Question

in addition to the facts in the diagram, which statement is necessary to prove that the two triangles are congruent by the sas criterion?
a. ac = de
b. m∠bac = m∠fde
c. bc = ef
d. m∠cab = m∠edf

Explanation:

Step1: Recall SAS congruence criterion

The SAS (Side - Angle - Side) criterion for triangle congruence states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent. In the given triangles, we already have one pair of sides equal (marked in the diagram) and one pair of angles equal (marked in the diagram). We need the other pair of sides that include the given equal - angles to be equal.

Step2: Analyze the options

For the SAS criterion, we need the sides that include the equal angles to be equal. The equal angles are $\angle ABC$ and $\angle DFE$. The sides that include $\angle ABC$ are $AB$ and $BC$, and the sides that include $\angle DFE$ are $DF$ and $EF$. So we need $BC = EF$.

Answer:

C. $BC = EF$