What is the value of $\angle PRQ$ if  $\angle PSR:\angle PQR=1:2 ?$

Concept: Every angle of a cyclic quadrilateral is on a circle (circumscribed circle), and the sum of any two opposite angles of a cyclic quadrilateral is 180˚.

Here, PQRS is a cyclic quadrilateral.

Then, sum of any pair of opposite angles of cyclic quadrilateral is 180˚. 

$\Rightarrow \angle PSR+\angle PQR={180}^{°}$

$\Rightarrow k+2k={180}^{{ }^{°}}$

$\Rightarrow k={60}^{°}$

So, 

$\angle PSR={60}^{°} \&\angle PQR={120}^{°}$

By angle sum property of triangle,

In $\Delta PQR,\angle PQR+\angle PRQ+\angle RPQ={180}^{°}$

$\Rightarrow \angle PRQ+{120}^{°}+{50}^{°}={180}^{°}$

Hence, the value of  $\angle PRQ={10}^{°}$