Question

1. since ∠arw ≅ ∠mrs because vertical angles are congruent, what additional information would be needed to prove the triangles congruent by aas? ∠a ≅ ∠m aw ≅ ms ∠w ≅ ∠m

Explanation:

Step1: Recall AAS congruence

AAS (Angle - Angle - Side) congruence states that 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, then the two triangles are congruent.
We already have \( \angle ARW\cong\angle MRS\) (vertical angles). Also, \( AR = MR\) (marked in the figure).

Step2: Analyze the required angle

For AAS, we need another pair of angles.
If \( \angle W\cong\angle M\), then in \(\triangle ARW\) and \(\triangle MRS\), we have \( \angle ARW\cong\angle MRS\), \( \angle W\cong\angle M\) (two angles) and \( AR = MR\) (a non - included side).

Answer:

\(\angle W\cong\angle M\)