Question

In the square PQRS, equilateral ΔOPQ is drawn. Prove that ΔOPS ≅ ΔOQR.

Answer 1

Given:

PQRS is a square.

ΔOPQ is equilateral.

To Prove:

ΔOPS ≅ ΔOQR.

Proof:

1. Since PQRS is a square.

All sides are equal.

i.e., PQ = QR = RS = SP.

2. ΔOPQ is equilateral 

⇒ OP = PQ = OQ

3. Consider ΔOPS and ΔOQR.

We need to show they are congruent.

4. In ΔOPS and ΔOQR:

PS = QR   ...(Since PS and QR are sides of the square, they are equal)

OP = OQ   ...(Since ΔOPQ is equilateral, OP = OQ)

∠OPS = ∠OQR   ...(Angles are equal because of the properties of the square and the equilateral triangle sharing vertex O)

5. By the SAS (Side-Angle-Side) congruence criterion. 

ΔOPS ≅ ΔOQR

Hence proved.