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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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Solution
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 angle between tangent and chord is equal to angle made by chord in 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 opposite to equal angles and equal.
    \therefore  PR = QR
Hence R bisects the arc PRQ
Hence Proved

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