Question

In the given figure, PAQ and PBR are tangents to the circle with centre 'O' at the points A and B respectively. If T is a point on the circle such that $\angle$ QAT = 45° and $\angle$ TBR = 65°, then find $\angle$ ATB.

• A. 70°

Answer 1

Answer: (A)

70°

Answer 2

Solution Explanation:

1. By the tangent–chord theorem, at point A the angle between the tangent (AQ) and chord (AT) equals the inscribed angle in the alternate segment. Thus,
     ∠QAT = 45° = ∠ABT.
2. Similarly, at point B the angle between the tangent (BR) and chord (BT) equals the inscribed angle in the alternate segment. That is,
     ∠TBR = 65° = ∠BAT.
3. In triangle ABT, the sum of angles is 180°. Therefore,
     ∠ATB = 180° – (∠BAT + ∠ABT) = 180° – (65° + 45°) = 70°.