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In the given figure, $TP$ and $TQ$ are two tangents. If $\angle PTQ = 50^\circ$, then find the measure of $\angle OPQ$.
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$TP = TQ$ (Tangents drawn from an exterior point to a circle are equal)
Since angles opposite to equal sides of a triangle are equal
$\therefore \angle TPQ = \angle TQP$ [$\frac{1}{2}$ mark]
In $\Delta TPQ$, $50^\circ + 2 \angle TPQ = 180^\circ$
$\Rightarrow \angle TPQ = 65^\circ$ [$1$ mark]
$\angle OPQ = 90^\circ - 65^\circ = 25^\circ$ [$\frac{1}{2}$ mark]
Since angles opposite to equal sides of a triangle are equal
$\therefore \angle TPQ = \angle TQP$ [$\frac{1}{2}$ mark]
In $\Delta TPQ$, $50^\circ + 2 \angle TPQ = 180^\circ$
$\Rightarrow \angle TPQ = 65^\circ$ [$1$ mark]
$\angle OPQ = 90^\circ - 65^\circ = 25^\circ$ [$\frac{1}{2}$ mark]