In the following figure, $\mathrm{ABCD}$ is a parallelogram. A circle through vertices $\mathrm{A}$ and $\mathrm{B}$ meets side $\mathrm{BC}$ at point $\mathrm{P}$ and side $\mathrm{AD}$ at point $\mathrm{Q}$ Show that quadrilateral $\mathrm{PCDQ}$is cyclic.

Prove that:

The parallelogram, inscribed in a circle, is a rectangle.

Prove that:

The rhombus, inscribed in a circle, is a square.

In the given figure, $\mathrm{AB}=\mathrm{AC} .$ Prove that $\mathrm{DECB}$is an isosceles trapezium.

The figure given below, shows a circle with centre O.
Given : $\angle \mathrm{AOC}=\mathrm{a}$ and $\angle \mathrm{ABC}=\mathrm{b}$

Find the relationship between $\mathrm{a}$and $\mathrm{b}.$

The figure given below, shows a circle with centre $\mathrm{O}.$

Find the measure of angle $\mathrm{OAB}$, if $\mathrm{OABC}$ is a parallelogram.

In the given figure, $\mathrm{RS}$ is a diameter of the circle. $\mathrm{NM}$is parallel to $\mathrm{RS}$and $\angle \mathrm{MRS}=29°.$ Calculate :

$\angle \mathrm{RNM},$