If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

If a line intersects two concentric circles (circles with the same centre) with centre $O$ at $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D},$ prove that $\mathrm{A}\mathrm{B}=\mathrm{C}\mathrm{D}.$

In the given figure, $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are three points on a circle with a centre $\mathrm{O}$ such that $\mathrm{\angle }\mathrm{B}\mathrm{O}\mathrm{C}=30°$ and $\mathrm{\angle }\mathrm{A}\mathrm{O}\mathrm{B}=60°$. If $\mathrm{D}$ is a point on the circle other than the arc$\mathrm{A}\mathrm{B}\mathrm{C},$ find $\mathrm{\angle }\mathrm{A}\mathrm{D}\mathrm{C}.$

In the given figure, $\mathrm{\angle }\mathrm{P}\mathrm{Q}\mathrm{R}=100°,$ where $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ are points on a circle with centre $\mathrm{O}.$ Find $\mathrm{\angle }\mathrm{O}\mathrm{P}\mathrm{R}$.

In the given figure, $\mathrm{\angle }\mathrm{A}\mathrm{B}\mathrm{C}=69°, \mathrm{\angle }\mathrm{A}\mathrm{C}\mathrm{B}=31°$, find $\mathrm{\angle }\mathrm{B}\mathrm{D}\mathrm{C}.${write only numerical value without degrees symbol}

In the given figure, $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$ are four points on a circle. $\mathrm{A}\mathrm{C}$ and $\mathrm{B}\mathrm{D}$ intersect at a point $\mathrm{E}$ such that $\mathrm{\angle }\mathrm{B}\mathrm{E}\mathrm{C}=130°$ and $\mathrm{\angle }\mathrm{E}\mathrm{C}\mathrm{D}=20°.$ If $\mathrm{\angle }\mathrm{B}\mathrm{A}\mathrm{C}=\mathrm{k}°$, then find the value of $\mathrm{k}$.