Does the relation “is parallel to” have a

24. Given: ,

, and Find: HINT: There is a line through C parallel to both and .

25. Given: and

Find: See “Hint” in Exercise 24.

26. Given: ,

See figure on page 79. Prove:

27. In triangle ABC, line t is

drawn through vertex A in such a way that . a Which pairs of are ? b What is the sum of , , and ? c What is the sum of measures of the of ? In Exercises 28 to 30, write a formal proof of each theorem.

28. If two parallel lines are cut by a transversal, then the

alternate exterior angles are congruent.

29. If two parallel lines are cut by a transversal, then the

exterior angles on the same side of the transversal are supplementary.

30. If a transversal is perpendicular to one of two parallel

lines, then it is also perpendicular to the other line. 䉭ABC ∠s m ∠5 m ∠4 m ∠1 ⬵ ∠s t 7 BC s ⬜ t r ⬜ t r 7 s m ∠ACD m ∠BAC + m∠CDE = 93° Í AB 7 Í DE Í DE Í AB m ∠ACD m ∠EDC = 54° m ∠BAC = 42° Í AB 7 Í DE Let represent the statement “If P, then Q.” The following statements are related to this conditional statement. NOTE: Recall that represents the negation of P. ~P P : Q

31. Suppose that two lines are cut by a transversal in such a

way that corresponding angles are not congruent. Can those two lines be parallel?

32. Given: Line 艎 and point P

not on 艎 Construct:

33. Given: Triangle ABC with

three acute angles Construct:

34. Given: Triangle MNQ with

obtuse Construct:

35. Given: Triangle MNQ with

obtuse Construct: HINT: Extend .

36. Given: A line m and a

point T not on m NQ MR ⬜ NQ ∠MNQ NE ⬜ MQ ∠MNQ BD ⬜ AC Í PQ ⬜ A D B E C Exercises 24, 25 1 t 2 3

5 4

A B C P B C A M Q N Exercises 34, 35 Conditional Converse Inverse Contrapositive Law of Negative Inference Indirect Proof Indirect Proof 2.2 KEY CONCEPTS m T Suppose that you do the following: i Construct a perpendicular line r from T to line m. ii Construct a line s perpendicular to line r at point T. What is the relationship between lines s and m? Conditional or Implication If P, then Q. Converse of Conditional If Q, then P. Inverse of Conditional If not P, then not Q. Contrapositive of Conditional If not Q, then not P. ~Q : ~P ~P : ~Q Q : P P : Q For example, consider the following conditional statement. If Tom lives in San Diego, then he lives in California. This true statement has these related statements: Converse: If Tom lives in California, then he lives in San Diego. false Inverse: If Tom does not live in San Diego, then he does not live in California. false Contrapositive: If Tom does not live in California, then he does not live in San Diego. true In general, the conditional statement and its contrapositive are either both true or both false Similarly, the converse and the inverse are also either both true or both false. See our textbook website for more information about the conditional and related statements. EXAMPLE 1 For the conditional statement that follows, give the converse, the inverse, and the contrapositive. Then classify each as true or false. If two angles are vertical angles, then they are congruent angles. Solution CONVERSE: If two angles are congruent angles, then they are vertical angles. false INVERSE: If two angles are not vertical angles, then they are not congruent angles. false CONTRAPOSITIVE: If two angles are not congruent angles, then they are not vertical angles. true 쮿 Venn Diagrams can be used to explain why the conditional statement and its contrapositive are equivalent. The relationship “If P, then Q” is repre- sented in Figure 2.11. Note that if any point is selected outside of , then it cannot possibly lie in set P. THE LAW OF NEGATIVE INFERENCE CONTRAPOSITION Consider the following circumstances, and accept each premise as true:

1. If Matt cleans his room, then he will go to the movie. 2. Matt does not get to go to the movie.

What can you conclude? You should have deduced that Matt did not clean his room; if he had, he would have gone to the movie. This “backdoor” reasoning is based on the fact that the truth of implies the truth of . ~Q : ~P P : Q ~Q P : Q Q that is, ~Q ~Q : ~P P : Q Exs. 1, 2 LAW OF NEGATIVE INFERENCE CONTRAPOSITION ‹ ~ P ~ Q P : Q P Q Figure 2.11 “If P, then Q” and “If not Q, then not P” are equivalent.