Question

ABCDE is a cyclic pentagon with centre of its circumcircle at point O such that AB = BC = CD and angle ABC = 120°.

Calculate:

1. ∠BEC
2. ∠BED 

Answer 1

i. Join OC and OB.

AB = BC = CD and ∠ABC = 120°

∴ ∠BCD = ∠ABC = 120°

OB and OC are the bisectors of ∠ABC and ∠BCD respectively.

∴ ∠OBC = ∠BCO = 60°

In ΔBOC,

∠BOC = 180° – (∠OBC + ∠BOC)

`=>` ∠BOC = 180° – (60° + 60°)

`=>` ∠BOC = 180° – 120° = 60°

Arc BC subtends ∠BOC at the centre and ∠BEC at the remaining part of the circle.

∴ `∠BEC = 1/2 ∠BOC = 1/2 xx 60^circ = 30^circ`

ii. In cyclic quadrilateral BCDE,

∠BED + ∠BCD = 180°

`=>` ∠BED + 120° = 180°

∴ ∠BED = 180° – 120° = 60°