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.

EXAMPLE 4 Prove the Converse of the Hinge Theorem Write an indirect proof of Theorem 5.14. GIVEN : AB DE BC EF AC > DF PROVE: m B > m E Proof : Assume temporarily that m B > m E. Then, it follows that either m B = m E or m B < m E.

EXAMPLE 4 Prove the Converse of the Hinge Theorem Case 1 If m B = m E, then B E. So, ABC DEF by the SAS Congruence Postulate and AC =DF. Case 2 If m B < m E, then AC < DF by the Hinge Theorem. Both conclusions contradict the given statement that AC > DF. So, the temporary assumption that m B > m E cannot be true. This proves that m B > m E.

Daily Homework Quiz 3. Suppose you want to write an indirect proof of this statement: “In ABC, if m A > 90° then ABC is not a right triangle.” What temporary assumption should start your proof? Assume ABC is a right triangle. ANSWER