Question

In square PQRS, ∆OPQ is a equilateral triangle.
Prove that: ∆OPS =~ ∆OQR​

Answer 1

Answer:

Step-by-step explanation:

Given:

Quadrilateral PQRS is a square.

ΔOPQ is a equilateral triangle.

To prove:

ΔOPS≅ΔOQR

Proof:

1) Quadrilateral PQRS is a square..........given

∴ PQ=QR=RS=PS........Property of sides of a square

Let each side of a square be 'a' units

∴ PQ=QR=RS=PS=a

2) ΔOPQ is a equilateral triangle.....given

∴OP=OQ=PQ..........property of sides of an equilateral triangle.

3) From points 1) and 2),

It can be said that OP=OQ=PQ=a

4)  ΔOPQ is a equilateral triangle.....given

∴ ∠OPQ=∠OQP=∠POQ=60°.......property of angles of an equilateral triangle.

5) Quadrilateral PQRS is a square..........given

∴ ∠PQR=∠QRS=∠RSP=∠SPQ=90°.....property of angles of a square.

6) ∠SPQ=∠SPO+∠OPQ.........angle addition postulate.

From 4) and 5), ∠SPQ=90° and ∠OPQ=60°

∴ 90°= ∠SPO+60°

∠SPO= 90°-60°=30°

7) Similarly we can prove ∠RQO to be equal to 30°.

8) In the triangles ΔOPS and ΔOQR,

seg. PS=seg. QR......both equal to 'a'...from point 1)

seg. OP=seg. OQ.....both equal to 'a'...from point 2)

∠SPO=∠RQO....both equal to 30°....from points 6) and 7)

∴  ΔOPS≅ΔOQR......SAS test of congruence.

Hence proved.