Question

which of the following relationships proves why $\triangle bea$ and $\triangle dae$ are congruent?
hl
sss
asa
sas
$\overline{ab} \cong \overline{ed}$
$\angle bae \cong \angle dea$
$\overline{ae} \cong \overline{ae}$

Explanation:

To determine the congruence of \(\triangle BEA\) and \(\triangle DAE\), we analyze the given information:

• We know \(\overline{AB} \cong \overline{ED}\) (a side).
• \(\angle BAE \cong \angle DEA\) (the included angle between the sides).
• \(\overline{AE} \cong \overline{AE}\) (a common side, so it is congruent to itself).

The SAS (Side - Angle - Side) congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Here, we have two sides (\(\overline{AB}\) and \(\overline{AE}\) in \(\triangle BEA\); \(\overline{ED}\) and \(\overline{AE}\) in \(\triangle DAE\)) and the included angle (\(\angle BAE\) in \(\triangle BEA\); \(\angle DEA\) in \(\triangle DAE\)) that are congruent.

HL (Hypotenuse - Leg) is for right - angled triangles, SSS (Side - Side - Side) requires three pairs of congruent sides, and ASA (Angle - Side - Angle) requires two angles and the included side. Since we have two sides and the included angle, SAS is the correct criterion.

Answer:

D. SAS