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  • A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ.

A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ.

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According to question

To Prove: R bisects the arc PRQ
Here    \angle Q_{3}= \angle Q_{1}     …..(i)    (\because  Alternate interior angles)
We know that the angle between the tangent and chord is equal to the angle made by the chord in the alternate segment.

\therefore \angle Q_{3}= \angle Q_{2} …..(ii)             
From equation (i) and (ii)
\angle Q_{1}= \angle Q_{2}
We know that sides are opposite to equal angles and equal.
\therefore  PR = QR
Hence R bisects the arc PRQ
Hence Proved

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