37. Prove that the angles made by a tangent to a circle with any chordthrough the point of contact are respectively equal to angles in thealternate segments of the circle.

37. Prove that the angles made by a tangent to a circle with any chord

1.	37. Prove that the angles made by a tangent to a circle with any chordthrough the point of contact are respectively equal to angles in thealternate segments of the circle.
Answer» Let PQ be the chord of a circle with center OLet AP and AQ be the tangents at points P and Q respectively.Let us assume that both the tangents meet at point A.Join points O, P. Let OA meets PQ at RHere we have to prove that ∠APR = ∠AQRConsider, ΔAPR and ΔAQRAP = AQ [Tangents drawn from an internal point to a circle are equal]∠PAR = ∠QARAR = AR [Common side]∴ ΔAPR ≅ ΔAQR [SAS congruence criterion]Hence ∠APR = ∠AQR [CPCT]