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Find the true statements

Source: 1976 AHSME Problem 16

May 18, 2014
geometryAMC

Problem Statement

In triangles ABCABC and DEFDEF, lengths AC, BC, DF,AC,~BC,~DF, and EFEF are all equal. Length ABAB is twice the length of the altitude of △DEF\triangle DEF from FF to DEDE. Which of the following statements is (are) true?
<spanclass=′latex−bold′>I.</span>∠ACB and ∠DFE must be complementary.<span class='latex-bold'>I. </span>\angle ACB \text{ and }\angle DFE\text{ must be complementary.}
<spanclass=′latex−bold′>II.</span>∠ACB and ∠DFE must be supplementary.<span class='latex-bold'>II. </span>\angle ACB \text{ and }\angle DFE\text{ must be supplementary.}
<spanclass=′latex−bold′>III.</span>The area of △ABC must equal the area of △DEF.{<span class='latex-bold'>III. </span>\text{The area of }\triangle ABC\text{ must equal the area of }\triangle DEF.}
<spanclass=′latex−bold′>IV.</span>The area of △ABC must equal twice the area of △DEF.{<span class='latex-bold'>IV. </span>\text{The area of }\triangle ABC\text{ must equal twice the area of }\triangle DEF.}
<spanclass=′latex−bold′>(A)</span><spanclass=′latex−bold′>II.</span>only<spanclass=′latex−bold′>(B)</span><spanclass=′latex−bold′>III.</span>only<span class='latex-bold'>(A) </span><span class='latex-bold'>II. </span>\text{only}\qquad<span class='latex-bold'>(B) </span><span class='latex-bold'>III. </span>\text{only}\qquad
<spanclass=′latex−bold′>(C)</span><spanclass=′latex−bold′>IV.</span>only<spanclass=′latex−bold′>(D)</span>I. and <spanclass=′latex−bold′>III.</span>only<spanclass=′latex−bold′>(E)</span><spanclass=′latex−bold′>II.</span>and <spanclass=′latex−bold′>III.</span>only<span class='latex-bold'>(C) </span><span class='latex-bold'>IV. </span>\text{only}\qquad<span class='latex-bold'>(D) </span>\text{I. }\text{and }<span class='latex-bold'>III. </span>\text{only}\qquad <span class='latex-bold'>(E) </span><span class='latex-bold'>II. </span>\text{and }<span class='latex-bold'>III. </span>\text{only}
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