Question

in the diagram below of △alb and △rmq are right triangles, (overline{ae} cong overline{rq}), and (overline{le} cong overline{mq}).
△ael can be proven congruent to △lqm by
hl
aaa
ssa
aas

Explanation:

Step1: Identify Triangle Type

Both \(\triangle AEL\) and \(\triangle RQM\) are right triangles (right angles at \(A\) and \(R\) respectively).

Step2: Analyze Given Congruences

We know \(\overline{AE} \cong \overline{RQ}\) (one leg) and \(\overline{LE} \cong \overline{MQ}\) (the other leg).

Step3: Recall Congruence Theorems for Right Triangles

The Hypotenuse - Leg (HL) theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, the triangles are congruent. Also, for right triangles, if two legs are congruent, we can use HL (since in right triangles, the hypotenuse can be derived from the legs via Pythagoras, but more directly, HL applies when we have a leg and hypotenuse, but here we have two legs. Wait, actually, in right triangles, if we have two legs congruent, the triangles are congruent by SAS, but HL is a special case for right triangles. Wait, the given congruences: \(\overline{AE}\) and \(\overline{RQ}\) are legs, \(\overline{LE}\) and \(\overline{MQ}\) are legs? Wait, no, looking at the triangles: \(\triangle AEL\) has right angle at \(A\), so legs are \(AE\) and \(AL\), hypotenuse \(LE\). \(\triangle RQM\) has right angle at \(R\), legs are \(RQ\) and \(RM\), hypotenuse \(QM\). Wait, the problem says \(\overline{AE} \cong \overline{RQ}\) and \(\overline{LE} \cong \overline{MQ}\). So \(AE\) (leg of \(\triangle AEL\)) \(\cong RQ\) (leg of \(\triangle RQM\)), and \(LE\) (hypotenuse of \(\triangle AEL\)) \(\cong MQ\) (hypotenuse of \(\triangle RQM\)). So by HL (Hypotenuse - Leg) theorem, the two right triangles are congruent. The other options: AAA (angle - angle - angle) only shows similarity, not congruence. SSA is not a valid congruence criterion. AAS is for non - right triangles (or right triangles but with two angles and a non - included side). So the correct criterion is HL.

Answer:

HL (The option corresponding to HL, which is the first option in the given choices: HL)