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EXAMPLE 4

AB DE. BC EF. 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. Write an indirect proof of Theorem 5.14 . GIVEN :. AC > DF. PROVE:.

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EXAMPLE 4

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

  2. If m B = m E, then B E. So, ABC DEF by the SAS Congruence Postulate and AC =DF. If m B <m E, then AC < DFby the Hinge Theorem. Both conclusions contradict the given statement that AC > DF. So, the temporary assumption that m B > m Ecannot be true. This proves that m B > m E. EXAMPLE 4 Prove the Converse of the Hinge Theorem Case 1 Case 2

  3. 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 for Example 4 GUIDED PRACTICE 5. Write a temporary assumption you could make to prove the Hinge Theorem indirectly. What two cases does that assumption lead to? SOLUTION

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