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Q 12. Prove that a cyclic parallelogram is a rectangle.

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Given: ABCD is a cyclic quadrilateral.

To prove: ABCD is a rectangle.

Proof :

In cyclic quadrilateral ABCD.
$\angle A+\angle C=180^{\circ}$------1(sum of either pair of opposite angles of a cyclic quadrilateral)
$\angle A=\angle C$------2(opposite angles of a parallelogram are equal )
From 1 and 2,
$\angle A+\angle A=180^{\circ}$
$\Rightarrow 2 \angle A=180^{\circ}$
$\Rightarrow \angle A=90^{\circ}$
We know that a parallelogram with one angle right angle is a rectangle.
Hence, $A B C D$ is a rectangle.

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