In the figure below, points $A, B, C$and $D$ lies on a circle. If $\angle ABC=69°$ and $\angle ACB=31°$, find $\angle BDC$ in degrees

In the figure below, chords $AB$ and $CD$ of a circle intersects at $P$. If $\angle CPB=120°$ and $\angle ACP=70°$, then find the value of $x$ in degrees.

In the figure below $O$ is the centre of the circle. If $\angle AOC=110°$, find $\angle ABC$ in degrees

In the figure below, $C$ is a point on the minor arc $AB$ of a circle with center $O$. If $\angle ACB=x°$ and $\angle AOB=y°$, then express $y$ in terms of $x$. Find $x$ if $OACB$ is a parallelogram.

In the figure below, $O$ is the centre of the circle. If $\angle AOC=130°$, find the value of $x$ in degrees.

In the figure below, $AB$ is the diameter of the circle. If $\angle ACD=35°$, find the value of $x$ in degrees

In the adjoining figure, two circles with centres $O$, $C$ intersect at points $A,B$. The centre $C$ of the smaller circle lies on the circumference of the other circle. If $\angle \mathrm{AOB}=80°$, find $\angle \mathrm{APB}.$

$O$ is the circumcentre of the triangle $ABC$ and $D$ is mid-point of the base $BC$. Prove that $\angle BOD=\angle \mathrm{A}.$

In the adjoining figure, $AB$ and $CD$ are equal chords. $AD$ and $BC$ intersect at $E$. Prove that $\mathrm{AE}=\mathrm{CE}$ and $\mathrm{BE}=\mathrm{DE}$