In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If $∠ T P Q = 70 °,$ find $∠ T R Q .$

45°

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50°

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55°

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60°

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Solution

The correct option is C 55°
Construction: Join OT and OQ.

OT and OQ are perpendicular to PT and PQ [Since radii are perpendicular to the tangents.]

In quadrilateral OTPQ,
$∠ O T P + ∠ T P Q + ∠ O Q P + ∠ T O Q = 360 ° \Rightarrow 90 ° + 70 ° + 90 ° + ∠ T O Q = 360 ° \Rightarrow ∠ T O Q = 110 °$

$Hence, ∠ T R Q = \frac{1}{2} ∠ T O Q = 55 °$ (Inscribed angle theorem)

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