Class Greeting

Objective: The students will write Indirect Proofs.

Vocabulary indirect proof

So far you have written proofs using direct reasoning. In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction.

When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or a theorem. Helpful Hint

Example 1: Writing an Indirect Proof Write an indirect proof that if a > 0, then Given: a > 0 (Identify the conjecture to be proven) Prove: (Assuming the opposite of the conclusion.) Assume (Reason logically to reach a contradiction.)     However, this contradicts the given information that a > 0; thus the assumption is false. Therefore, must be true.

Check It Out! Example 1 Write an indirect proof that a triangle cannot have two right angles. Step 1 Identify the conjecture to be proven. Given: A triangle. Prove: A triangle cannot have two right angles. Step 2 Assume the opposite of the conclusion. Assume a triangle has two right angles.

Check It Out! Example 1 Continued Step 3 Use direct reasoning to lead to a contradiction. m1 + m2 + m3 = 180° by the Triangle Sum Theorem So, 90° + 90° + m3 = 180° by Substitution. Then 180° + m3 = 180° by Simplifying and m3 = 0° by Subtraction. However, by the Protractor Postulate, a triangle cannot have an angle with a measure of 0°. Thus the assumption that a triangle can have two right angles is false. Therefore, a triangle cannot have two right angles is true.

Mini-Quiz Indirect Proof: Step 1 Assume that . Write an indirect proof. Step 1 Assume that . Given: Prove: Step 2 Substitute –2 for y in the equation. Substitution Multiply. Add. False Step 3 The assumption leads to a contradiction. Therefore, the assumption that must be false, which means that must be true. Example 3-2a

Lesson Summary: Objective: The students will write Indirect Proofs.

Preview of the Next Lesson: Objective: The students will apply the Triangle Inequality Theorem.

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