Question

what additional information could be used to prove \\( \triangle a b c \cong \triangle m q r \\) using sas? select two options \\( m \angle a = 64 ^ { \circ } \\) and \\( a b = m q = 31 \mathrm { cm } \\) \\( c b = m q = 29 \mathrm { cm } \\) \\( m \angle q = 56 ^ { \circ } \\) and \\( \overline { c b } \cong \overline { r q } \\) \\( m \angle r = 60 ^ { \circ } \\) and \\( \overline { a b } \cong \overline { m q } \\) \\( a b = q r = 31 \mathrm { 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 the included angle of the other triangle.

Step2: Analyze the first option (\(m\angle A = 64^{\circ}\) and \(AB = MQ = 31\space cm\))

In \(\triangle ABC\), if \(m\angle A=64^{\circ}\), \(AB = 31\space cm\) and in \(\triangle MQR\), \(MQ = 31\space cm\) and \(m\angle M = 64^{\circ}\). Also, assume \(AC = MR\) (given by the single tick marks in the figure). The included angles \(\angle A\) and \(\angle M\) with sides \(AB - AC\) and \(MQ - MR\) respectively satisfy the SAS criterion.

Step3: Analyze the second option (\(m\angle R = 60^{\circ}\) and \(\overline{AB}\cong\overline{MQ}\))

If \(m\angle R = 60^{\circ}\), and \(\overline{AB}\cong\overline{MQ}\), and since \(AC = MR\) (from the figure, single tick marks) and \(\angle C=\angle R = 60^{\circ}\) (if \(m\angle R = 60^{\circ}\)). The sides \(AC - AB\) and \(MR - MQ\) with included angles \(\angle C\) and \(\angle R\) satisfy the SAS criterion.

Step4: Analyze the other options

• For \(CB = MQ = 29\space cm\): There is no information about the included angles between the sides.
• For \(m\angle Q = 56^{\circ}\) and \(\overline{CB}\cong\overline{RQ}\): There is no proper side - angle - side correspondence.
• For \(AB = QR = 31\space cm\): There is no information about the included angles.

Answer:

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