1976 AMC 12 Question 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?

$\mathrm{I}.$ $\angle ACB$ and $\angle DFE$ must be complementary.

$\mathrm{II}.$ $\angle ACB$ and $\angle DFE$ must be supplementary.

$\mathrm{III}.$ The area of $\triangle ABC$ must equal the area of $\triangle DEF.$

$\mathrm{IV}.$ The area of $\triangle ABC$ must equal twice the area of $\triangle DEF.$