Question

what additional information could be used to prove that the triangles are congruent using aas? select two options.

□ ∠c ≅ ∠q

□ ⎔{cb} ≅ ⎔{qm}

□ ac = 3.9 cm and rq = 3.9 cm

□ m∠c = 35° and m∠q = 35°

□ ab = 2.5 cm and mq = 2.5 cm

Explanation:

Step1: Recall AAS (Angle - Angle - Side) congruence criterion

AAS 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.

Step2: Analyze each option

• For \(\angle C\cong\angle Q\):

We already know \(\angle A=\angle R = 29^{\circ}\) and \(\angle B=\angle M=116^{\circ}\). If \(\angle C\cong\angle Q\), but we still don't have a non - included side.

• For \(\overline{CB}\cong\overline{QM}\):

We have \(\angle A=\angle R = 29^{\circ}\) and \(\angle B=\angle M = 116^{\circ}\). The side \(CB\) in \(\triangle ABC\) and \(QM\) in \(\triangle RMQ\) are non - included sides. By AAS, if \(\overline{CB}\cong\overline{QM}\), \(\triangle ABC\cong\triangle RMQ\).

• For \(AC = 3.9cm\) and \(RQ=3.9cm\):

\(AC\) is adjacent to \(\angle A\) and \(RQ\) is adjacent to \(\angle R\). They are not non - included sides for the pairs of angles \(\angle A,\angle B\) (in \(\triangle ABC\)) and \(\angle R,\angle M\) (in \(\triangle RMQ\)).

• For \(m\angle C = 35^{\circ}\) and \(m\angle Q=35^{\circ}\):

Since the sum of angles in a triangle is \(180^{\circ}\), in \(\triangle ABC\), \(\angle C=180-(29 + 116)=35^{\circ}\), in \(\triangle RMQ\), \(\angle Q=180-(29 + 116)=35^{\circ}\). But we still need a non - included side.

• For \(AB = 2.5cm\) and \(MQ=2.5cm\):

\(\angle A=\angle R = 29^{\circ}\), \(\angle B=\angle M = 116^{\circ}\), and \(AB\) (non - included side in \(\triangle ABC\)) and \(MQ\) (non - included side in \(\triangle RMQ\)). By AAS, if \(AB = MQ\), \(\triangle ABC\cong\triangle RMQ\).

Answer:

\(\overline{CB}\cong\overline{QM}\), \(AB = 2.5cm\) and \(MQ = 2.5cm\)