Two tangents TP and TQ are drawn to a circle with center O from an external point T. Then ∠PTQ= ____ ∠OPQ.

# Two tangents TP and TQ are drawn to a circle with center O from an external point T. Then ∠PTQ= ____ ∠OPQ.

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### Solution:

TP = TQ …(1)
⇒∠TQP=∠TPQ …(2)
OP is perpendicular to TP
⇒∠OPT= ${90}^{\circ }$
⇒∠OPQ + ∠TPQ= ${90}^{\circ }$
⇒∠TPQ= ${90}^{\circ }$−∠OPQ …(3)
In triangle PTQ,
⇒∠TPQ + ∠PQT + ∠QTP = ${180}^{\circ }$
${90}^{\circ }$−∠OPQ +∠TPQ + ∠QTP =
⇒ 2(${90}^{\circ }$−∠OPQ) + ∠QTP = ${180}^{\circ }$
−2∠OPQ + ∠PTQ = ${180}^{\circ }$
⇒ 2∠OPQ = ∠PTQ
⇒ 2(${90}^{\circ }$− ∠OPQ) + ∠QTP = ${180}^{\circ }$
− 2∠OPQ + ∠PTQ = ${180}^{\circ }$
⇒ 2∠OPQ = ∠PTQ
Two tangents TP and TQ are drawn to a circle with center O from an external point T. Then ∠PTQ= 2∠OPQ.

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