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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. - Mathematics

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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.

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
∴ ∠ABC = 180° - ∠CDA = 180° - (180° - ∠PAD) = ∠PAD
And ∠DCB = 180° - ∠BAD = 180° - (180° - ∠PAD) = ∠PAD
∴ ∠ABC = ∠DCB = ∠PAD = ∠PAD
That means AD || BC

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APPEARS IN

Selina Solution for Concise Mathematics for Class 10 ICSE (2020 (Latest))
Chapter 17: Circles
Exercise 17(A) | Q: 28 | Page no. 260
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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. Concept: Cyclic Properties.
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