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Question
Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.
Solution
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] [Equation 1]
Since, at point R, there is a tangent.
∠RPQ = ∠1 ...[Angles in alternate segments are equal] [Equation 2]
From equation 1 and equation 2
∠1 = ∠2
Hence Proved.
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