Question

are the triangles congruent? why or why not?
yes, all the angles of each of the triangles are acute.
yes, they are congruent by either asa or aas.
no, ∠b is not congruent to ∠q.
no, the congruent sides do not correspond.

Explanation:

Step1: Calculate angle \(C\) in \(\triangle ABC\)

The sum of angles in a triangle is \(180^{\circ}\). So, \(\angle C=180^{\circ}-(42^{\circ} + 53^{\circ})=85^{\circ}\)

Step2: Analyze congruence criteria

In \(\triangle ABC\) and \(\triangle RMQ\), \(\angle A=\angle R = 42^{\circ}\), \(\angle C=\angle Q=85^{\circ}\), and the side between \(\angle A\) and \(\angle C\) (in \(\triangle ABC\)) and the side between \(\angle R\) and \(\angle Q\) (in \(\triangle RMQ\)) are congruent (marked as equal). By ASA (Angle - Side - Angle) criterion (two angles and the included side are congruent) or AAS (Angle - Angle - Side, since if two angles are congruent, the third angle is also congruent as sum of angles in a triangle is \(180^{\circ}\) and we can use non - included side) the triangles are congruent.

Answer:

Yes, they are congruent by either ASA or AAS.