Question

In the given figure, O is the centre of the circle. The tangents at B and D intersect each other at point P. If AB is parallel to CD and ∠ABC = 55°, find:

1. ∠BOD
2. ∠BPD

Answer 1

In the given figure, O is the centre of the circle AB || CD, ∠ABC = 55° tangents at B and D are drawn which meet at P.

∵ AB || CD

∴ ∠ABC = ∠BCD  ...(Alternate angles)

∴ ∠ABC = 55°   ...(Given)

i. Now arc BD subtands ∠BOD at the centre and ∠BCD at the remaining part of the circle.

∴ ∠BOD = 2∠BCD

= 2 × 55°

= 110°

ii. In quadrilateral OBPD,

∠OBP = ∠ODP = 90°  ...(∵ BP and DP are tangents)

∴ ∠BOD + ∠BPD = 180°

`\implies` 110° + ∠BPD = 180°

`\implies` ∠BPD = 180° – 110° = 70°

Hence, ∠BOD = 110° and ∠BPD = 70°