Lines r and s are cut by

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. Exs. 3, 4 In Example 2 and in all indirect proofs, the first statement takes the form “Suppose that . . .” or “Assume that . . .” By its very nature, such a statement cannot be supported even though every other state- ment in the proof can be justified; thus, when a contradiction is reached, the finger of blame points to the supposition. Having reached a contradiction, we may say that the claim involving has failed and is false; thus, our only recourse is to conclude that Q is true. Following is an outline of this technique. ~Q When the bubble displayed on the level is not centered, the board used in construction is neither vertical nor horizontal. Like the Law of Detachment from Section 1.1, the Law of Negative Inference Law of Contraposition is a form of deduction. Whereas the Law of Detachment characterizes the method of “direct proof ” found in preceding sections, the Law of Negative Infer- ence characterizes the method of proof known as indirect proof. INDIRECT PROOF You will need to know when to use the indirect method of proof. Often the theorem to be proved has the form , in which Q is a negation and denies some claim. For instance, an indirect proof might be best if Q reads in one of these ways: c is not equal to d 艎 is not perpendicular to m However, we will see in Example 4 of this section that the indirect method can be used to prove that line 艎 is parallel to line m. Indirect proof is also used for proving ex- istence and uniqueness theorems; see Example 5. The method of indirect proof is illustrated in Example 2. All indirect proofs in this book are given in paragraph form as are some of the direct proofs. In any paragraph proof, each statement must still be justified. Because of the need to order your statements properly, writing this type of proof may have a positive impact on the essays you write for your other classes P : Q EXAMPLE 2 GIVEN: In Figure 2.12, is not perpendicular to PROVE: and are not complementary PROOF: Suppose that and are complementary. Then because the sum of the measures of two complementary is 90. We also know that , by the Angle- Addition Postulate. In turn, by substitution. Then is a right angle. In turn, . But this contradicts the given hypothesis; therefore, the supposition must be false, and it follows that and are not complementary. 쮿 ∠2 ∠1 BA ⬜ BD ∠ABD m ∠ABD = 90° m ∠1 + m∠2 = m∠ABD ∠s m ∠1 + m∠2 = 90° ∠2 ∠1 ∠2 ∠1 BD BA 1 2 A C B D Figure 2.12 Geometry in the Real World