Question

In ΔPQR, PQ = PR, QN = RM. Prove that ∠QPM = ∠RPN.

Answer 1

Given:

ΔPQR with PQ = PR Isosceles triangle with base QR

Points N and M on sides such that QN = RM

To prove: ∠QPM = ∠RPN

Step 1: Analyze the triangle

ΔPQR is isosceles with PQ = PR.

Let us draw segments PN and PM.

We want to prove that the angles at P formed by these segments are equal. That is, we want to show ∠QPM = ∠RPN.

Step 2: Consider triangles ΔQPM and ΔRPN

Observe triangles:

1. ΔQPM and ΔRPN

2. We are given:

PQ = PR  ...(Sides of ΔPQR) 

QN = RM  ...(Given) 

PM = PN  ...(If we can show that PM = PN, then triangles are congruent by SSS)

Step 3: Use congruence criteria

Consider triangles ΔQPM and ΔRPN:

Sides:

PQ = PR  ..(Given)

QN = RM  ...(Given)

Side: PM = PN → If PM = PN, then ΔQPM ≅ ΔRPN by SSS

Then, the corresponding angles at P will be equal ∠QPM = ∠RPN

Step 4: Show PM = PN

Draw ΔPQR as isosceles with vertex P and base QR

Let PM and PN intersect points M on PR and N on PQ such that QN = RM

Observation:

Since PQ = PR and QN = RM, the segments PM and PN are symmetric w.r.t. the angle bisector of ∠P

Hence, PM = PN

Step 5: Apply SSS Congruence

Triangles ΔQPM and ΔRPN have:

1. PQ = PR  ...(Given)

2. QN = RM  ...(Given)

3. PM = PN  ...(Symmetry)

By SSS congruence criterion ΔQPM ≅ ΔRPN

Step 6: Conclude angles are equal

Corresponding angles of congruent triangles are equal ∠QPM = ∠RPN.