Let C be a circle with centre O, and R be an external point to C. Let RP and RQ be two tangents to circle C with P and Q being the points of tangency, respectively. If ∠PRQ = 60°, then find ∠POQ.

The correct option is 120°.

Step-by-step Explanation:

1. Identify the given geometric properties:

We are given a circle $C$ with centre $O$ and an external point $R$.

$RP$ and $RQ$ are two tangents drawn from the external point $R$ to the circle at points of tangency $P$ and $Q$, respectively.

The angle between the two tangents is given as $\angle PRQ=60°$.

2. Recall the property of radius and tangent:

The radius of a circle is always perpendicular to the tangent at the point of contact.

Therefore, $OP\perp RP$ and $OQ\perp RQ$.

This implies that:

$\angle OPR=90°$

$\angle OQR=90°$

3. Use the sum of angles in a quadrilateral:

Consider the quadrilateral $OPRQ$.

The sum of all interior angles of a quadrilateral is $360°$.

$\angle POQ+\angle OPR+\angle PRQ+\angle OQR=360°$

4. Substitute the known angle values into the equation:

$\angle POQ+90°+60°+90°=360°$

$\angle POQ+240°=360°$

$\angle POQ=360°-240°$

$\angle POQ=120°$

Thus, the measure of $\angle POQ$ is 120°.