Prove that the perpendicular at the point of contact of the tangent to a circle passes through the centre.

In the given figure, two tangents $\mathrm{RQ}$, and $\mathrm{RP}$ are drawn from an external point $\mathrm{R}$to the circle with centre $\mathrm{O}$. If $\angle \mathrm{PRQ}=120°$ , then prove that$\mathrm{OR}=\mathrm{PR}+\mathrm{RQ}$.

In the given figure, $\mathrm{O}$ is the centre of the circle. $\mathrm{PA}$ and $\mathrm{PB}$ are tangents. Show that $\mathrm{AOBP}$ is cyclic quadrilateral.

In the given figure, $PQ$ is chord of a circle with centre $O$and $PT$ is a tangent. If
$\angle QPT=60°$, find the $\angle PRQ$.

In the given figure, $\mathrm{PA}$ and $\mathrm{PB}$ are two tangents to the circle with centre $\mathrm{O}$. If $\angle \mathrm{APB}=60°$, then find the measure of $\angle \mathrm{OAB}$.