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Choice C is correct. Since angles $B$ and $E$ each have the same measure and angles $C$ and $F$ each have the same measure, triangles $ABC$ and $DEF$ are similar, where side $BC$ corresponds to side $EF$. To determine whether two similar triangles are congruent, it is sufficient to determine whether one pair of corresponding sides are congruent. Therefore, to determine whether triangles $ABC$ and $DEF$ are congruent, it is sufficient to determine whether sides $BC$ and $EF$ have equal length. Thus, the lengths of $BC$ and $EF$ are sufficient to determine whether triangle $ABC$ is congruent to triangle $DEF$.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect. The given information is sufficient to determine that triangles $ABC$ and $DEF$ are similar, but not whether they are congruent.