PQ is a tangent drawn from an external point P to a circle with center O and QOR is the diameter of the circle. If $\angle \mathrm{POR}$=120°, what is the measure of $\angle \mathrm{OPQ}$?

It is given that PQ is a tangent drawn from an external point P to a circle with centre O and QOR is the diameter of the circle. We need to find the measure of $\angle \mathrm{OPQ}$ if $\mathrm{\angle }\mathrm{POR}={120}^{\circ }$.

From $△\mathrm{OPQ}$, according to the exterior angle theorem,

$\mathrm{\angle }\mathrm{POR}=\mathrm{\angle }\mathrm{OPQ}+\mathrm{\angle }\mathrm{OQP}$..................(i)

Since PQ is a tangent and QOR is the diameter of the circle, it can be said that $\mathrm{\angle }\mathrm{OQP}={90}^{\circ }$.

On substituting the values in equation (i) we get,

$\begin{array}{l}\Rightarrow {120}^{\circ }=\mathrm{\angle }\mathrm{OPQ}+{90}^{\circ }\\ \Rightarrow \mathrm{\angle }\mathrm{OPQ}={30}^{\circ }\end{array}$     

Hence, the measure of $\mathrm{\angle }\mathrm{OPQ}\text{ is }{30}^{\circ }\text{. }$