Chapter Test

In the figure given below, $O$ is the centre of the circle. Prove that $\angle AOC=2\left(\angle ACB+\angle BAC\right)$.

In the figure given below, $O$ is the centre of the circle. Prove that $x+y=z$.

In the figure given below, sides $AB$ and $DC$ of a cyclic quadrilateral are produced to meet at a point $P$ and the sides $AD$ and $BC$ are produced to meet at a point $Q$. If $\angle ADC=75°$ and $\angle BPC=50°$, find $\angle BAD$ and $\angle CQD$.

In the figure given below, $AB$ and $XY$ are diameters of a circle with centre $O$. If $\angle APX=30°$, find $\angle OAX$.

In the figure given below, $AP$ and $BP$ are tangents to the circle with centre $O$. If $\angle CBP=25°$ and $\angle CAP=40°$, find $\angle ADB$.

In the figure given below, $AP$ and $BP$ are tangents to the circle with centre $O$. If $\angle CBP=25°$ and $\angle CAP=40°$, find $\angle AOB$.

In the figure given below, $AP$ and $BP$ are tangents to the circle with centre $O$. If $\angle CBP=25°$ and $\angle CAP=40°$, find $\angle ACB$.

In the figure given below, $AP$ and $BP$ are tangents to the circle with centre $O$. If $\angle CBP=25°$ and $\angle CAP=40°$, find $\angle APB$.