12. Prove that a cyclic parallelogram is a rectangle.

Given: PQRS is a parallelogram inscribed in a circle.

To prove: PQRS is a rectangle.

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.