Question

In the given figure, O is the centre of the circle. The tangents at B and D intersect at point P. If AB || CD and ∠ABC = 50°, find:

1. ∠BOD 
2. ∠BPD

Answer 1

Given:

O is the centre of the circle.

Tangents at B and D intersect at P.

AB || CD.

∠ABC = 50°. Need to find: (i) ∠BOD, (ii) ∠BPD

To find ∠BOD:

Since AB || CD and BC is the transversal, ∠ABC = ∠BCD = 50° (Alternate interior angles are equal).

In the circle with centre O, the angle subtended by chord BD at the centre O is twice the angle subtended at any point on the circumference. Thus,

∠BOD = 2 × ∠BCD

= 2 × 50°

= 100°.

To find ∠BPD:

The tangents at B and D intersect at P. The angle between two tangents drawn from an external point is supplementary to the angle subtended by the chord joining the points of tangency at the centre.

Hence,

∠BPD = 180° − ∠BOD

= 180° − 100°

= 80°.