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12. Prove that a cyclic parallelogram is a rectangle.
Answer :

Given: PQRS is a parallelogram inscribed in a circle.

To prove: PQRS is a rectangle.
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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.