2-2 Conditional Statements (pp. 81–87) EXAMPLES ■
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Write a conditional statement from the sentence “A rectangle has congruent diagonals.” If a figure is a rectangle, then it has congruent diagonals. Write the inverse, converse, and contrapositive of the conditional statement “If m∠1 = 35°, then ∠1 is acute.” Find the truth value of each. Converse: If ∠1 is acute, then m∠1 = 35°. Not all acute angles measure 35°, so this is false. Inverse: If m∠1 ≠ 35°, then ∠1 is not acute. You can draw an acute angle that does not measure 35°, so this is false. Contrapositive: If ∠1 is not acute, then m∠1 ≠ 35°. An angle that measures 35° must be acute. So this statement is true.
EXERCISES Write a conditional statement from each Venn diagram. 15. 16. ��������
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Determine if each conditional is true. If false, give a counterexample. 17. If two angles are adjacent, then they have a common ray. 18. If you multiply two irrational numbers, the product is irrational. Write the converse, inverse, and contrapositive of each conditional statement. Find the truth value of each. 19. If ∠X is a right angle, then m∠X = 90°. 20. If x is a whole number, then x = 2.
2-3 Using Deductive Reasoning to Verify Conjectures (pp. 88–93) EXAMPLES ■
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Determine if the conjecture is valid by the Law of Detachment or the Law of Syllogism. Given: If 5c = 8y, then 2w = -15. If 5c = 8y, then x = 17. Conjecture: If 2w = -15, then x = 17. Let p be 5c = 8y, q be 2w = -15, and r be x = 17. Using symbols, the given information is written as p → q and p → r. Neither the Law of Detachment nor the Law of Syllogism can be applied. The conjecture is not valid. Draw a conclusion from the given information. Given: If two points are distinct, then there is one line through them. A and B are distinct points. Let p be the hypothesis: two points are distinct. Let q be the conclusion: there is one line through the points. The statement “A and B are distinct points” matches the hypothesis, so you can conclude that there is one line through A and B.
EXERCISES Use the true statements below to determine whether each conclusion is true or false. Sue is a member of the swim team. When the team practices, Sue swims. The team begins practice when the pool opens. The pool opens at 8 A.M. on weekdays and at 12 noon on Saturday. 21. The swim team practices on weekdays only. 22. Sue swims on Saturdays. 23. Swim team practice starts at the same time every day. Use the following information for Exercises 24–26. The expression 2.15 + 0.07x gives the cost of a long-distance phone call, where x is the number of minutes after the first minute. If possible, draw a conclusion from the given information. If not possible, explain why. 24. The cost of Sara’s long-distance call is $2.57. 25. Paulo makes a long-distance call that lasts ten minutes. 26. Asa’s long-distance phone bill for the month is $19.05. Study Guide: Review
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