Question

triangle congruence: asa and aas
are the triangles congruent? why or why not?
diagrams of triangles abc and rqm with angle and side markings
options:
no, the congruent sides do not correspond.
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.

Explanation:

Step1: Analyze Triangle ABC

In triangle \( ABC \), we know \( \angle A = 42^\circ \), \( \angle B = 53^\circ \). Using the triangle angle - sum property (\( \angle A+\angle B+\angle C = 180^\circ \)), we calculate \( \angle C=180^\circ-(42^\circ + 53^\circ)=85^\circ \). Also, \( AB \) is marked as equal to \( RM \) (the marked sides).

Step2: Analyze Triangle RQM

In triangle \( RQM \), \( \angle R = 42^\circ \), \( \angle Q=85^\circ \), and \( RM \) is marked equal to \( AB \). The side \( RM \) is between \( \angle R \) and \( \angle Q \) in triangle \( RQM \), and the side \( AB \) is between \( \angle A \) and \( \angle B \) in triangle \( ABC \) (wait, no, actually in triangle \( ABC \), the side \( AB \) is between \( \angle A \) and \( \angle B \), and in triangle \( RQM \), the side \( RM \) is between \( \angle R \) and \( \angle M \)? Wait, no, let's re - check. Wait, in triangle \( ABC \): angles are \( \angle A = 42^\circ \), \( \angle B = 53^\circ \), \( \angle C = 85^\circ \), side \( AB \) is equal to \( RM \). In triangle \( RQM \): \( \angle R=42^\circ \), \( \angle Q = 85^\circ \), so \( \angle M=180^\circ-(42^\circ + 85^\circ)=53^\circ \). Now, we have \( \angle A=\angle R = 42^\circ \), \( AB = RM \) (marked), and \( \angle B=\angle M = 53^\circ \). So by ASA (Angle - Side - Angle) congruence criterion, triangle \( ABC\cong\triangle RQM \).

Step3: Evaluate the Options

• Option 1: "No, the congruent sides do not correspond." This is wrong because we have a corresponding side and angles.
• Option 2: "Yes, all the angles of each of the triangles are acute." Just because triangles are acute does not mean they are congruent.
• Option 3: "No, \( \angle B \) is not congruent to \( \angle Q \)." But \( \angle B = 53^\circ \), \( \angle Q=85^\circ \)? Wait, no, wait we made a mistake earlier. Wait, in triangle \( ABC \), \( \angle B = 53^\circ \), in triangle \( RQM \), \( \angle M=180-(42 + 85)=53^\circ \). So \( \angle B=\angle M = 53^\circ \), \( \angle A=\angle R = 42^\circ \), and \( AB = RM \). So the correct congruence: \( \angle A=\angle R \), \( AB = RM \), \( \angle B=\angle M \), so by ASA (since the side is between the two angles), the triangles are congruent by either ASA or AAS (because if we consider \( \angle A=\angle R \), \( \angle C=\angle Q = 85^\circ \), and \( AB = RM \), it can also be AAS).
• Option 4: "Yes, they are congruent by either ASA or AAS." This is correct.

Answer:

Yes, they are congruent by either ASA or AAS.