If $PQ$ is a chord of a circle with radius $r$ units and $R$ is a point on the circle such that $\angle \mathrm{PRQ}=90°$, then find the length of $PQ$.

In the adjoining figure, $O$ is the centre of a circle. If the length of chord $PQ$ is equal to the radius of the circle, then find $\angle PRQ.$

In the adjoining figure, $O$ is the centre of a circle and $A,B$ and $C$ are points on the circle. If $\angle \mathrm{AOB}=100°$ and $\angle \mathrm{BOC}=120°$, then find $\angle ABC$

In the adjoining figure, $O$ is the centre of a circle. Find the value of $x$.

In the adjoining figure, $ABCD$ is a quadrilateral inscribed in a circle. If $\angle DAC=40°$ and $\angle BDC=45°$, then find $\angle BCD.$

In the adjoining figure,  chords $AB$ and $CD$ of a circle when produced meet at $P$. If $\angle \mathrm{APD}=35°$ and $\angle BCD=25°$, then find $\angle \mathrm{ADC}.$

In the adjoining figure, $AB$ and $CD$ are diameter of a circle with centre $O$. If $\angle \mathrm{OBD}=35°$, then find the value of $x$.

In the adjoining figure, $ABCD$ is a cyclic quadrilateral. If the angles are as given in the figure, then find the values of $x$ and $y$.