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

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.

Doubt by Muskan

Solution : 

Given : PQ is a chord. A tangent AB which is parallel to the chord PQ at point R of the circle.

To Prove : Arc PR = Arc QR

Construction : Join OR which intersect PQ at O.

Proof : We know, tangent at any point of the circle is perpendicular to the radius through the point of contact.
 ∴ OR⊥AB

Because PQ||AB
OS⊥PQ [corresponding Angles]

In ∆OSP and ∆OSQ
∠OSP = ∠OSQ [Each 90°]
OP = OQ [Radii of the same circle]
OS = OS [common]
∆OSP ≅ ∆OSQ [by RHS]
∠POS = ∠QOS [ by CPCT]
Arc PR = Arc QR
Hence R bisects the arc PRQ.