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Geometry and Trigonometry Difficulty: Hard

In triangles $A B C$ and $D E F$ , angles $B$ and $E$ each have measure $27 degree$ and angles $C$ and $F$ each have measure $41 degree$ . Which additional piece of information is sufficient to determine whether triangle $A B C$ is congruent to triangle $D E F$ ?

The measure of angle $A$

The length of side $A B$

The lengths of sides $B C$ and $E F$

No additional information is necessary.

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Explanation

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 $A B C$ and $D E F$ are similar, where side $B C$ corresponds to side $E F$ . 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 $A B C$ and $D E F$ are congruent, it is sufficient to determine whether sides $B C$ and $E F$ have equal length. Thus, the lengths of $B C$ and $E F$ are sufficient to determine whether triangle $A B C$ is congruent to triangle $D E F$ .

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 $A B C$ and $D E F$ are similar, but not whether they are congruent.