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$O$ is the center of a circle and $PT$ , $P Q$ are the tangents to the circle from an external point $P$ . If the measure of $∠ T P Q = 70 °$ then $∠ T R Q =$ [[1]].

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Detailed Solution

O

is the center of a circle and

PT

P Q

are the tangents to the circle from an external point

P

. If the measure of

∠ T P Q = 70 °

then

∠ T R Q =

55 °

.
Given is the following figure,
Question Image

∠ TPQ = 70 °

PT

and

PQ

are tangents to the circle from an external point

P

.
Extending the two radii

OT

and

OQ

we get the following figure,
Question Image

∠ T = ∠ Q = 90 °

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

∠ O,

∠ O + ∠ T + ∠ Q + ∠ P = 360 ° ∠ O = 360 ° − 90 ° − 90 ° − 70 ° ∠ O = 360 ° − 250 ° ∠ O = 110 °

∠ O

is an angle formed by an arc at the center of the circle, therefore always double the angle at any other point on the circle that will be

∠ T R Q

∴ 2 ∠ T R Q = ∠ O ⇒ ∠ T R Q = 110 ° 2 ⇒ ∠ T R Q = 55 °

Thus, the measure of

∠ T R Q

55 °

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