Question

PQRS is a quadrilateral, PR and QS intersect each other at O. In which of the following cases, PQRS is a parallelogram?

(a) ∠P = 100°, ∠Q = 80°, ∠R = 95°

(b) ∠P =85°, ∠Q = 85°, ∠R = 95°

(c) PQ = 7 cm, QR = 7 cm, RS = 8 cm, SP = 8 cm

(d) OP = 6.5 cm, OQ = 6.5 cm, OR = 5.2 cm, OS = 5.2 cm

Answer 1

Let us analyze each case one by one.

We have a quadrilateral named PQRS, with diagonals PR and QS intersecting at O.

(a) ,,

By angle sum property of a quadrilateral, we get:

Clearly,

And

Thus we have PQRS a quadrilateral with opposite angles are equal.

Therefore,

PQRS is a parallelogram.

(b) ,,

By angle sum property of a quadrilateral, we get:

Clearly,

And

Thus we have PQRS a quadrilateral with opposite angles are not equal.

Therefore,

PQRS is not a parallelogram.

(c) ,,,

Clearly,

And

Thus we have PQRS a quadrilateral with opposite sides are not equal

Therefore,

PQRS is not a parallelogram.

(c) ,,,

We know that the diagonals of a parallelogram bisect each other.

But, here we have

And

Therefore,

PQRS is not a parallelogram.

Hence, the correct choice is (a).