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 \(PRQ\).
Proof:
Given: Chord \(PQ\) is parallel to tangents at \(R\).
Alternate interior angles are equal.
\(\angle 1 = \angle 2\) - - - - (i)
Angle between tangent and chord is equal to angle made by chord in the alternate segment.
\(\angle 1 = \angle 3\) - - - - (ii)
Thus, \(\angle 2 = \angle 3\).
Sides opposite to angles are equal.
\(PR = \)
Since equal chords subtends equal arcs in a circle.
Thus, arc \(PR =\) arc \(RQ\).
Hence, \(R\) bisects arc \(PRQ\).