Question

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

Explanation:

Step1: Find third - angle of first triangle

In $\triangle ABC$, using angle - sum property of a triangle ($A + B+C=180^{\circ}$), if $A = 42^{\circ}$ and $B = 53^{\circ}$, then $C=180-(42 + 53)=85^{\circ}$.

Step2: Analyze congruence criteria

In $\triangle ABC$ and $\triangle RMQ$, $\angle A=\angle R = 42^{\circ}$, $\angle C=\angle M = 85^{\circ}$, and there is a pair of non - included sides (or included side if we consider the common side concept in the context of the congruence rules). So, they are congruent by either AAS (Angle - Angle - Side) or ASA (Angle - Side - Angle) as we have two pairs of equal angles and a pair of sides (either included or non - included depending on how we look at it).

Answer:

Yes, they are congruent by either ASA or AAS.