Question

triangle congruence
is △mnl ≅ △qnl? why or why not?
no, ∠m is not congruent to ∠nlq.
yes, they are both right triangles.
yes, they are congruent by either asa or aas.
no, there are no congruent sides.

Explanation:

Step1: Find the measure of ∠MLN

In right - triangle \( \triangle MNL\), using the angle - sum property of a triangle (\(180^{\circ}\)).
Since \( \angle MNL = 90^{\circ}\) and \( \angle M=68^{\circ}\), then \( \angle MLN=180^{\circ}-\angle M - \angle MNL=180^{\circ}-68^{\circ}-90^{\circ}=22^{\circ}\).

Step2: Find the measure of ∠Q

In right - triangle \( \triangle QNL\), using the angle - sum property of a triangle (\(180^{\circ}\)).
Since \( \angle QNL = 90^{\circ}\) and \( \angle NLQ = 22^{\circ}\), then \( \angle Q=180^{\circ}-\angle NLQ-\angle QNL=180^{\circ}-22^{\circ}-90^{\circ}=68^{\circ}\).

Step3: Check congruence criteria

• \( \angle M=\angle Q = 68^{\circ}\)
• \( \angle MNL=\angle QNL = 90^{\circ}\)
• \( NL = NL\) (common side)

By AAS (Angle - Angle - Side) congruence criterion (two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle) or ASA (Angle - Side - Angle, since \( \angle MLN=\angle NLQ = 22^{\circ}\), \( NL = NL\), \( \angle MNL=\angle QNL = 90^{\circ}\)), \( \triangle MNL\cong\triangle QNL\).

Answer:

Yes, they are congruent by either ASA or AAS.