Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Question

Philoid · Accepted Answer

Let QR be a chord in a circle with center O and ∠1 and ∠2 are the angles made by tangent at point R and Q with chord respectively.

To Prove : ∠1 = ∠2

Let P be another point on the circle, then, join PQ and PR.

Since, at point Q, there is a tangent.

∠RPQ = ∠2 [angles in alternate segments are equal] [Eqn 1]

Since, at point R, there is a tangent.

∠RPQ = ∠1 [angles in alternate segments are equal] [Eqn 2]

From Eqn 1 and Eqn 2

∠1 = ∠2

Hence Proved .