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 $PRQ$.

Answer 1

Given: Chord $PQ$ is parallel to tangents at $R$.
To prove: $R$ bisects the arc $PRQ$

Proof: Since $\angle 1=\angle 2$.....(i) [alternate interior angles]
$\angle 1=\angle 3$ ......(ii) [Angle between tangent and chord is equal to angle made by chord in alternate segment]
From equation(i) and (ii), we get $\angle 2=\angle 3$
$\Rightarrow$ $PR=QR$ [sides opposite to equal angles are equal]
Since, equal chords subtends equal arcs in a circle.
Thus, arc $PR=$ arc $RQ$
Hence, $R$ bisects arc $PRQ$.