Question

 In ABCD is a cyclic quadrilateral; O is the centre of the circle. If BOD = 160°, find the measure of BPD.

Answer 1

Consider the arc BCD of the circle. This arc makes an angle ∠BOD = 160° at the centre of the circle and ∠BAD at a point A on the circumference.

∴ ∠BAD = `1/2`∠BOD = 80°
Now, ABPD is a cyclic quadrilateral.
∴ ∠BAD + ∠BPD = 180°
⇒ 80° + ∠BPD = 180°
⇒ ∠BPD = 180° - 80° 
⇒ ∠BPD = 100°
⇒ ∠BCD = 100°        ....(∵∠BPD and ∠BCD are angles in the same segment. ∴∠BCD = ∠BPD = 100°)