Question

ABCD is a cyclic quadrilateral in which AB and DC on being produced, meet at P such that PA = PD. Prove that AD is parallel to BC.

Answer 1

Let ABCD be the given cyclic quadrilateral

Also, PA = PD  ...(Given)

∴ ∠PAD = ∠PDA   ...(1)

∴ ∠BAD = 180° – ∠PAD

And ∠CDA = 180° – PDA

= 180° – ∠PAD   ...(From (1))

We know that the opposite angles of a cyclic quadrilateral are supplementary

∴ ∠ABC = 180° – ∠CDA

= 180° – (180° – ∠PAD)

= ∠PAD

And ∠DCB = 180° – ∠BAD

= 180° – (180° – ∠PAD)

= ∠PAD

∴ ∠ABC = ∠DCB = ∠PAD = ∠PAD

That means AD || BC