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Question
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.
Solution
Given: Chord PQ is parallel to tangent at R.
To prove: R bisects the arc PRQ.
Proof: ∠1 = ∠2 ...[Alternative interior angles]
∠1 = ∠3 ...[Angle between tangent and chord is equal to angle made by chord in alternative segment]
∴ ∠2 = ∠3
⇒ PR = QR ...[Sides opposite to equal angles are equal]
⇒ PR = QR
So, R bisects PQ.
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