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In the given figure, $TP$ and $TQ$ are tangents at points $P$ and $Q$ of the circle respectively. If reflex $\angle POQ = 250^{\circ}$, find the measure of each angle of quadrilateral $POQT$.
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Solution: $\angle POQ = 360^{\circ} - 250^{\circ} = 110^{\circ}$ [1 mark]
As tangent is perpendicular to the radius through the point of contact.
$\angle OPT = \angle OQT = 90^{\circ}$ [1 mark]
In Quadrilateral $OPTQ$
$\angle OPT + \angle OQT + \angle POQ + \angle PTQ = 360^{\circ}$
$\angle PTQ = 70^{\circ}$ [1 mark]
As tangent is perpendicular to the radius through the point of contact.
$\angle OPT = \angle OQT = 90^{\circ}$ [1 mark]
In Quadrilateral $OPTQ$
$\angle OPT + \angle OQT + \angle POQ + \angle PTQ = 360^{\circ}$
$\angle PTQ = 70^{\circ}$ [1 mark]