Question

What additional information could be used to prove ΔABC ≅ ΔMQR using SAS? Check all that apply.

Answer 1

To prove that ΔABC ≅ ΔMQR using SAS, we show that two sides with the intersection angle are congruent.

From the diagram, it is shown that CA is congruent to RM.

From the first option, given that m∠A = 64° and AB = MQ = 31 cm, then we have CA = RM, AB = MQ, and CAB = RMQ (i.e. m∠A = m∠M = 64°).
This shows that the first option is correct.

From the second option, given that CB = MQ = 29 cm, then we have CA = RM, CB = MQ, but ACB is not congruent to RMQ.
Thus the second option in not correct.

From the third option, m∠Q = 56° and CB ≅ RQ, then we have CA = RM, CB = RQ, ACB = 60°, but we do not know the value of MRQ.
Thus the third option is not correct.

From the fourth option, m∠R = 60° and AB ≅ MQ, then we have CA = RM, AB = MQ, RMQ = 64°, but we do not know the value of CAB.
Thus the fourth option is not correct.

From the fifth option, AB = QR = 31 cm, then we have CA = RM, AB = QR, but we do not know the value of CAB or MRQ.
Thus, the fifth option is not correct.

Therefore, the additional information that could be used to prove ΔABC ≅ ΔMQR using SAS is m∠A = 64° and AB = MQ = 31 cm

Answer 2

Answer: 1) m∠A = 64° and AB = MQ = 31 cm

3) m∠Q = 56° and CB ≅ RQ

Step-by-step explanation:

By the SAS postulate of congruence, Two triangles are called congruent if any pair of corresponding sides and their included angles are congruent in both triangles.

1) If m∠A = 64° and AB = MQ = 31 cm is given,

Then we can write,

AB ≅ MQ

∠ A ≅ ∠ M

And,  AC ≅  RM

Where AB, and AC are corresponding to MQ and RM respectively and Angle A and angle M are included angles in triangles ABC and MQR.

⇒ Δ ABC ≅ Δ MQR

2) If CB = MQ = 29 cm is given,

Then We have,

CB ≅ MQ

and AC ≅ RM

But the included angles ∠C and ∠M of these corresponding sides are not congruent.

⇒ By second option we can not prove, triangles ABC and MQR are congruent.

3) If m∠Q = 56° and CB ≅ RQ is given,

Then, We have,

m∠R = 60°

⇒ ∠C ≅ ∠R

Where CB, and AC are corresponding to RQ and RM respectively and Angle C and angle R are included angles in triangles ABC and MQR.

⇒ Δ ABC ≅ Δ MQR

4) If m∠R = 60° and AB ≅ MQ  is given,

∠C ≅ ∠R

But, Angle C and R are not the included angle of congruent corresponding sides of triangles ABC and MQR.

⇒ we can not prove, triangles ABC and MQR are congruent.

5) If AB = QR = 31 cm is given,

Then there are not any pair of congruent angles in triangles ABC and MQR.

⇒ we can not prove triangle ABC and MQR are congruent.