1976 AHSME Problem 16

In triangles ABC and DEF, lengths AC,~BC,~DF, and EF are all equal. Length AB is twice the length of the altitude of \triangle DEF from F to DE. Which of the following statements is (are) true?

\begin{array}{l}\textbf{I. }\angle ACB \text{ and }\angle DFE\text{ must be complementary.}\\ \textbf{II. }\angle ACB \text{ and }\angle DFE\text{ must be supplementary.}\\ \textbf{III. }\text{The area of }\triangle ABC\text{ must equal the area of }\triangle DEF.\\ \textbf{IV. }\text{The area of }\triangle ABC\text{ must equal twice the area of }\triangle DEF.\end{array}

\textbf{(A) }\textbf{II. }\text{only}\qquad \textbf{(B) }\textbf{III. }\text{only}\qquad \textbf{(C) }\textbf{IV. }\text{only}\qquad \textbf{(D) }\text{I. }\text{and }\textbf{III. }\text{only}\qquad \textbf{(E) }\textbf{II. }\text{and }\textbf{III. }\text{only}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.