Given: A line m and a

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 Exs. 5–7 To prove the statement or to complete the proof problem of the form Given: P Prove: Q by the indirect method, use the following steps:

1. Suppose that is true.

2. Reason from the supposition until you reach a contradiction. 3. Note that the supposition claiming that is true must be false and that Q must therefore be true. Step 3 completes the proof. ~Q ~Q P : Q STRATEGY FOR PROOF 왘 Method of Indirect Proof General Rule: The first statement of an indirect proof is generally “SupposeAssume the opposite of the Prove statement.” Illustration: See Example 3, which begins “Assume that .” 7 m STRATEGY FOR PROOF 왘 The First Line of an Indirect Proof The contradiction that is discovered in an indirect proof often has the form . Thus, the assumed statement has forced the conclusion , asserting that is true. Then the desired theorem the contrapositive of is also true. ~Q : ~P P : Q ~Q : ~P ~P ~Q ~P EXAMPLE 3 Complete a formal proof of the following theorem: If two lines are cut by a transversal so that corresponding angles are not congruent, then the two lines are not parallel. GIVEN: In Figure 2.13, 艎 and m are cut by transversal t PROVE: PROOF: Assume that . When these lines are cut by transversal t, the corresponding angles including and are congruent. But by hypothesis. Thus, the assumed statement, which claims that , must be false. It follows that . 7 m 7 m ∠1 ⬵ ∠5 ∠5 ∠1 7 m 7 m ∠1 ⬵ ∠5 The versatility of the indirect proof is shown in the final examples of this section. The indirect proofs preceding Example 4 contain a negation in the conclusion Prove; the proofs in the final illustrations use the indirect method to arrive at a positive conclusion. Exs. 8, 9 EXAMPLE 4 GIVEN: In Figure 2.14, plane T intersects parallel planes P and Q in lines 艎 and m, respectively PROVE: PROOF: Assume that 艎 is not parallel to m. Then 艎 and m intersect at some point A. But if so, point A must be on both planes P and Q, which means that planes P and Q intersect; but P and Q are parallel by hypothesis. Therefore, the assumption that 艎 and m are not parallel must be false, and it follows that . 쮿 7 m 7 m m 1 2 4 3 5 6 8 7 t Figure 2.13 T m P Q Figure 2.14 쮿