Question

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

Explanation:

Step1: Calculate \(\angle C\)

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 \(\triangle RMQ\), \(\angle A=\angle R = 42^{\circ}\), \(AB = RM\) (marked as equal), \(\angle C=\angle Q = 85^{\circ}\).
For ASA (Angle - Side - Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. Here, if we consider the side \(AB\) (in \(\triangle ABC\)) and \(RM\) (in \(\triangle RMQ\)) as the included sides between the pairs of angles \((\angle A,\angle C)\) and \((\angle R,\angle Q)\) respectively.
For AAS (Angle - Angle - Side): If two angles and a non - included side of one triangle are equal to two angles and the corresponding non - included side of another triangle, the triangles are congruent. Since we have two pairs of equal angles and a pair of equal sides.

Answer:

Yes, they are congruent by either ASA or AAS.