Question

in triangles abc and def, angles b and e each have measure 27° and angles c and f each have measure 41°. which additional piece of information is sufficient to determine whether triangle abc is congruent to triangle def?
a) the measure of angle a
b) the length of side ab
c) the lengths of sides bc and ef
d) no additional information is necessary

Explanation:

Step1: Recall congruence criteria

For two triangles, we have ASA (angle - side - angle), AAS (angle - angle - side), SAS (side - angle - side), SSS (side - side - side) and HL (hypotenuse - leg for right - triangles) congruence criteria. Given $\angle B=\angle E = 27^{\circ}$ and $\angle C=\angle F=41^{\circ}$, we need a side - length between the known angles or corresponding to a known angle - pair.

Step2: Analyze each option

• Option A: The measure of $\angle A$ only gives the third - angle of $\triangle ABC$ (since the sum of angles in a triangle is $180^{\circ}$). But just knowing angles does not prove congruence (AAA is not a congruence criterion).
• Option B: The length of $AB$ is not between the known angles $\angle B$ and $\angle C$ and does not directly help in proving congruence with the given information about $\triangle DEF$.
• Option C: If we know the lengths of $BC$ and $EF$, we have the ASA (angle - side - angle) congruence criterion. $\angle B=\angle E$, $BC = EF$ and $\angle C=\angle F$. So, this information is sufficient to prove $\triangle ABC\cong\triangle DEF$.
• Option D: We need more information than just the two pairs of equal angles.

Answer:

C. The lengths of sides $BC$ and $EF$