17・139 In $\triangle A B C$ and $\triangle D E F$, $A C, B C, D F$ and $E F$ are all equal, and $A B$ is twice the length of the altitude from $D$ to $E$ in $\triangle D E F$. Which of the following statements is/are true:
I. $\angle A C B$ and $\angle D F E$ must be complementary.
II. $\angle A C B$ and $\angle D F E$ must be supplementary.
III. The area of $\triangle A B C$ must be equal to the area of $\triangle D E F$.
N. The area of $\triangle A B C$ must be twice the area of $\triangle D E F$. The correct answer is
(A) only II.
(B) only III.
(C) only IV.
(D) only I and III.
(E) only II and III.
(22nd American High School Mathematics Examination, 1976)
[Solution] Let $CG$ and $FH$ be the altitudes on $AB$ and $DE$ respectively. Since $AG=HF$, and $AC=DF$, then Rt $\triangle AGC \cong \text{Rt} \triangle FHD$.
Therefore, $\angle ACB + \angle DFE = 2 \angle ACG + 2 \angle DFH = 180^{\circ}$,

thus
$$
S_{\triangle ABC} = 2 S_{\triangle ACG} = 2 S_{\triangle DFH} = S_{\triangle DEF}.
$$

Hence, the answer is $(E)$.