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$P T$ is a tangent to a circle with centre $O$ . $P Q$ is a chord of the circle such that $∠ Q P T = 70 °$ , then the measure of $∠ P O Q =$ [[1]].

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Detailed Solution

P T

is a tangent to a circle with centre

O

P Q

is a chord of the circle such that

∠ Q P T = 70 °

, then the measure of

∠ P O Q =

140 °

.
Given that

P Q

is a chord of the circle such that

∠ Q P T = 70 °

TP

is the tangent to the circle from an external point

T

∠ PBT = 30 °

∠ P = 90 °

[radius drawn to the point of contact of a tangent]
Calculating

∠ O P Q

∠ O P Q + ∠ Q P T = ∠ O P T ∠ O P Q = 90 ° − 70 ° ∠ O P Q = 20 °

∠ O P Q and ∠ O Q P

are equal. [radius of the circle],
Calculating

∠ P O Q

,
∠OPQ + ∠OQP + ∠POQ

= 180^{\circ}

∠POQ

= 180^{\circ} - 20^{\circ} - 20^{\circ}

∠POQ

= 140^{\circ}

The measure of

∠ P O Q

140 °

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