Question

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

Explanation:

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

For two triangles \(\triangle ABC\) and \(\triangle MQR\) to be congruent by SAS, we need two sides and the included angle of one triangle to be equal to the corresponding two sides and included angle of the other triangle.

Step2: Analyze each option

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

In \(\triangle ABC\), if \(m\angle A = 64^{\circ}\), and we know that in \(\triangle MQR\), \(\angle M = 64^{\circ}\). Also, if \(AB = MQ\) (given). If we assume \(AC = MR\) (the equal sides in the given triangles' markings for the first triangle \(AC\) and for the second triangle \(MR\) are equal as per the single - tick marks). Then \(\triangle ABC\cong\triangle MQR\) by SAS (\(AC = MR\), \(\angle A=\angle M = 64^{\circ}\), \(AB = MQ\))

• Option 2: \(CB = MQ = 29\mathrm{cm}\)

\(CB\) and \(MQ\) are not corresponding sides for an SAS congruence. The sides for SAS must be adjacent to the included angle.

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

We need to relate sides and angles such that the angle is between two sides. Here, we don't have the proper side - angle - side relationship.

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

If \(m\angle R=60^{\circ}\), and we know that in \(\triangle ABC\), \(\angle C = 60^{\circ}\). Also, if \(AB = MQ\) (given). If we assume \(AC = MR\) (the equal sides in the given triangles' markings for the first triangle \(AC\) and for the second triangle \(MR\) are equal as per the single - tick marks). Then \(\triangle ABC\cong\triangle MQR\) by SAS (\(AC = MR\), \(\angle C=\angle R = 60^{\circ}\), \(AB = MQ\))

• Option 5: \(AB = QR = 31\mathrm{cm}\)

\(AB\) and \(QR\) are not corresponding sides for an SAS congruence. The sides for SAS must be adjacent to the included angle.

Answer:

\(m\angle A = 64^{\circ}\) and \(AB = MQ = 31\mathrm{cm}\); \(m\angle R = 60^{\circ}\) and \(\overline{AB}\cong\overline{MQ}\)