Find the measurement of......

Question

Find the measurement of unknown angle in the given circle with centre at $O$.

My Attempt $1$. $\angle QRS=\angle PQR$

$2$. $\angle PQR=\angle PMR$.

I am struck here. Please help me with this problem..

Thanks

Answer 1

We have that $\angle PMQ = 90^{\circ}$ as $PQ$ is the diameter of the circle. Now additionally we have that $\angle QMS = \angle PMR$. So using this we have:

$$\angle PMR = \frac{90^{\circ} - 60^{\circ}}{2} = 15^{\circ}$$