Question

In the figure, $PQ$ is a chord of a circle with center $O$ and $PT$ is a tangent. If $\angle QRT={60}^o$, find $\angle OPQ$.

Answer 1

Let $y$ be the on the tangent $PT$.

Now,

$\angle QPT+\angle QPX={180}^o\left(\text{From}\text{linear}\text{pair}\right)$

$\Rightarrow \angle QPX={180}^o-\angle QPT$

$\Rightarrow \angle QPX={180}^o-{60}^o$

$\Rightarrow \angle QPX={120}^o$

So, again,

$\angle QPX=\angle PRQ$

$\Rightarrow \angle PRQ={120}^o$

Hence, this is the answer.