Question

is $\triangle mnl \cong \triangle qnl$? why or why not?
(options:
yes, they are congruent by either asa or aas.
yes, they are both right triangles.
no, $\angle m$ is not congruent to $\angle nlq$
no, there are no congruent sides.)

Explanation:

Step1: Analyze Triangle Angles

In $\triangle MNL$, $\angle M = 58^\circ$, $\angle MNL = 90^\circ$, so $\angle MLN = 180^\circ - 58^\circ - 90^\circ = 32^\circ$. In $\triangle QNL$, $\angle QNL = 90^\circ$, $\angle NLQ = 32^\circ$ (given), so $\angle Q = 180^\circ - 90^\circ - 32^\circ = 58^\circ$.

Step2: Check Congruence Conditions

• $\angle MNL = \angle QNL = 90^\circ$ (right angles).
• $NL$ is common to both triangles.
• $\angle MLN = \angle QLN = 32^\circ$ (or $\angle M = \angle Q = 58^\circ$). So by ASA (angle - side - angle) or AAS (angle - angle - side), $\triangle MNL \cong \triangle QNL$.

Answer:

Yes, they are congruent by either ASA or AAS.