Question

In a quadrilateral ABCD, AO and BO are bisectors of angle A and angle B respectively. Show that:

∠AOB = `1/2` (∠C + ∠D)

Answer 1

Given: AO and BO are the bisectors of ∠A and ∠B respectively.
∠1 = ∠4 and ∠3 = ∠5 ……..(i)

To prove : ∠AOB = `1/2` (∠C + ∠D)

Proof: In quadrilateral ABCD

∠A + ∠B + ∠C + ∠D = 360°

`1/2`(∠A + ∠B + ∠C + ∠D) = 180° …………(ii)

Now in ∆AOB

∠1 + ∠2 + ∠3 = 180° ………(iii)

Equating equation (ii) and equation (iii), we get

∠1 + ∠2 + ∠3 = ∠A + ∠B + `1/2` (∠C + ∠D)

∠1 + ∠2 + ∠3 = ∠1 + ∠3 +`1/2` (∠C + ∠D)

∠2 = `1/2` (∠C + ∠D)

∠AOB = `1/2` (∠C + ∠D)

Hence proved.