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:

1. First, analyze the triangles: Both \(\triangle MNL\) and \(\triangle QNL\) are right triangles (since \(\angle MNL=\angle QNL = 90^\circ\)).
2. For \(\triangle MNL\), we know \(\angle M = 58^\circ\), so \(\angle MLN=180^\circ - 90^\circ - 58^\circ=32^\circ\).
3. In \(\triangle QNL\), \(\angle NLQ = 32^\circ\), and \(NL\) is a common side (so \(NL = NL\) by reflexive property).
4. Now, check congruence criteria:

• By ASA: \(\angle MNL=\angle QNL = 90^\circ\), \(NL = NL\), \(\angle MLN=\angle QLN = 32^\circ\).
• By AAS: \(\angle M = 58^\circ\), \(\angle MNL=\angle QNL = 90^\circ\), \(NL = NL\).

1. The option "Yes, they are both right triangles" is incorrect because being right triangles alone doesn't prove congruence. The option "No, \(\angle M\) is not congruent to \(\angle NLQ\)" is wrong as \(\angle M = 58^\circ\) and \(\angle MLN = 32^\circ=\angle NLQ\), and we can relate angles. The option "No, there are no congruent sides" is wrong because \(NL\) is a congruent side. So the correct option is the first one.

Answer:

A. Yes, they are congruent by either ASA or AAS.