Question

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

Explanation:

Step1: Find angle - measures

In right - triangle $MNL$, $\angle M = 58^{\circ}$ and $\angle MNL=90^{\circ}$. Using the angle - sum property of a triangle ($\angle M+\angle NML+\angle MLN = 180^{\circ}$), we find $\angle MLN=180^{\circ}-90^{\circ}-58^{\circ}=32^{\circ}$. In right - triangle $QNL$, $\angle QLN = 32^{\circ}$ and $\angle QNL = 90^{\circ}$. Also, $LN$ is common to both triangles.

Step2: Apply congruence criteria

We have $\angle MNL=\angle QNL = 90^{\circ}$, $LN = LN$ (common side), $\angle MLN=\angle QLN = 32^{\circ}$ (ASA - Angle - Side - Angle) or we can also consider $\angle M=\angle Q$ (since in right - triangles with one equal non - right angle, the other non - right angles are equal), $\angle MLN=\angle QLN$, $LN = LN$ (AAS - Angle - Angle - Side).

Answer:

Yes, they are congruent by either ASA or AAS.