Question

10TH GRADE GEOMETRY PLS HELLLLLLLLP!

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


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

Answer 1

Answer: m∠A = 64° and AB = MQ = 31 cm

m∠Q = 56° and CB ≅ RQ

Explanation:

SAS postulate of congruence says that if two sides and the included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

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

Then in ΔABC and ΔMQR

AB ≅ MQ =31 cm

∠ A ≅ ∠ M= 64°

AC ≅ RM [given in picture]

⇒ Δ ABC ≅ Δ MQR [by SAS postulate]

• If CB = MQ = 29 cm

Then we don't have enough information to prove triangles congruent by SAS postulate.

• If m∠Q = 56° and CB ≅ RQ

Then, by angle sum property m∠R = 60°

⇒ ∠C ≅ ∠R

AC ≅ RM [given in picture]

⇒ Δ ABC ≅ Δ MQR [by SAS postulate]

• If m∠R = 60° and AB ≅ MQ is given,

⇒∠C ≅ ∠R

Then we don't have enough information to prove triangles congruent by SAS postulate.

• AB = QR = 31 cm

Then we don't have enough information to prove triangles congruent by SAS postulate.

Answer 2

Answer: m∠A = 64° and AB = MQ = 31 cm

m∠Q = 56° and CB ≅ RQ

Explanation:

According to SAS postulate of congruence, Triangles are congruent if any pair of corresponding sides and their included angles are congruent in both triangles.

First Option: 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.

Thus, Δ ABC ≅ Δ MQR

Second Option: 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.

Thus, by second option we can not prove, triangles ABC and MQR are congruent.

Third Option: 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.

Thus, Δ ABC ≅ Δ MQR

Fourth Option: 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.

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

Fifth Option: If AB = QR = 31 cm is given,

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

Thus, we can not prove triangle ABC and MQR are congruent.