90
In the given figure, TQ and TR are tangents to the circle with centre O. Prove that $\angle$ QTR $= 2 \angle$ OQR.
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Since TQ = TR
$\therefore \angle$TQR $= \angle$TRQ
Also, $\angle$QTR $= 180^\circ - (\angle$TQR $+ \angle$TRQ)
$\Rightarrow \angle$QTR $= 180^\circ - 2 \angle$TQR --- (1)
Now TQ $\perp$ OQ
$\therefore \angle$OQR $= 90^\circ - \angle$TQR
Using (1), we get
$2 \angle$OQR $= 180^\circ - 2\angle$TQR $= \angle$QTR
$\therefore \angle$TQR $= \angle$TRQ
Also, $\angle$QTR $= 180^\circ - (\angle$TQR $+ \angle$TRQ)
$\Rightarrow \angle$QTR $= 180^\circ - 2 \angle$TQR --- (1)
Now TQ $\perp$ OQ
$\therefore \angle$OQR $= 90^\circ - \angle$TQR
Using (1), we get
$2 \angle$OQR $= 180^\circ - 2\angle$TQR $= \angle$QTR