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:

1. First, analyze triangle \(ABC\): In \(\triangle ABC\), we know \(\angle A = 42^\circ\), \(\angle B = 53^\circ\). The sum of angles in a triangle is \(180^\circ\), so \(\angle C=180^\circ - 42^\circ - 53^\circ = 85^\circ\). Also, \(AB\) has a mid - segment mark, indicating some side - length relationship.
2. Then, analyze triangle \(MRQ\): In \(\triangle MRQ\), \(\angle R = 42^\circ\), \(\angle Q = 85^\circ\). The side \(MR\) has a mid - segment mark (same as \(AB\)'s mark, implying \(AB = MR\) or corresponding sides are equal).
3. Now, check congruence criteria:

• For ASA (Angle - Side - Angle): In \(\triangle ABC\) and \(\triangle MRQ\), \(\angle A=\angle R = 42^\circ\), \(AB = MR\) (corresponding sides), \(\angle C=\angle Q = 85^\circ\). So ASA can be applied.
• For AAS (Angle - Angle - Side): We have two angles equal (\(\angle A=\angle R = 42^\circ\), \(\angle C=\angle Q = 85^\circ\)) and a non - included side (e.g., \(BC\) and \(RQ\) or \(AC\) and \(MQ\)) equal (since the sum of angles is equal and one side is equal), so AAS can also be applied.
• Option 1: Just because angles are acute does not mean triangles are congruent. There are many acute - angled non - congruent triangles. So this is wrong.
• Option 3: \(\angle B = 53^\circ\), \(\angle Q = 85^\circ\)? No, \(\angle Q = 85^\circ\) and \(\angle C = 85^\circ\), \(\angle B=53^\circ\), but we have \(\angle A=\angle R\), \(\angle C=\angle Q\) and corresponding sides. So this is wrong.
• Option 4: The congruent sides do correspond. The side with the mid - segment mark in \(AB\) and \(MR\) correspond, and the angles at the ends of these sides correspond. So this is wrong.

Answer:

Yes, they are congruent by either ASA or AAS.