Question

what additional information could be used to prove (\triangle abccong\triangle mqr) using sas? select two options.(mangle a = 64^{circ}) and (ab = mq = 31 cm)(cb = mq = 29 cm)(mangle q = 56^{circ}) and (overline{cb}congoverline{rq})(mangle r = 60^{circ}) and (overline{ab}congoverline{mq})(ab = qr = 31 cm)

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

The SAS criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Analyze each option

• Option 1: \(m\angle A = 64^{\circ}\) and \(AB = MQ=31\) cm

In \(\triangle ABC\) and \(\triangle MQR\), we know \(\angle A\) and \(AB\). But we don't know the relationship of the sides adjacent to \(\angle A\) and the corresponding sides in \(\triangle MQR\) to form the SAS condition.

• Option 2: \(CB = MQ = 29\) cm

Just knowing one side is equal (and no information about the included angles) is not sufficient for SAS.

• Option 3: \(m\angle Q=56^{\circ}\) and \(\overline{CB}\cong\overline{RQ}\)

If \(m\angle Q = 56^{\circ}\), in \(\triangle ABC\), using the angle - sum property of a triangle (\(180-(60 + 64)=56^{\circ}\)). If \(\angle B=\angle Q = 56^{\circ}\) (since in \(\triangle ABC\), \(\angle B=180-(60 + 64)=56^{\circ}\)) and \(\overline{CB}\cong\overline{RQ}\), and we already know from the given angles \(\angle A=\angle M = 64^{\circ}\). But we need to check the side - angle - side. If \(\angle B=\angle Q\), \(CB = RQ\) and \(AB = MQ\) (by some relation). Wait, using the angle - sum property of a triangle:
In \(\triangle ABC\), \(\angle B=180-(60 + 64)=56^{\circ}\). If \(m\angle Q = 56^{\circ}\) and \(\overline{CB}\cong\overline{RQ}\), and we know \(\angle A=\angle M = 64^{\circ}\). If we assume \(AB = MQ\) (by the problem's need for SAS).

• Option 4: \(m\angle R = 60^{\circ}\) and \(\overline{AB}\cong\overline{MQ}\)

If \(m\angle R = 60^{\circ}\), then \(\angle C=\angle R = 60^{\circ}\). If \(\overline{AB}\cong\overline{MQ}\) and \(\angle A=\angle M = 64^{\circ}\), then by SAS (\(\angle A=\angle M\), \(AB = MQ\), \(\angle C=\angle R\)), \(\triangle ABC\cong\triangle MQR\)

• Option 5: \(AB = QR = 31\) cm

Just knowing one side is equal (and no information about the included angles) is not sufficient for SAS

Answer:

\(m\angle Q = 56^{\circ}\) and \(\overline{CB}\cong\overline{RQ}\), \(m\angle R = 60^{\circ}\) and \(\overline{AB}\cong\overline{MQ}\)