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Prove that a cyclic parallelogram is a rectangle.


Answer :

Given:
PQRS is a parallelogram inscribed in a circle.

To prove: PQRS is a rectangle.

image

Proof:
Since, PQRS is a cyclic quadrilateral.

Thus, \(\angle{P}\) + \(\angle{R}\) = \(180^\circ\) ...(i)
(Since, Sum of opposite angles in a cyclic quadrilateral is \(180^\circ\))

But, \(\angle{P}\) = \(\angle{R}\) ...(ii)
(Since, in a parallelogram, opposite angles are equal)

from eq. (i) and (ii), we get,

\(\angle{P}\) = \(\angle{R}\) = \(90^\circ\)

Similarly,
\(\angle{Q}\) = \(\angle{S}\) = \(90^\circ\)

Thus, Each angle of PQRS is \(90^\circ\).

Hence, it is proved that PQRS is a rectangle.

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