Question

which of the following relationships proves why $\triangle abe$ and $\triangle dbc$ are congruent?
hl
sss
asa
sas

Explanation:

To determine the congruence of \(\triangle ABE\) and \(\triangle DBC\), we analyze the given information: \(\overline{AE} \cong \overline{DC}\), \(\angle A \cong \angle D\), and \(\angle E \cong \angle C\). The ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. In \(\triangle ABE\) and \(\triangle DBC\), we have two pairs of congruent angles (\(\angle A \cong \angle D\), \(\angle E \cong \angle C\)) and the included side \(\overline{AE} \cong \overline{DC}\), which matches the ASA criterion. HL (Hypotenuse - Leg) is for right triangles, SSS (Side - Side - Side) requires three congruent sides, and SAS (Side - Angle - Side) requires two sides and the included angle; none of these match the given information as well as ASA.

Answer:

ASA (the option labeled "ASA")