In the adjoining figure, $ABC$ and $DEF$ are straight lines. Prove that $AD$ is parallel to $CF$.

In the adjoining figure, $ABC$ is an isosceles triangle in which $AB=AC$ and circle passing through $B$ and $C$ intersects sides $AB$ and $AC$ at points $D$ and $E$. Prove that $DE\left|\right|BC$.

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 $\angle \mathrm{ADC}.$