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.

GUIDED PRACTICE for Example 4 5. Write a temporary assumption you could make to prove the Hinge Theorem indirectly. What two cases does that assumption lead to? SOLUTION The third side of the first is less than or equal to the third side of the second; Case 1: Third side of the first equals the third side of the second. is less than the third side of the second. Case 2: Third side of the first