Chapter 2-Reasoning and Proof

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1 Chapter 2-Reasoning and Proof Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Identify the hypothesis and conclusion of this conditional statement: If two lines intersect at right angles, then the two lines are perpendicular. a. Hypothesis: The two lines are perpendicular. Conclusion: Two lines intersect at right angles. b. Hypothesis: Two lines intersect at right angles. Conclusion: The two lines are perpendicular. c. Hypothesis: The two lines are not perpendicular. Conclusion: Two lines intersect at right angles. d. Hypothesis: Two lines intersect at right angles. Conclusion: The two lines are not perpendicular. 2. Write this statement as a conditional in if-then form: All triangles have three sides. a. If a triangle has three sides, then all triangles have three sides. b. If a figure has three sides, then it is not a triangle. c. If a figure is a triangle, then all triangles have three sides. d. If a figure is a triangle, then it has three sides. 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. b. Jonah Lincoln lives in Springfield, Illinois. c. Billy Jones lives in Chicago, Illinois. d. Erin Naismith lives in Springfield, Massachusetts. 4. Draw a Draw a Venn diagram to illustrate this conditional: Cars are motor vehicles.

2 a. Motor vehicles Cars 5. 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 b. hypothesis; conclusion; conditional c. conditional; hypothesis; conclusion d. hypothesis; conditional; conclusion b. c. Motor vehicles Cars Cars 6. Which choice shows a true conditional with the hypothesis and conclusion identified correctly? a. Yesterday was Monday if tomorrow is Thursday. Hypothesis: Tomorrow is Thursday. Conclusion: Yesterday was Monday. b. If tomorrow is Thursday, then yesterday was Tuesday. Hypothesis: Yesterday was Tuesday. Conclusion: Tomorrow is not Thursday. c. If tomorrow is Thursday, then yesterday was Tuesday. Hypothesis: Yesterday was Tuesday. Conclusion: Tomorrow is Thursday. d. Yesterday was Tuesday if tomorrow is Thursday. Hypothesis: Tomorrow is Thursday. Conclusion: Yesterday was Tuesday. Motor vehicles d. Motor Cars vehicles 7. What is the conclusion of the following conditional? A number is divisible by 3 if the sum of the digits of the number is divisible by 3. a. The number is odd. b. The sum of the digits of the number is divisible by 3. c. If the sum of the digits of a number is divisible by 3, then the number is divisible by 3. d. The number is divisible by 3.

3 8. What is the converse of the following conditional? If a point is in the first quadrant, then its coordinates are positive. a. If a point is in the first quadrant, then its coordinates are positive. b. If a point is not in the first quadrant, then the coordinates of the point are not positive. c. If the coordinates of a point are positive, then the point is in the first quadrant. 9. What is the converse and the truth value of the converse of the following conditional? If an angle is a right angle, then its measure is 90. a. If an angle is not a right angle, then its measure is 90. False b. If an angle is not a right angle, then its measure is not 90. True c. If an angle has measure 90, then it is a right angle. False d. If an angle has measure 90, then it is a right angle. True d. If the coordinates of a point are not positive, then the point is not in the first quadrant. 10. Which conditional has the same truth value as its converse? a. If x = 7, then. b. If a figure is a square, then it has four sides. c. If x 17 = 4, then x = 21. d. If an angle has measure 80, then it is acute. 11. For the following true conditional statement, write the converse. If the converse is also true, combine the statements as a biconditional. If x = 3, then x 2 = 9. a. If x 2 = 9, then x = 3. True; x 2 = 9 if and only if x = 3. b. If x 2 = 3, then x = 9. False c. If x 2 = 9, then x = 3. True; x = 3 if and only if x 2 = 9. d. If x 2 = 9, then x = 3. False 12. 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 two lines are parallel, they do not intersect. If two lines do not intersect, they are parallel. a. One statement is false. If two lines do not intersect, they could be skew.. b. One statement is false. If two lines are parallel, they may intersect twice. c. Both statements are true. Two lines are parallel if and only if they do not intersect. d. Both statements are true. Two lines are not parallel if and only if they do not intersect. 13. 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 measure is 90, the angle may be a vertical angle. b. One statement is false. If an angle is a right angle, its measure may be 180. c. Both statements are true. An angle is a right angle if and only if its measure is 90. d. Both statements are true. The measure of angle is 90 if and only if it is not a right angle. 14. Write the two conditional statements that make up the following biconditional. I drink juice if and only if it is breakfast time. a. I drink juice if and only if it is breakfast time. It is breakfast time if and only if I drink juice. b. If I drink juice, then it is breakfast time. If it is breakfast time, then I drink juice. c. If I drink juice, then it is breakfast time. I drink juice only if it is breakfast time. d. I drink juice. It is breakfast time. 15. When a conditional and its converse are true, you can combine them as a true. a. counterexample b. biconditional c. unconditional d. hypothesis 16. 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. The statement is not reversible. b. Reversible; if two lines intersect at right angles, then they are perpendicular. c. Reversible; if two lines are perpendicular, then they intersect at right angles. d. Reversible; two lines intersect at right angles if and only if they are perpendicular.

4 17. Is the statement a good definition? If not, find a counterexample. A square is a figure with two pairs of parallel sides and four right angles. a. The statement is a good definition. b. No; a rhombus is a counterexample. c. No; a rectangle is a counterexample. d. No; a parallelogram is a counterexample. a. converse b. conditional c. biconditional d. counterexample 19. Which statement provides a counterexample to the following faulty definition? A square is a figure with four congruent sides. a. A six-sided figure can have four sides congruent. b. Some triangles have all sides congruent. c. A square has four congruent angles. d. A rectangle 18. One way to show that a statement is NOT a good has four sides. definition is to find a. 20. Which biconditional is NOT a good definition? a. A whole number is odd if and only if the number is not divisible by 2. b. An angle is straight if and only if its measure is 180. c. A whole number is even if and only if it is divisible by 2. d. A ray is a bisector of an angle if and only if it splits the angle into two angles. Fill in each missing reason. 21. Given:,, and. Find x. P R Q S Drawing not to scale x 5 + x 11 = 100 2x 16 = 100 2x = 116 x = 58 a. b. Substitution Property c. Simplify d. e. Division Property of Equality a. Angle Addition Postulate; Subtraction Property of Equality b. Protractor Postulate; Addition Property of Equality c. Angle Addition Postulate; Addition Property of Equality d. Protractor Postulate; Subtraction Property of Equality 22. Given: Prove: a. a. Given

5 b. Symmetric Property of Equality c. Subtraction Property of Equality d. Division Property of Equality e. Reflexive Property of Equality b. a. Given b. Substitution Property c. Subtraction Property of Equality d. Division Property of Equality e. Symmetric Property of Equality c. a. Given b. Substitution Property c. Subtraction Property of Equality d. Division Property of Equality e. Reflexive Property of Equality d. a. Given b. Substitution Property c. Subtraction Property of Equality d. Addition Property of Equality e. Symmetric Property of Equality 23. Name the Property of Equality that justifies the statement: If p = q, then. a. Reflexive Property b. Multiplication Property c. Symmetric Property d. Subtraction Property 24. Which statement is an example of the Addition Property of Equality? a. If p = q then b. If p = q then. c. If p = q then d. p = q 25. Name the Property of Congruence that justifies the statement: If. a. Symmetric Property b. Transitive Property c. Reflexive Property d. none of these 26. Name the Property of Congruence that justifies the statement: If. a. Transitive Property b. Symmetric Property c. Reflexive Property d. none of these Use the given property to complete the statement. 27. Transitive Property of Congruence If. a. b. c. d. 28. Multiplication Property of Equality If, then. a. b. c. d. 29. Substitution Property of Equality If, then. a. b. c. d. (7x 8) (6x + 11) Drawing not to scale a. 19 b. 125 c. 19 d Find 30. bisects = 7x. =. Find a. 50 b. 125 c. 75 d Find the value of x. Drawing not to scale

6 a. 37 b. 143 c. 27 d Find the values of x and y. a. x = 15, y = 17 b. x = 112, y = 68 c. x = 68, y = 112 d. x = 17, y = 15 4y 7x Short Answer Drawing not to scale 34. Write the converse of the statement. If the converse is true, write true; if not true, provide a counterexample. If x = 4, then x 2 = Write the converse of the given true conditional and decide whether the converse is true or false. If the converse is true, combine it with the conditional to form a true biconditional. If the converse is false, give a counterexample. If the probability that an event will occur is 0, then the event is impossible to occur. Fill in each missing reason. 36. Given: 2x 6x + 8 (2x) Drawing not to scale 6(x 3) Drawing not to scale 38. Complete the paragraph proof. Given: Prove: are supplementary, and are supplementary. 37. Given:

7 40. Write the conditional statement that the Venn diagram illustrates. Quadrilaterals By the definition of supplementary angles, (a) and (b). Then by (c). Subtract from each side. You get (d), or (e). Squares 39. Solve for x. Justify each step. Essay 41. Given: are complementary, and are complementary. Prove: Fill in each missing reason. 42. Given: Prove:

8 Drawing not to scale Other 43. a. Write the following conditional in if-then form. b. Write its converse in if-then form. c. Determine the truth value of the original conditional and its converse. Explain why each of them is true or false, and provide a counterexample(s) for any false statement(s). On a number line, the points with coordinates 2 and 5 are 7 units apart. 44. Write the two conditional statements that form the given biconditional. Then decide whether the biconditional is a good definition. Explain. Three points are collinear if and only if they are coplanar.

9 Chapter 2-Reasoning and Proof Answer Section MULTIPLE CHOICE 1. ANS: B REF: 2-1 Conditional Statements OBJ: Conditional Statements 2. ANS: D REF: 2-1 Conditional Statements OBJ: Conditional Statements 3. ANS: D REF: 2-1 Conditional Statements OBJ: Conditional Statements 4. ANS: A REF: 2-1 Conditional Statements OBJ: Conditional Statements 5. ANS: C REF: 2-1 Conditional Statements OBJ: Conditional Statements 6. ANS: D REF: 2-1 Conditional Statements OBJ: Conditional Statements 7. ANS: D REF: 2-1 Conditional Statements OBJ: Conditional Statements 8. ANS: C REF: 2-1 Conditional Statements OBJ: Converses 9. ANS: D REF: 2-1 Conditional Statements OBJ: Converses 10. ANS: C REF: 2-1 Conditional Statements OBJ: Converses 11. ANS: D REF: 2-2 Biconditionals and Definitions OBJ: Writing Biconditionals 12. ANS: A REF: 2-2 Biconditionals and Definitions OBJ: Writing Biconditionals 13. ANS: C REF: 2-2 Biconditionals and Definitions OBJ: Writing Biconditionals 14. ANS: B REF: 2-2 Biconditionals and Definitions OBJ: Writing Biconditionals 15. ANS: B REF: 2-2 Biconditionals and Definitions OBJ: Writing Biconditionals 16. ANS: D REF: 2-2 Biconditionals and Definitions OBJ: Recognizing Good Definitions 17. ANS: C REF: 2-2 Biconditionals and Definitions OBJ: Recognizing Good Definitions 18. ANS: D REF: 2-2 Biconditionals and Definitions OBJ: Recognizing Good Definitions 19. ANS: A REF: 2-2 Biconditionals and Definitions OBJ: Recognizing Good Definitions 20. ANS: D REF: 2-2 Biconditionals and Definitions OBJ: Recognizing Good Definitions 21. ANS: C REF: 2-4 Reasoning in Algebra 22. ANS: B REF: 2-4 Reasoning in Algebra 23. ANS: D REF: 2-4 Reasoning in Algebra 24. ANS: B REF: 2-4 Reasoning in Algebra 25. ANS: A REF: 2-4 Reasoning in Algebra 26. ANS: A REF: 2-4 Reasoning in Algebra 27. ANS: C REF: 2-4 Reasoning in Algebra 28. ANS: C REF: 2-4 Reasoning in Algebra 29. ANS: C REF: 2-4 Reasoning in Algebra 30. ANS: D REF: 2-4 Reasoning in Algebra 31. ANS: C REF: 2-5 Proving Angles Congruent OBJ: Theorems About Angles STA: CA GEOM 1.0 CA GEOM 2.0 CA GEOM 4.0

10 32. ANS: A REF: 2-5 Proving Angles Congruent OBJ: Theorems About Angles STA: CA GEOM 1.0 CA GEOM 2.0 CA GEOM ANS: A REF: 2-5 Proving Angles Congruent OBJ: Theorems About Angles STA: CA GEOM 1.0 CA GEOM 2.0 CA GEOM 4.0 SHORT ANSWER 34. ANS: If x 2 = 16, then x = 4. False; if x 2 = 16, then x can be equal to 4. REF: 2-1 Conditional Statements OBJ: Converses 35. ANS: If an event is impossible, the probability of the event is 0. True An event is impossible if and only if the probability of the event is zero. REF: 2-2 Biconditionals and Definitions 36. ANS: a. Angle Addition Postulate b. Substitution Property c. Distributive Property d. Simplify e. Addition Property of Equality f. Division Property of Equality REF: 2-4 Reasoning in Algebra STA: CA GEOM 1.0 CA GEOM ANS: a. Segment Addition Postulate b. Substitution c. Simplify d. Subtraction Property of Equality e. Division Property of Equality OBJ: Writing Biconditionals OBJ: Connecting Reasoning in Algebra and Geometry REF: 2-4 Reasoning in Algebra OBJ: Connecting Reasoning in Algebra and Geometry STA: CA GEOM 1.0 CA GEOM ANS: a. 180 b. 180 c. Transitive Property (or Substitution Property) d. e. REF: 2-5 Proving Angles Congruent OBJ: Theorems About Angles STA: CA GEOM 1.0 CA GEOM 2.0 CA GEOM ANS: Given Addition Property of Equality Simplify

11 x = 27 Division Property of Equality Simplify REF: 2-4 Reasoning in Algebra OBJ: Connecting Reasoning in Algebra and Geometry STA: CA GEOM 1.0 CA GEOM ANS: If a figure is a square, then it is a quadrilateral. REF: 2-1 Conditional Statements OBJ: Conditional Statements ESSAY 41. ANS: [4] By the definition of complementary angles, and. By the Transitive Property of Equality (or Substitution Property),. By the Subtraction Property of Equality,, and by the definition of congruent angles. OR equivalent explanation [3] one step missing OR one incorrect justification [2] two steps missing OR two incorrect justifications [1] correct steps with no explanations REF: 2-5 Proving Angles Congruent OBJ: Theorems About Angles STA: CA GEOM 1.0 CA GEOM 2.0 CA GEOM ANS: [4] a. Given b. Substitution Property c. Vertical Angles Theorem d. Substitution Property [3] three parts correct [2] two parts correct [1] one part correct REF: 2-5 Proving Angles Congruent OBJ: Theorems About Angles STA: CA GEOM 1.0 CA GEOM 2.0 CA GEOM 4.0 OTHER 43. ANS: a. On a number line, if points have coordinates 2 and 5, then they are 7 units apart. b. On a number line, if points are 7 units apart, then they have coordinates 2 and 5. c. The original conditional is true by the Ruler Postulate. The converse is false. The points 0 and 7 and 7 units apart, but their coordinates are not 2 and 5. REF: 2-1 Conditional Statements OBJ: Converses 44. ANS: If three points are collinear, then they are coplanar.

12 If three points are coplanar, then they are collinear. The biconditional is not a good definition. Three coplanar points might not lie on the same line. REF: 2-2 Biconditionals and Definitions OBJ: Recognizing Good Definitions

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