Ch 2 Practice. Multiple Choice

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1 Ch 2 Practice Multiple Choice 1. For the conditional statement, write the converse and a biconditional statement. If a figure is a right triangle with sides a, b, and c, then a 2 + b 2 = c 2. a. Converse: If a 2 + b 2 c 2, then the figure is not a right triangle with sides a, b, and c. Biconditional: A figure is a right triangle with sides a, b, and c if and only if a 2 + b 2 = c 2. b. Converse: If a 2 + b 2 c 2, then the figure is not a right triangle with sides a, b, and c. Biconditional: A figure is not a right triangle with sides a, b, and c if and only if a 2 + b 2 c 2 c. Converse: If a 2 + b 2 = c 2, then the figure is a right triangle with sides a, b, and c. Biconditional: A figure is a right triangle with sides a, b, and c if and only if a 2 + b 2 = c 2. d. Converse: If a figure is not a right triangle with sides a, b, and c, then a 2 + b 2 c 2. Biconditional: A figure is a right triangle with sides a, b, and c if and only if a 2 + b 2 = c Identify the hypothesis and conclusion of this conditional statement: If today is Wednesday, then tomorrow is Thursday. a. Hypothesis: Today is Wednesday. Conclusion: Tomorrow is not Thursday. b. Hypothesis: Tomorrow is Thursday. Conclusion: Today is Wednesday. c. Hypothesis: Tomorrow is not Thursday. Conclusion: Today is Wednesday. d. Hypothesis: Today is Wednesday. Conclusion: Tomorrow is Thursday. 3. Which statement is a counterexample for the following conditional? If you live in Springfield, then you live in Illinois. a. Sara Lucas lives in Springfield. c. Billy Jones lives in Chicago, Illinois. b. Jonah Lincoln lives in Springfield, Illinois. d. Erin Naismith lives in Springfield, Massachusetts. 4. Use the Law of Detachment to draw a conclusion from the two given statements. The doctor recommends rest if the patient has the flu. The doctor recommends rest. a. The patient does not have the flu. b. If the doctor recommends rest, the patient has the flu. c. The patient has the flu. d. not possible 5. One way to show that a statement is NOT a good definition is to find a. a. counterexample c. conditional b. biconditional d. converse 6. Which statement is the Law of Syllogism? a. If p q and q r are true statements, then r p is a true statement. b. If p q is a true statement and q is true, then p is true. c. if p q and q r are true statements, then p r is a true statement. d. If p q is a true statement and p is true, then q is true. 1

2 7. What is the inverse of the following conditional? If a point is in the fourth quadrant, then its coordinates are negative. a. If the coordinates of a point are not negative, then the point is not in the fourth quadrant. b. If a point is in the fourth quadrant, then its coordinates are negative. c. If the coordinates of a point are negative, then the point is in the fourth quadrant. d. If a point is not in the fourth quadrant, then the coordinates of the point are not negative. 8. Name the Property of Congruence that justifies the statement: If M N and N O, then M O. a. Symmetric Property c. Reflexive Property b. Transitive Property d. none of these 9. BD bisects ABC. m ABC = 9x. m ABD = 4x+ 6. Find m DBC. a. 42 b. 12 c. 54 d Find the values of x and y. a. x = 18, y = 10 c. x = 10, y = 18 b. x = 30, y = 150 d. x = 150, y = Determine if the conditional statement is true. If false, give a counterexample. If a figure has four sides, then it is a square. a. False; A rectangle has four sides, and it is not a square. b. True. 12. There is a myth that a duck s quack does not echo. A group of scientists observed a duck in a special room, and they found that the quack does echo. Therefore, the myth is false. Is the conclusion a result of inductive or deductive reasoning? a. Since the conclusion is based on a pattern of observation, it is a result of inductive reasoning. b. Since the conclusion is based on logical reasoning from scientific research, it is a result of deductive reasoning. c. Since the conclusion is based on logical reasoning from scientific research, it is a result of inductive reasoning. d. Since the conclusion is based on a pattern of observation, it is a result of deductive reasoning. 2

3 13. Determine if the conjecture is valid by the Law of Detachment. Given: If Tommy makes cookies tonight, then Tommy must have an oven. Tommy has an oven. Conjecture: Tommy made cookies tonight. a. The conjecture is valid, because Tommy could have an oven but he could make something besides cookies tonight. b. The conjecture is not valid, because Tommy could have an oven but he could make something besides cookies tonight. c. The conjecture is valid, because if Tommy didn t have an oven then he didn t make cookies tonight d. The conjecture is not valid, because if Tommy didn t have an oven then he didn t make cookies tonight. 14. Determine if the conjecture is valid by the Law of Syllogism. Given: If you are in California, then you are in the west coast. If you are in Los Angeles, then you are in California. Conjecture: If you are in Los Angeles, then you are in the west coast. a. Yes, the conjecture is valid. b. No, the conjecture is not valid. 15. Use the Law of Syllogism to draw a conclusion from the given information. Given: If two lines are perpendicular, then they form right angles. If two lines meet at a 90 angle, then they are perpendicular. Two lines meet at a 90 angle. a. Conclusion: The lines form a right angle. b. Conclusion: The lines are parallel. c. Conclusion: The lines are perpendicular and meet at a 90 angle. d. Conclusion: The lines meet at a 90 angle. 16. Write the conditional statement and converse within the biconditional. A rectangle is a square if and only if all four sides of the rectangle have equal lengths. a. Converse: If all four sides of the rectangle have equal lengths, then it is a square. Conditional: If a rectangle is a square, then its four sides have equal lengths. b. Conditional: If all four sides have equal lengths, then all four angles are 90. Converse: If all four angles are 90, then all four sides have equal lengths. c. Conditional: If a rectangle is a square, then it is also a rhombus. Converse: If a rectangle is a rhombus, then it is also a square. d. Conditional: If a rectangle is not a square, then its sides are of different lengths. Converse: If the sides are of different lengths, then the rectangle is not a square. 17. When a conditional and its converse are true, you can combine them as a true. a. counterexample c. hypothesis b. unconditional d. biconditional 18. Another name for an if-then statement is a. Every conditional has two parts. The part following if is the and the part following then is the. a. conditional; conclusion; hypothesis c. conditional; hypothesis; conclusion b. hypothesis; conclusion; conditional d. hypothesis; conditional; conclusion 3

4 19. Write a justification for each step, given that EG= FH. EG = FH Given information EG= EF+FG [1] FH = FG + GH Segment Addition Postulate EF+FG=FG+GH [2] EF=GH Subtraction Property of Equality a. [1] Angle Addition Postulate [2] Subtraction Property of Equality b. [1] Segment Addition Postulate [2] Substitution Property of Equality c. [1] Segment Addition Postulate [2] Definition of congruent segments d. [1] Substitution Property of Equality [2] Transitive Property of Equality 20. Give the reason for the last statement in the proof. Statement 1 is a supplement of 2 3 is a supplement of ? a. Congruent Complements Theorem b. Congruent Supplements Theorem c. Linear Pair Postulate d. Vertical Angles Congruence Theorem 21. State a counterexample to disprove the following conjecture: A square is a figure with four right angles. a. Rectangles also have four right angles. b. A triangle has three angles. c. A line connects two points. d. Octagons are figures. Reason Given Given Given 4

5 22. Use the given paragraph proof to write a two-column proof. Given: BAC is a right angle. 1 3 Prove: 2 and 3 are complementary. Paragraph proof: Since BAC is a right angle, m BAC=90 by the definition of a right angle. By the Angle Addition Postulate, m BAC=m 1+m 2. By substitution, m 1+m 2=90. Since 1 3, m 1=m 3 by the definition of congruent angles. Using substitution, m 3+m 2=90. Thus, by the definition of complementary angles, 2 and 3 are complementary. Complete the proof. Two-column proof: Statements Reasons 1. BAC is a right angle Given 2. m BAC=90 2. Definition of a right angle 3. m BAC=m 1+m 2 3. [1] 4. m 1+m 2=90 4. Substitution 5. m 1=m 3 5. [2] 6. m 3+m 2=90 6. Substitution 7. 2 and 3 are complementary. 7. Definition of complementary angles a. [1] Substitution [2] Definition of congruent angles c. [1] Angle Addition Postulate [2] Definition of congruent angles b. [1] Substitution [2] Definition of equality d. [1] Angle Addition Postulate [2] Definition of equality 23. What is the converse of the following conditional? If a number is divisible by 10, then it is divisible by 5. a. If a number is divisible by 5, then it is divisible by 10. b. If a number is not divisible by 5, then it is not divisible by 10. c. If a number is not divisible by 10, then a number is not divisible by 5. d. If a number is divisible by 10, then it is divisible by 5. 5

6 24. Determine whether the conditional and its converse are both true. If both are true, combine them as a biconditional. If either is false, give a counterexample. If an angle is a right angle, its measure is 90. If an angle measure is 90, the angle is a right angle. a. One statement is false. If an angle is a right angle, its measure may be 180. b. One statement is false. If an angle measure is 90, the angle may be a vertical angle. c. Both statements are true. The measure of angle is 90 if and only if it is not a right angle. d. Both statements are true. An angle is a right angle if and only if its measure is Decide whether the following definition of perpendicular is reversible. If it is, state the definition as a true biconditional. Two lines that intersect at right angles are perpendicular. a. Reversible; if two lines are perpendicular, then they intersect at right angles. b. The statement is not reversible. c. Reversible; two lines intersect at right angles if and only if they are perpendicular. d. Reversible; if two lines intersect at right angles, then they are perpendicular. 26. Which statement provides a counterexample to the following faulty definition? A square is a figure with four congruent sides. a. A square has four congruent angles. b. Some triangles have all sides congruent. c. A rectangle has four sides. d. A six-sided figure can have four sides congruent. 27. Which statement is the Law of Detachment? a. If p q is a true statement and q is true, then q p is true. b. If p q and q r are true, then p r is a true statement. c. If p q is a true statement and q is true, then p is true. d. If p q is a true statement and p is true, then q is true. 28. Use the Law of Syllogism to draw a conclusion from the two given statements. If three points lie on the same line, they are collinear. If three points are collinear, they lie in the same plane. a. The three points are collinear. b. If three points lie in the same line, they lie in the same plane. c. The three points lie in the same plane. d. If three points do not lie in the same plane, they do not lie on the same line. 29. FH bisects EFG, m EFH= (6x 5), and m HFG= (3x+ 4). Find m EFH. a. m EFH = 23 c. m EFH = 26 b. m EFH = 13 d. m EFH = An angle measures 2 degrees more than 3 times its complement. Find the measure of its complement. a. 22 c. 23 b. 272 d M is the midpoint of AN, A has coordinates (4, 4), and M has coordinates ( 3, 1). Find the coordinates of N. a. ( 9, 3) c. ( 1 2, 21 2 ) b. (1, 5) d. ( 10, 2) 32. Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from R(3, 6) to W( 7, 7). a. 3.0 units c. 1.7 units b. 9.7 units d units 6

7 33. Find the distance between the points ( 4, 6) and ( 1, 5). a. 146 b. 146 c. 10 d Let p be you are a senior and let q be you are in high school. Write the inverse. Then decide whether it is true or false. a. If you are a senior, then you are in high school; true b. If you are in high school, then you are a senior; false c. If you are not a senior, then you are not in high school; false d. If you are not in high school, then you are not a senior; true 35. Let p be an animal is a dog and let q be an animal is a golden retriever. Write the contrapositive. Then decide whether it is true or false. a. If an animal is a golden retriever, then it is a dog; true b. If an animal is not a golden retriever, then it is not a dog; false c. If an animal is a dog, then it is a golden retriever; false d. If an animal is not a dog, then it is not a golden retriever; true Find a counterexample to disprove the conjecture. 36. Conjecture AB divides CAD into two angles. So, AB is an angle bisector of CAD. a. m CAB = 20 and m DAB = 20 c. m CAB = 70 and m DAB = 20 b. m CAB=20 and m CAD= 40 d. m DAB=60 and m CAD=120 Use the Law of Detachment to determine what you can conclude from the given information, if possible. 37. If it is a nice day today, then your parents will take you to the beach. Your parents take you to the beach. a. It is a nice day. b. not possible 38. If the roads are icy, then school will be closed. School is closed. a. The roads are icy. b. not possible Name the property that the statement illustrates. 39. CD CD a. Symmetric Property of Segment Congruence b. Transitive Property of Segment Congruence c. Reflexive Property of Segment Congruence 40. If MN XY and XY RS, then MN RS a. Reflexive Property of Segment Congruence b. Symmetric Property of Segment Congruence c. Transitive Property of Segment Congruence 7

8 If possible, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements. 41. If a figure is a square, then it is a rectangle. If a figure is a rectangle, then it has four right angles. a. If a figure is has four right angles, then it is a square. c. If a figure is a square, then it has four right angles. b. If a figure is a rectangle, then it is a square. d. not possible 42. If school is canceled this morning, then you can sleep late. If the power goes out this morning, then school will be canceled. a. If you sleep late this morning, then school will be canceled. c. If school is canceled this morning, then the power will go out. b. If power goes out this morning, then you can sleep late. d. not possible 43. If you are thirteen years old, then you are a teenager. If you are nineteen years old, then you are a teenager. a. If you are a teenager, then you are thirteen years old. b. If you are not a teenager, then you are not thirteen years old. c. If you are a teenager, then you are nineteen years old. d. not possible Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. 44. Every day after school, your mom gives you a snack. So, when you get home from school today, you will get a snack. a. inductive reasoning; A pattern is used. c. deductive reasoning; A pattern is used. b. inductive reasoning; Facts are used. d. deductive reasoning; Facts are used. 45. All mammals are warm-blooded. Snakes are not warm-blooded. Your pet snake is not a mammal. a. inductive reasoning; A pattern is used. c. deductive reasoning; A pattern is used. b. inductive reasoning; Facts are used. d. deductive reasoning; Facts are used. 46. All multiples of 10 are divisible by is a multiple of 10. So, 40 is divisible by 5. a. inductive reasoning; A pattern is used. c. deductive reasoning; A pattern is used. b. inductive reasoning; Facts are used. d. deductive reasoning; Facts are used. 8

9 State the law of logic that is illustrated. 47. If you get an A on your math test, then you can choose where to go to dinner. If you can choose where to dinner, then you can go to your favorite restaurant. If you get an A on your math test, then you can go to your favorite restaurant. a. Law of Detachment b. Law of Syllogism 48. If you miss class the day before a holiday break, then you will not get bonus points. You miss class the day before Thanksgiving break. You do not get bonus points. a. Law of Detachment b. Law of Syllogism Matching Match each vocabulary term with its definition. a. conjecture e. biconditional statement b. inductive reasoning f. hypothesis c. deductive reasoning g. counterexample d. conclusion h. conditional statement 49. an example that proves that a conjecture or statement is false 50. a statement that is believed to be true 51. the part of a conditional statement following the word then 52. the part of a conditional statement following the word if 53. the process of reasoning that a rule or statement is true because specific cases are true 54. a statement that can be written in the form if p, then q, where p is the hypothesis and q is the conclusion 55. Find the midpoint and distance between A(1, -4) and B(-3, -8) 56. Refer to the following statement: Two lines are perpendicular if and only if they intersect to form a right angle. A. Is this a biconditional statement? B. Is the statement true? 57. Given the following statements, can you conclude that Eileen watches TV on Thursday night? (1) If it is Thursday night, Eileen stays at home. (2) If Eileen stays at home, she watches TV. 9

10 58. Provide the reasons for each statement in the proof. Given: AB=DE Prove: AD= BE Statement AB=DE? AB+BD=DE+BD? AB+BD=AD, DE+BD=BE? AD= BE? Reason 59. Provide reasons for each statement in the proof. Given: 3 4 Prove: 1 2 Statement Reason ? ; ? ? ? True or False: 60. Vertical angles are always equal to each other. 61. If two angles are supplements of the same angle, then they must be equal in measure. 62. If two angles are complements of the same angle, then they must be equal in measure is complementary to 2; 3 is complementary to 2. What theorem, property, or postulate allows you to state that 1 3? 10

11 64. Find the distance between the points ( 5, 9) and ( 9, 12). 65. Determine the coordinates of the midpoint of CG and find the approximate distance CG for the points CÊ Ë Á 2, 1ˆ and GÊ Ë Á7, 4ˆ. 66. Find the midpoint of the segment with endpoints (4, 3) and ( 8, 15). 67. Determine whether the conclusion is valid. If so, state the law of logic illustrated. If not, find a counterexample. Statement If you live in Detroit, then you live in Michigan. Statement If you live in Michigan, then it is cold in December. Conclusion If you live in Detroit, then it is cold in December. 68. Three planes head out on a search-and-rescue mission. Plane M flies due north, plane E veers to the east at an angle of ( 5x + 15), and plane W veers to the west at an angle of ( 8x). Plane M bisects the angle formed by the other two planes. Find the value of x. Then write a two-column proof to support your answer. Other 69. Identify the hypothesis and conclusion of the statement. If today is Wednesday, then tomorrow is Thursday. 70. Tell whether the following statement is True or False. If it is false, give a counterexample. "If a number is even, then it is a multiple of 4." 54 & 55: Decide whether inductive or deductive reasoning is used to reach the conclusion. Explain your reasoning. Also, if it is deductive reasoning determine if the reasoning follows the law of detachment. 71. The rule in your house is that you must complete all of your homework in order to watch television (no exceptions allowed). You watched your favorite television show Tuesday night. Therefore, you completed all of your homework on Tuesday. 72. For the past 6 weeks, your aunt has asked you to watch your cousin on Wednesday night. You conclude that you will be asked to watch your cousin next Wednesday. 73. Given the conditional statement if it is 95 degrees then it is hot, write the inverse, converse and the contrapositive. Evaluate each and determine if the statement can be written as a true biconditional. 11

12 Rewrite the definition as a single biconditional statement. 74. If a ray divides an angle into two congruent angles, then the ray is an angle bisector. 75. If an angle has a measure of 180, then it is a straight angle. 12

13 ID: A Ch 2 Practice Answer Section MULTIPLE CHOICE 1. C 2. D 3. D 4. D 5. A 6. C 7. D 8. B 9. C 10. A 11. A 12. A 13. B 14. A 15. A 16. A 17. D 18. C 19. B 20. B 21. A 22. C 23. A 24. D 25. C 26. D 27. D 28. B 29. B 30. A 31. D 32. D 33. D 34. C 35. B 36. C 37. B 38. B 1

14 ID: A 39. C 40. C 41. C 42. B 43. D 44. A 45. D 46. D 47. B 48. A MATCHING 49. G 50. A 51. D 52. F 53. B 54. H SHORT ANSWER 55. see teacher 56. A. yes B. yes 57. yes Statement AB=DE Reason Given 58. AB+BD=DE+BD AB+BD=AD, DE+BD= BE AD = BE Statement Addition property of equality Segment addition postulate Substitution property of equality Reason Given ; VerticalAngles Congruence Theorem Transitive Property of Congruence Transitive Property of Congruence 60. True 61. True 62. True 63. Congruent Complements Theorem units 2

15 ID: A Ê midpoint = 2, 5 ˆ Ë Á 2 distance = ( 2, 9) 67. valid; Law of Syllogism 68. x= 5; Given AM bisects WAE Prove x= 5 STATEMENTS REASONS 1. AM bisects WAE. 1. Given 2. WAM EAM 2. Definition of angle bisector 3. m WAM = m EAM 3. Definition of congruent angles 4. 8x= 5x Substitution Property of Equality 5. 3x= Subtraction Property of Equality 6. x= 5 6. Division Property of Equality OTHER 69. hypothesis: today is Wednesday, conclusion: tomorrow is Thursday 70. False; sample counterexample: Deductive reasoning; the conclusion is reached by using laws of logic and the facts about house rules and what you did that day. It can not be determined by detachment. 72. Inductive reasoning; the conclusion is based on a pattern that has developed over the last 6 weeks. 73. S: if 95 them hot True C: if hot then 95 False, could be hot and 96 I: if ~95 then ~hot False, could be 97 and hot C: if ~hot then ~95 True Can not be written as a true biconditional because the converse is false. 74. A ray divides an angle into two congruent angles if and only if it is an angle bisector 75. An angle has a measure of 180 if and only if it is a straight angle. 3