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.

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Solution

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 so,

∠ABC = 180 – ∠CDA = 180 – ( 180 – ∠PAD) = ∠PAD

And ∠DCB = 180 – ∠BAD = 180 – ( 180 – ∠PAD) = ∠PAD

⇒ ∠ABC = ∠DCB = ∠PAD = ∠PDA

That means AD || BC

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