Geometry

Relationships in Triangles Chapter 5 Relationships in Triangles

Section 5-4 Indirect Proof Objective: Use indirect proof with algebra. Use indirect proof with geometry.

Section 5.3 Indirect Proof Until now the proofs we have written have been direct proofs. Sometimes it is difficult or even impossible to find a direct proof. In that case it may be possible to reason indirectly. Indirect reasoning is commonplace in every day life.

Indirect Proof With Algebra When using indirect reasoning, we assume that the conclusion is false and then show that this assumption leads to a contradiction of the hypothesis, or some other accepted fact, such as a definition, postulate, theorem, or corollary. Since all other steps in the proof are logical correct, the assumption has been proven false, so the original conclusion must be true. A proof of this type is called an indirect proof or a proof by contradiction.

Steps for writing an Indirect Proof 1. Assume that the conclusion is false. 2. Show that this assumption leads to a contradiction of the hypothesis, or some other fact, such as a definition, postulate, theorem, or corallary. 3. Point out that because the false conclusion leads to an incorrect statement, the original conclusion must be true.

Example: State the assumption you would make to start an indirect proof of each statement. 1.) If mA = 50, then mB = 40 Assume that mB ≠ 40

Example: State the assumption you would make to start an indirect proof of each statement. 2.) If EF = GH, then EF and GH aren’t parallel. Assume that EF and GH are parallel.

Example: State the assumption you would make to start an indirect proof of each statement. 2.) Given: AB || CD Prove: mA = mB Assume that mA ≠ mB.

Example: Given: n is an integer and n² is even Prove: n is even. 1.) Assume that n is odd. 2.) If n² = n x n, then odd x odd = odd. This contradicts the given information that n² is even. 3.) The assumption leads to the contradiction of the known fact that n² is even. Therefore, the assumption that n is odd must be false, which means that n is even must be true.

Example: Given: m1 ≠ m2 Prove: j || k 1.) Assume j || k. 2.) If j || k, then 1  2 because they are corresponding angles. Thus, m1 = m2. 3.) The assumption leads to the contradiction that m1 ≠ m2. Therefore, the assumption that j || k must be false. Therefore, j || k.

End of Section 5.4