5.2.1 Indirect Proof and Inequalities in One Triangle Key Objectives • Write indirect proofs. • Apply inequalities in one triangle. Key Terms • In an indirect proof , or proof by contradiction, the conclusion is assumed to be false. Theorems, Postulates, Corollaries, and Properties
• Theorem If two sides of a triangle are not congruent, then the larger angle is opposite of the longer side. • Theorem If two angles of a triangle are not congruent, then the longer side is opposite of the larger angle. • Triangle Inequality Theorem The sum of any two side lengths of a triangle is greater than the third side length.
Writing an Indirect Proof 1. Identify the conjecture to be proven. 2. Assume the opposite (the negative) of the conclusion is true.
3. Use direct reasoning to show that the assumption leads to a contradiction. 4. Conclude that, since the assumption is false, the original conjecture must be true. Example 1 Writing an Indirect Proof The first step in writing an indirect proof is to identify the conclusion of the conjecture to be proven and then to assume that the opposite of that conclusion is true. In other words, treat the negation of the conjecture’s conclusion as given information, but do not negate the conjecture’s hypothesis. Then, use direct reasoning to show a contradiction of either the given information (the conjecture’s hypothesis), a theorem, a postulate, or a definition.
In this example, the conclusion of the conjecture is that △ ABC is not a right triangle. So, the negation of the conclusion is that △ ABC is a right triangle. Assume that the negation is true. Use the negation of the conclusion along with the given information, that △ ABC is obtuse, to find a contradiction. Here, the Protractor Postulate is contradicted. Once the contradiction is found, the negation of the conclusion can be assumed to be false. If the statement “ △ ABC is a right triangle” is false, then the negation of that statement must be true. So, “ △ ABC is not a right triangle” must be true.
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