Question

Given: bisects ∠MRQ; ∠RMS ≅ ∠RQS

Triangles R M S and R Q S share side R S. Point N is on line R S. Lines are drawn from point M to point N and from point Q to point N. Angles M R S and S R Q are congruent.

Which relationship in the diagram is true?

ΔMNR ≅ ΔMNS by ASA
ΔRMS ≅ ΔRQS by AAS
ΔSNQ ≅ ΔSNM by SSS
ΔQNR ≅ ΔMNR by HL

Answer 1

Answer:

ΔRMS ≅ ΔRQS by AAS

Step-by-step explanation:

See the diagram attached.

Given that ∠ RMS = ∠ RQS and N is any point on RS and ∠ MRS = ∠ SRQ.

Therefore, between Δ RMS and Δ RQS, we have

(i) ∠ RMS = ∠ RQS {Given}

(ii) ∠ MRS = ∠ SRQ {Also given} and

(iii) RS is the common side.

So, by angle-angle-side i.e. AAS criteria we can write ΔRMS ≅ ΔRQS. (Answer)

Answer 2

Based on the information given regarding the triangle, the relationship in the diagram that is is true is B. ΔRMS ≅ ΔRQS by AAS

What is a congruent triangle?

It should be noted that a congruent triangle simply means when two triangles have the same sides and the same angles.

In this case, the information that can be depicted include:

• ∠ RMS = ∠ RQS

• ∠ MRS = ∠ SRQ

• RS is the common side.

Therefore, using the angle-angle-side, we can deduce that ΔRMS ≅ ΔRQS.

In conclusion, the correct option is B.

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