player may be fouled while scoring a two-point shot and be allowed to shoot one free throw, or a player may be fouled behind the three-point line and be allowed to shoot three free throws.when Katsu left to get in the concession line before the end of the first half of a basketball game, her school s team was winning 28 to 26. When she returned during halftime, the opposing team s score was 29 and Katsu s team s score was 28. Katsu concluded that a player on the visiting team had made a three-point basket. Prove or disprove her assumption using an indirect proof. showing this assumption to be logically impossible or that there is an alternative conclusion, you prove your assumption false and the original conclusion true. For this problem, assume that a player for the other team earned three points with the combination of a two-point basket and a foul shot.. We know that the other team scored 3 points, and Katsu thinks that they made a three-point shot. We also know that a player can score 3 points by making a basket and a foul shot. Step 1 Assume that a player for the other team made a two-point basket and a foul shot. Step 2 The other team s score before Katsu left was 26, so their score after a two-point basket and a foul shot would be or 29. Step 3 The score is correct when we assume that the other team made a two-point basket and a foul shot, so Katsu s assumption may not be correct. The other team could have made a three point basket or a two-point basket and a foul shot. esolutions Manual - Powered by Cognero Page 15

16 34. GAMES A computer game involves a knight on a quest for treasure. At the end of the journey, the knight approaches the two doors shown below. A servant tells the knight that one of the signs is true and the other is false. Use indirect reasoning to determine which door the knight should choose. Explain your reasoning. showing this assumption to be logically impossible or that there is an alternative conclusion, you prove your assumption false and the original conclusion true. For this problem, assume that each sign is true one and a time. If the sign on the right is true, then the sign on the left must be false. However, it doesn't contradict the sign on the right so it must also be true. This contradicts the given information that one sign is false. If the sign on the left is true, then the sign on the right must be false. This doesn't contradict any information on the signs so this must be thr true case. The door on the left. If the sign on the door on the right were true, then both signs would be true. But one sign is false, so the sign on the door on the right must be false. esolutions Manual - Powered by Cognero Page 16

17 35. SURVEYS Luisa s local library conducted an online poll of teens to find out what activities teens participate in to preserve the environment. The results of the poll are shown in the graph below. a. Prove the following statement. More than half of teens polled said that they recycle to preserve the environment. b. If 400 teenagers were polled, verify that 92 of them said that they participate in Earth Day. showing this assumption to be logically impossible or that there is an alternative conclusions, you prove your assumption false and the original conclusion true. For this problem, assume that less than 50% recycle. a. Indirect Proof: Step 1 50% is half, and the statement says more than half of the teens polled said that they recycle, so assume that less than 50% recycle. Step 2 The data shows that 51% of teens said that they recycle, and 51% > 50%, so the number of teens that recycle is not less than half. Step 3 This contradicts the data given. Therefore, the assumption is false, and the conclusion more than half of the teens polled said they recycle must be true. b. esolutions Manual - Powered by Cognero Page 17

18 36. CARS James, Hector, and Mandy all have different color cars. Only one of the statements below is true. Use indirect reasoning to determine which statement is true. Explain. (1) James has a red car. (2) Hector does not have a red car. (3) Mandy does not have a blue car. showing this assumption to be logically impossible or that there is an alternative conclusions, you prove your assumption false and the original conclusion true. For this problem, assume that each statement is not true and look for a contradiction to your assumption. Statement 2; sample answer: If you assume that statement 2 is not true, or Hector has a red car, it contradicts statement 1 James has a red car, so statement 2 is true. If you assume that statements 1 and 3 are not true, neither of the other remaining statements contradicts the assumption. esolutions Manual - Powered by Cognero Page 18

19 Determine whether each statement about the shortest distance between a point and a line or plane can be proved using a direct or indirect proof. Then write a proof of each statement. 37. Given: line p Prove: is the shortest segment from A to line p. showing this assumption to be logically impossible or that there is an alternative conclusions, you prove your assumption false and the original conclusion true. For this problem, assume that A to p and prove it false. Indirect proof Given: Prove: is the shortest segment from A to p. is not the shortest segment from Indirect Proof: Step 1 Assume is not the shortest segment from A to p. Step 2 Since is not the shortest segment from A to p, there is a point C such that is the shortest distance. is a right triangle with hypotenuse, the longest side of since it is across from the largest angle in by the Angle Side Relationships in Triangles Theorem. Step 3 This contradicts the fact that is the shortest side, must be true. is the shortest side. Therefore, the assumption is false, and the conclusion, esolutions Manual - Powered by Cognero Page 19

20 38. Given: plane M Prove: is the shortest segment from P to plane M. A direct proof is more appropriate for this proof, as the conclusion is directly related to the given information. Given: plane M Prove: is the shortest segment from P to plane M. Proof: By definition, is perpendicular to plane M if it is perpendicular to every line in M that intersects it. But since the perpendicular segment from a point to a line is the shortest segment from the point to the line, that perpendicular segment is the shortest segment from the point to each of these lines. Therefore, segment from P to M. is the shortest 39. NUMBER THEORY In this problem, you will make and prove a conjecture about a number theory relationship. a. Write an expression for the sum of the cube of a number and three. b. Create a table that includes the value of the expression for 10 different values of n. Include both odd and even values of n. c. Write a conjecture about n when the value of the expression is even. d. Write an indirect proof of your conjecture. When trying to indirectly prove that is even, you must assume this statement is false and look for a contradiction. In order to prove an expression is even, you let n = 2k, where k some integer and substitute it in for n in your expression. If you can simplify your expression to a form of 2k, where k is any expression of k, then your expression is even. If it simplifies to some form of 2k+1, then your expression is odd. a. n b. Sample answer: esolutions Manual - Powered by Cognero Page 20

21 . Sample answer: c. Sample answer: When n is even, n is odd. d. Indirect Proof: Step 1 Assume that n is even. Let n = 2k, where k some integer. Step 2 Since k is an integer, 4k is also an integer. Therefore, n is odd. Step 3 This contradicts the given information that n is even. Therefore, the assumption is false, so the conclusion that n is odd must be true. 40. WRITING IN MATH Explain the procedure for writing an indirect proof. Sample answer: First identify the statement you need to prove and assume temporarily that this statement is false by assuming that the opposite of the statement is true. Next, reason logically until you reach a contradiction. Finally, point out that the statement you wanted to prove must be true because the contradiction proves that the temporary assumption you made was false. esolutions Manual - Powered by Cognero Page 21

22 41. OPEN ENDED Write a statement that can be disproved using indirect proof. Include the indirect proof of your statement. For this problem, you might consider choosing something like a definition or a theorem that can be easily proven true by contradiction. Sample answer: is scalene. Given: ; ; ; Prove: is scalene. Indirect Proof: Step 1 Assume that is not scalene. Case 1: Step 2 If is isosceles. is isosceles, then AB = BC, BC = AC, or AB = AC. Step 3 This contradicts the given information, so is not isosceles. Case 2: is equilateral. In order for a triangle to be equilateral, it must also be isosceles, and Case 1 proved that Thus, is not equilateral. Therefore, is scalene. is not isosceles. esolutions Manual - Powered by Cognero Page 22

23 42. CHALLENGE If x is a rational number, then it can be represented by the quotient for some integers a and b, if. An irrational number cannot be represented by the quotient of two integers. Write an indirect proof to show that the product of a nonzero rational number and an irrational number is an irrational number. For this problem, assume that xy is a rational number, or in the form for some integers a and b,. Substitute this into xy, also a rational number of the form for some integers c and d,. Set the two forms of xy equal to each other and simplify the expression for y. Consider if it is in the form of a rational number. If so, you have reached a contradiction to your assumption. Given: x is a nonzero rational number and y is an irrational number. Prove: xy is irrational. Indirect Proof: Step 1 Since we are given that x is a nonzero rational, for some integers a and b,. Substituting,. Assume that xy is a rational number. Then for some integers c and d,. Step 2 Since a, b, c, and d are integers and a, is the quotient of two integers. Therefore, y is a rational number. This contradicts the given statement that y is an irrational number. Step 3 Since the assumption that xy is a rational number leads to a contradiction of the given, the original conclusion that xy is irrational must be true. esolutions Manual - Powered by Cognero Page 23

24 43. ERROR ANALYSIS Amber and Raquel are trying to verify the following statement using indirect proof. Is either of them correct? Explain your reasoning. If the sum of two numbers is even, then the numbers are even. Neither student is correct. Amber is incorrect because she doesn't find an counterexample to prove the statement false. Zero is not considered to be even or odd number so her example doesn't meet the conditions of the hypothesis. Raquel is incorrect, as well. Although she comes up with a good counterexample to the statement, she makes the wrong conclusion that the statement is true. Neither;sample answer: Since the hypothesis is true when the conclusion is false, the statement is false. 44. WRITING IN MATH Refer to Exercise 8. Write the contrapositive of the statement and write a direct proof of the contrapositive. How are the direct proof of the contrapositive of the statement and the indirect proof of the statement related? To prove that a statement is false, using an indirect proof, you must prove that a contradiction exists if the hypothesis is assumed to be true. If you assume that x is not an odd integer, or in other words, an even integer, then 5x-2 must also be even. If no contradiction exists, then the original statement must be true. If x is not an odd integer, then 5x 2 is not an odd integer. Sample answer: If x is not an odd integer, then it is an even integer. If x is an even integer, then 5x is also even because the product of any number and an even number is even. 5x 2 is also even, because two subtracted from an even number is even. Therefore, the statement If x is not an odd integer, then 5x 2 is not an odd integer is true; The direct proof of the contrapositive of the statement and the indirect proof of the statement make the same assumptions and reach the same conclusions. esolutions Manual - Powered by Cognero Page 24

25 45. SHORT RESPONSE Write an equation in slope-intercept form to describe the line that passes through the point (5, 3) and is parallel to the line represented by the equation 2x + y = 4. The slope of the line 2x + y = 4 is 2. So, the slope of line parallel to this line is also 2. The equation of the line parallel to the given line in is: 46. Statement: If and A is supplementary to C, then B is supplementary to C. Dia is proving the statement above by contradiction. She began by assuming that B is not supplementary to C. Which of the following definitions will Dia use to reach a contradiction? A definition of congruence B definition of a linear pair C definition of a right angle D definition of supplementary angles By using the definition of supplementary angles, Dia can reach a contradiction.if she assumes that and are not supplementary but and are supplementary, she can easily come to a contradiction when she substitutes in for, since they are congruent to each other. The correct choice is D. 47. List the angles of MNO in order from smallest to largest if = 9, = 7.5, and = 12. F N, O, M G O, M, N H O, N, M J M, O, N Sketch the triangle. The sides from shortest to longest are,,. The angles opposite these sides are M, O, N. So the angles from smallest to largest are M, O, N. Therefore, the correct choice is J. esolutions Manual - Powered by Cognero Page 25

26 48. SAT/ACT If b > a, which of the following must be true? A a > b B 3a > b C a 2 < b 2 D a 2 < ab E b > a SAT/ACT If b > a, which of the following must be true? A a > b This is true.multiply each side by 1. We will get, B 3a > b This would not be true if and b=4. C a 2 < b 2 This would not be true if a = 1 and b = 1. D a 2 < ab This would not be true if a = 1 and. E b > a This would not be true if a = 5 and b = 4. The answer is choice A. esolutions Manual - Powered by Cognero Page 26

27 49. PROOF Write a two-column proof. Given: bisects SRT. Prove: m SQR > m SRQ Since this proof ends with an inequality statement, you will need to incorporate one into this proof. This can be done by writing a statement using the Exterior Angle Theorem for triangle RQT. Then, creating an Inequality statement from this equation, comparing and. Finally, use the substitution property to replace with, since they are congruent. Given: bisects SRT. Prove: msqr > m SRQ Proof: Statements (Reasons) 1. bisects SRT. (Given) 2. (Definition of angle bisector) 3. m QRS = m QRT (Definition of congruent angles) 4. m SQR = m T + m QRT (Exterior Angle Theorem) 5. m SQR > m QRT (Definition of Inequality) 6. m SQR > m SRQ (Substitution) esolutions Manual - Powered by Cognero Page 27

28 COORDINATE GEOMETRY Find the coordinates of the circumcenter of each triangle with the given vertices. 50. D( 3, 3), E(3, 2), F(1, 4) The circumcenter of the triangle is the intersection point of the perpendicular bisectors. The slope of is or So, the slope of the perpendicular bisector of is 6. The midpoint of is Now, the equation of the perpendicular bisector to is: Find the equation of the perpendicular bisector to in the same way. Solve the equations to find the intersection point of the perpendicular bisectors. So, the coordinates of the circumcenter of are esolutions Manual - Powered by Cognero Page 28

29 51. A(4, 0), B( 2, 4), C(0, 6) The circumcenter of the triangle is the intersection point of the perpendicular bisectors. The slope of is or So, the slope of the perpendicular bisector of is The midpoint of is Now, the equation of the perpendicular bisector to is: Use the same way to find the equation of the perpendicular bisector to Solve the equations to find the intersection point of the perpendicular bisectors.. That is, So, the coordinates of the circumcenter of are esolutions Manual - Powered by Cognero Page 29

30 Find each measure. 52. m 1 Since, And, 53. m 4 Since, COORDINATE GEOMETRY Find the distance between each pair of parallel lines with the given equations. esolutions Manual - Powered by Cognero Page 30

31 54. The slope of a line perpendicular to both the lines will be 3. Consider the y-intercept of any of the two lines and write the equation of the perpendicular line through it. The y-intercept of the line is (0, 2). So, the equation of a line with slope 3 and a y-intercept of 2 is To find the point of intersection of the perpendicular and the second line, solve the two equations. Use the value of x to find the value of y. So, the point of intersection is ( 2, 4). Use the Distance Formula to find the distance between the points ( 2, 4) and (0, 2). Therefore, the distance between the two lines is about 6.3 units. esolutions Manual - Powered by Cognero Page 31

32 55. The slope of a line perpendicular to both the lines will be Consider the y-intercept of any of the two lines and write the equation of the perpendicular line through it. The y-intercept of the line is (0, 2). So, the equation of a line with slope and a y-intercept of 2 is To find the point of intersection of the perpendicular and the second line, solve the two equations. Use the value of x to find the value of y. So, the point of intersection is (2, 1). Use the Distance Formula to find the distance between the points (2, 1) and (0, 2). Therefore, the distance between the two lines is about 2.2 units. esolutions Manual - Powered by Cognero Page 32

33 56. RECYCLING Refer to the Venn diagram that represents the number of neighborhoods in a city with a curbside recycling program for paper or aluminum. a. How many neighborhoods recycle aluminum? b. How many neighborhoods recycle paper or aluminum or both? c. How many neighborhoods recycle paper and aluminum? a. (20 + 4) or 24 neighborhoods recycle aluminum. Twenty neighborhoods recycle aluminum only (the blue area) and 4 neighborhoods recycle both aluminum and paper (the green space). Since all of these neighborhoods are represented inside the aluminum circle, they all are recycling aluminum. b. ( ) or 36 neighborhoods recycle paper or aluminum or both. Twenty neighborhoods recycle aluminum only (the blue area) and 4 neighborhoods recycle both aluminum and paper ( the green space) and 12 neighborhoods recycle paper only (the yellow space). Since all of these neighborhoods are represented inside the aluminum circle, the paper circle or both, then they all are recycling aluminum or paper or both. c. 4 neighborhoods recycle both paper and aluminum.4 neighborhoods recycle both aluminum and paper (the green space). Determine whether each inequality is true or false > < < 115 esolutions Manual - Powered by Cognero Page 33

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