Find the value of $\angle PTR$ if $PQRS$ is a cyclic quadrilateral.

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˚.

From figure, $\angle POQ=2\angle PQR$

$\Rightarrow \angle PQR={65}^{°}$

Here, PQRS 

is a cyclic quadrilateral.$\begin{array}{l}\Rightarrow \angle PQR+\angle PSR={180}^{°}\\ \Rightarrow \angle PSR={115}^{°}\end{array}$

Then, sum of any pair of opposite angles of cyclic quadrilateral is ${180}^{°}$

Since angles subtended by an arc in the same segment are equal,

$\Rightarrow \angle PTR=\angle PSR={115}^{°}$

Hence, the value of $\angle PTR={115}^o$