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 $y$.

In the adjoining figure, $ABCD$ is a cyclic quadrilateral in which $AB\left|\right|DC$. If $\angle A=80°$, then find $\angle B$.

Two congruent circles intersect at points $P$ and $Q$. A line through $P$ and perpendicular to $PQ$ meets the circles at $A$ and $B$. Show that $AQ=BQ$.

If a circle is divided into three equal parts, then each part of the circle is a _____ arc.