Foundations of Math 3 -- Proof Practice

Transcription

1 Foundations of Math 3 -- Proof Practice 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. a. c. Motor vehicles Cars Cars Motor vehicles

2 b. d. Motor vehicles Cars Cars Motor vehicles 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 c. conditional; hypothesis; conclusion b. hypothesis; conclusion; conditional d. hypothesis; conditional; conclusion 6. 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. d. If the coordinates of a point are not positive, then the point is not in the first quadrant. 7. 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. 8. 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. 9. 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. 10. When a conditional and its converse are true, you can combine them as a true. a. counterexample c. unconditional

3 b. biconditional d. hypothesis 11. 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. 12. Use the Law of Detachment to draw a conclusion from the two given statements. If two angles are congruent, then they have equal measures. and are congruent. a. + = 90 c. is the complement of. b. = d. 13. Use the Law of Detachment to draw a conclusion from the two given statements. If not possible, write not possible. I can go to the concert if I can afford to buy a ticket. I can go to the concert. a. I can afford to buy a ticket. b. I cannot afford to buy the ticket. c. If I can go to the concert, I can afford the ticket. d. not possible 14. Which statement is the Law of Detachment? a. If is a true statement and q is true, then p is true. b. If is a true statement and q is true, then is true. c. If and are true, then is a true statement. d. If is a true statement and p is true, then q is true. 15. Use the Law of Syllogism to draw a conclusion from the two given statements. If a number is a multiple of 64,then it is a multiple of 8. If a number is a multiple of 8, then it is a multiple of 2. a. If a number is a multiple of 64, then it is a multiple of 2. b. The number is a multiple of 2. c. The number is a multiple of 8. d. If a number is not a multiple of 2, then the number is not a multiple of Which statement is the Law of Syllogism? a. If is a true statement and p is true, then q is true. b. If is a true statement and q is true, then p is true. c. if and are true statements, then is a true statement. d. If and are true statements, then is a true statement. Fill in each missing reason. 17. Given:,, and. Find x.

4 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 18. Given: Prove: a. a. Given 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 19. Given: Prove:

5 a. Given b. Addition Property of Equality c.? d. Substitution Property a. Addition Property of Equality c. Multiplication Property of Equality b. Angle Addition Postulate d. Division Property of Equality 20. Name the Property of Equality that justifies the statement: If p = q, then. a. Reflexive Property c. Symmetric Property b. Multiplication Property d. Subtraction Property Use the given property to complete the statement. 21. Transitive Property of Congruence If. a. c. b. d. 22. Multiplication Property of Equality If, then. a. c. b. d. 23. Substitution Property of Equality If, then. a. c. b. d. 24. Name an angle supplementary to

6 a. b. c. d. 25. What can you conclude from the information in the diagram? P S U R Q T a. b. c. d are vertical angles are adjacent angles is a right angle 3. are vertical angles is a right angle 3. are adjacent angles 26. Complete the paragraph proof. Given: and are right angles. Prove:

7 By the definition of,. By the Property,, or. a. complementary angles; Substitution b. right angles; Substitution c. complementary angles; Reflexive d. right angles; Symmetric 27. Find the values of x and y. 4y 7x Drawing not to scale a. x = 15, y = 17 c. x = 68, y = 112 b. x = 112, y = 68 d. x = 17, y = 15 Short Answer Fill in each missing reason. 28. Given: (2x) 6(x 3) Drawing not to scale

8 29. A plumber knows that if you shut off the water at the main valve, it is safe to remove the sink faucet. The plumber turns the main valve to the Off position. What conclusion can the plumber make? 30. Write the conditional statement that the Venn diagram illustrates.

9 Quadrilaterals Squares Answer Section Foundations of Math 3 -- Proof Practice MULTIPLE CHOICE 1. ANS: B PTS: 1 DIF: L1 REF: 2-1 Conditional Statements OBJ: Conditional Statements STA: NC 2.01 TOP: 2-1 Example 1 KEY: conditional statement hypothesis conclusion 2. ANS: D PTS: 1 DIF: L1 REF: 2-1 Conditional Statements OBJ: Conditional Statements STA: NC 2.01 TOP: 2-1 Example 2 KEY: hypothesis conclusion conditional statement 3. ANS: D PTS: 1 DIF: L1 REF: 2-1 Conditional Statements OBJ: Conditional Statements STA: NC 2.01 TOP: 2-1 Example 3 KEY: conditional statement counterexample 4. ANS: A PTS: 1 DIF: L1 REF: 2-1 Conditional Statements OBJ: Conditional Statements STA: NC 2.01 TOP: 2-1 Example 4 KEY: conditional statement Venn Diagram 5. ANS: C PTS: 1 DIF: L1 REF: 2-1 Conditional Statements OBJ: Conditional Statements STA: NC 2.01 TOP: 2-1 Example 1 KEY: conditional statement hypothesis conclusion

10 6. ANS: C PTS: 1 DIF: L1 REF: 2-1 Conditional Statements OBJ: Converses STA: NC 2.01 TOP: 2-1 Example 5 KEY: conditional statement coverse of a conditional 7. ANS: C PTS: 1 DIF: L1 REF: 2-1 Conditional Statements OBJ: Converses STA: NC 2.01 TOP: 2-1 Example 6 KEY: conditional statement coverse of a conditional truth value 8. ANS: A PTS: 1 DIF: L1 REF: 2-2 Biconditionals and Definitions OBJ: Writing Biconditionals NAT: NAEP G1c NAEP G5a CAT5.LV20.54 IT.LV16.CP IT.LV16.PS S9.TSK2.PRA S10.TSK2.PRA STA: NC 2.01 TOP: 2-2 Example 1 KEY: conditional statement coverse of a conditional biconditional statement counterexample MSC: NAEP G1c NAEP G5a CAT5.LV20.54 IT.LV16.CP IT.LV16.PS S9.TSK2.PRA S10.TSK2.PRA 9. ANS: B PTS: 1 DIF: L1 REF: 2-2 Biconditionals and Definitions OBJ: Writing Biconditionals NAT: NAEP G1c NAEP G5a CAT5.LV20.54 IT.LV16.CP IT.LV16.PS S9.TSK2.PRA S10.TSK2.PRA STA: NC 2.01 TOP: 2-2 Example 2 KEY: biconditional statement conditional statement MSC: NAEP G1c NAEP G5a CAT5.LV20.54 IT.LV16.CP IT.LV16.PS S9.TSK2.PRA S10.TSK2.PRA 10. ANS: B PTS: 1 DIF: L1 REF: 2-2 Biconditionals and Definitions OBJ: Writing Biconditionals NAT: NAEP G1c NAEP G5a CAT5.LV20.54 IT.LV16.CP IT.LV16.PS S9.TSK2.PRA S10.TSK2.PRA STA: NC 2.01 TOP: 2-2 Example 1 KEY: conditional statement biconditional statement MSC: NAEP G1c NAEP G5a CAT5.LV20.54 IT.LV16.CP IT.LV16.PS S9.TSK2.PRA S10.TSK2.PRA 11. ANS: D PTS: 1 DIF: L2 REF: 2-2 Biconditionals and Definitions OBJ: Recognizing Good Definitions NAT: NAEP G1c NAEP G5a CAT5.LV20.54 IT.LV16.CP IT.LV16.PS S9.TSK2.PRA S10.TSK2.PRA STA: NC 2.01 KEY: biconditional statement MSC: NAEP G1c NAEP G5a CAT5.LV20.54 IT.LV16.CP IT.LV16.PS S9.TSK2.PRA S10.TSK2.PRA 12. ANS: B PTS: 1 DIF: L1 REF: 2-3 Deductive Reasoning OBJ: Using the Law of Detachment STA: NC 2.01 TOP: 2-3 Example 2 KEY: deductive reasoning Law of Detachment

11 13. ANS: D PTS: 1 DIF: L2 REF: 2-3 Deductive Reasoning OBJ: Using the Law of Detachment STA: NC 2.01 TOP: 2-3 Example 3 KEY: deductive reasoning Law of Detachment 14. ANS: D PTS: 1 DIF: L1 REF: 2-3 Deductive Reasoning OBJ: Using the Law of Detachment STA: NC 2.01 TOP: 2-3 Example 2 KEY: Law of Detachment deductive reasoning 15. ANS: A PTS: 1 DIF: L1 REF: 2-3 Deductive Reasoning OBJ: Using the Law of Syllogism STA: NC 2.01 TOP: 2-3 Example 4 KEY: deductive reasoning Law of Syllogism 16. ANS: C PTS: 1 DIF: L1 REF: 2-3 Deductive Reasoning OBJ: Using the Law of Syllogism STA: NC 2.01 TOP: 2-3 Example 4 KEY: deductive reasoning Law of Syllogism 17. ANS: C PTS: 1 DIF: L1 REF: 2-4 Reasoning in Algebra STA: NC 2.01 TOP: 2-4 Example 1 KEY: Properties of Equality Angle Addition Postulate deductive reasoning 18. ANS: B PTS: 1 DIF: L2 REF: 2-4 Reasoning in Algebra STA: NC 2.01 KEY: Properties of Equality proof 19. ANS: B PTS: 1 DIF: L2 REF: 2-5 Proving Angles Congruent OBJ: Theorems About Angles NAT: NAEP G3g CAT5.LV20.55 CAT5.LV20.56 IT.LV16.CP S9.TSK2.GM S10.TSK2.GM TV.LV20.13 TV.LV20.14 STA: NC 2.02 KEY: Angle Addition Postulate deductive reasoning MSC: NAEP G3g CAT5.LV20.55 CAT5.LV20.56 IT.LV16.CP S9.TSK2.GM S10.TSK2.GM TV.LV20.13 TV.LV20.14

12 20. ANS: D PTS: 1 DIF: L1 REF: 2-4 Reasoning in Algebra STA: NC 2.01 TOP: 2-4 Example 3 KEY: Properties of Equality 21. ANS: C PTS: 1 DIF: L2 REF: 2-4 Reasoning in Algebra STA: NC 2.01 TOP: 2-4 Example 3 KEY: Properties of Congruence 22. ANS: C PTS: 1 DIF: L2 REF: 2-4 Reasoning in Algebra STA: NC 2.01 TOP: 2-4 Example 3 KEY: Properties of Equality 23. ANS: C PTS: 1 DIF: L2 REF: 2-4 Reasoning in Algebra STA: NC 2.01 TOP: 2-4 Example 3 KEY: Properties of Equality 24. ANS: C PTS: 1 DIF: L1 REF: 2-5 Proving Angles Congruent OBJ: Identifying Angle Pairs NAT: NAEP G3g CAT5.LV20.55 CAT5.LV20.56 IT.LV16.CP S9.TSK2.GM S10.TSK2.GM TV.LV20.13 TV.LV20.14 STA: NC 2.02 TOP: 2-5 Example 1 KEY: supplementary angles MSC: NAEP G3g CAT5.LV20.55 CAT5.LV20.56 IT.LV16.CP S9.TSK2.GM S10.TSK2.GM TV.LV20.13 TV.LV ANS: A PTS: 1 DIF: L1 REF: 2-5 Proving Angles Congruent OBJ: Identifying Angle Pairs NAT: NAEP G3g CAT5.LV20.55 CAT5.LV20.56 IT.LV16.CP S9.TSK2.GM S10.TSK2.GM TV.LV20.13 TV.LV20.14 STA: NC 2.02 TOP: 2-5 Example 2 KEY: vertical angles supplementary angles adjacent angles right angle congruent segments MSC: NAEP G3g CAT5.LV20.55 CAT5.LV20.56 IT.LV16.CP S9.TSK2.GM S10.TSK2.GM TV.LV20.13 TV.LV ANS: B PTS: 1 DIF: L1 REF: 2-5 Proving Angles Congruent OBJ: Theorems About Angles NAT: NAEP G3g CAT5.LV20.55 CAT5.LV20.56 IT.LV16.CP S9.TSK2.GM S10.TSK2.GM TV.LV20.13 TV.LV20.14 STA: NC 2.02 TOP: 2-5 Example 3 KEY: right angle proof multi-part question MSC: NAEP G3g CAT5.LV20.55 CAT5.LV20.56 IT.LV16.CP S9.TSK2.GM S10.TSK2.GM TV.LV20.13 TV.LV ANS: A PTS: 1 DIF: L1 REF: 2-5 Proving Angles Congruent

13 OBJ: Theorems About Angles NAT: NAEP G3g CAT5.LV20.55 CAT5.LV20.56 IT.LV16.CP S9.TSK2.GM S10.TSK2.GM TV.LV20.13 TV.LV20.14 STA: NC 2.02 TOP: 2-5 Example 4 KEY: Vertical Angles Theorem vertical angles supplementary angles multi-part question MSC: NAEP G3g CAT5.LV20.55 CAT5.LV20.56 IT.LV16.CP S9.TSK2.GM S10.TSK2.GM TV.LV20.13 TV.LV20.14 SHORT ANSWER 28. ANS: a. Angle Addition Postulate b. Substitution Property c. Distributive Property d. Simplify e. Addition Property of Equality f. Division Property of Equality PTS: 1 DIF: L2 REF: 2-4 Reasoning in Algebra STA: NC 2.01 TOP: 2-4 Example 1 KEY: proof deductive reasoning Properties of Equality multi-part question 29. ANS: It is safe to remove the sink faucet. PTS: 1 DIF: L1 REF: 2-3 Deductive Reasoning OBJ: Using the Law of Detachment STA: NC 2.01 TOP: 2-3 Example 1 KEY: deductive reasoning word problem 30. ANS: If a figure is a square, then it is a quadrilateral. PTS: 1 DIF: L2 REF: 2-1 Conditional Statements OBJ: Conditional Statements STA: NC 2.01 TOP: 2-1 Example 4 KEY: Venn Diagram conditional statement