Indirect Proofs

Steps for Writing an Indirect Proof Identify the conjecture to be proven. Assume the opposite of the conclusion is true. Use direct reasoning to show that the assumption leads to a contradiction. Conclude that since the assumptions is false, the original conjecture must be true.

So far you have written proofs using direct reasoning So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. 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.

Example 1

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’s interior angles add up to 180°. Prove: A triangle cannot have two right angles. Step 2 Assume the opposite of the conclusion. An angle 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° 90° + 90° + m3 = 180° 180° + m3 = 180° m3 = 0° However, by the Protractor Postulate, a triangle cannot have an angle with a measure of 0°.

Check It Out! Example 1 Continued Step 4 Conclude that the original conjecture is true. The assumption that a triangle can have two right angles is false. Therefore a triangle cannot have two right angles.

Exit Ticket Write an indirect proof that if a > 0, then

Exit Ticket: Writing an Indirect Proof Write an indirect proof that if a > 0, then Step 1 Identify the conjecture to be proven. Given: a > 0 Prove: Step 2 Assume the opposite of the conclusion. Assume

Example 1 Continued Step 3 Use direct reasoning to lead to a contradiction. Given, opposite of conclusion Zero Prop. of Mult. Prop. of Inequality 1  0 Simplify. However, 1 > 0.

Example 1 Continued Step 4 Conclude that the original conjecture is true. The assumption that is false. Therefore

Examples 1. Write an indirect proof of the following: If Lauren spends more than $100 on two spring jackets, then at least one of the jackets cost more than $50. 2. You know the inhabitants of Jamais always lie, while the inhabitants of Toujours always tell the truth. You meet a man who comes from one of the villages. How can you find out what village he is from by asking him only one question? OBJ: SWBAT work collaboratively in order to solve a logical thinking puzzle.

Lies?? http://www.math.harvard.edu/~knill/mathmovies/swf/shrek3_lies.html