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: Find angle \(C\) in \(\triangle ABC\)

Using the angle - sum property of a triangle (\(\angle A+\angle B+\angle C = 180^{\circ}\)). Given \(\angle A = 42^{\circ}\) and \(\angle B=53^{\circ}\), then \(\angle C=180^{\circ}-(42^{\circ} + 53^{\circ})=85^{\circ}\).

Step2: Analyze congruence criteria

In \(\triangle ABC\) and \(\triangle MRQ\), \(\angle A=\angle R = 42^{\circ}\), \(\angle C=\angle Q = 85^{\circ}\). Also, in \(\triangle MRQ\), the side between \(\angle R\) and \(\angle Q\) (by the markings) is congruent to the side between \(\angle A\) and \(\angle C\) (implied by the triangle structure).
For ASA (Angle - Side - Angle): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. For AAS (Angle - Angle - Side): If two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, the triangles are congruent. Here, we can use either ASA or AAS.

Answer:

Yes, they are congruent by either ASA or AAS.