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

Given: $\angle RPQ={50}^o$

$\angle PSR : \angle PQR = 1 : 2$

Let ∠PSR and ∠PQR be k and 2k respectively.

We know that the sum of either pair of opposite angles of a cyclic quadrilateral is 180^o.

So in cyclic quadrilateral PQRS,

$\angle PSR+\angle PQR={180}^o$

$\Rightarrow k+2k={180}^o$

$\Rightarrow 3k={180}^o$

$\Rightarrow k={60}^o$

$\therefore \angle PSR=k={60}^o$

And, $\angle PQR=2k=2\times {60}^o={120}^o$

Now in △PQR,

$\angle PQR+\angle PRQ+\angle RPQ={180}^o$

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

$\Rightarrow \angle PRQ={10}^o$

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