Question

what additional information could be used to prove δabc ≅ δmqr using sas? select two options. m∠a = 64° and ab = mq = 31 cm cb = mq = 29 cm m∠q = 56° and cb ≅ rq m∠r = 60° and ab ≅ mq ab = qr = 31 cm

Explanation:

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

SAS requires two pairs of corresponding sides to be congruent and the included angles to be congruent.

Step2: Analyze each option

• For the first option: $m\angle A = 64^{\circ}$ and $AB = MQ=31$ cm. But we don't know the included - side relationship for the angles, so this is not enough.
• For the second option: $CB = MQ = 29$ cm. This gives one pair of sides, but no information about the included angles.
• For the third option: If $m\angle Q = 56^{\circ}$, in $\triangle MQR$, since $m\angle M=64^{\circ}$, then $m\angle R=180-(64 + 56)=60^{\circ}$. And $CB\cong RQ$. We have $\angle C = 60^{\circ}$, $CB\cong RQ$ and $\angle R = 60^{\circ}$, which satisfies SAS.
• For the fourth option: If $m\angle R = 60^{\circ}$ and $AB\cong MQ$. We have $\angle C = 60^{\circ}$, $AB\cong MQ$ and the included angles ($\angle C$ and $\angle R$) are equal, which satisfies SAS.
• For the fifth option: $AB = QR = 31$ cm gives one pair of sides, but no information about the included angles.

Answer:

$m\angle Q = 56^{\circ}$ and $\overline{CB}\cong\overline{RQ}$; $m\angle R = 60^{\circ}$ and $\overline{AB}\cong\overline{MQ}$