In a quadrilateral ABCD, AO and BO are the bisectors of angle A and angle B. Prove that .

Question

Philoid · Accepted Answer

Given: AO and BO are bisectors of ∠A and ∠B

To Prove

Proof:

As AO and BO are bisectors, and

We know that,

Sum of angles of a triangle = 180°

For Δ AOB,

∠AOB + ∠ABO + ∠AOB = 180°

∠A+ ∠B + 2 ∠AOB = 360°

∠A+ ∠B = 360° – 2 ∠AOB ……eq(1)

We also know that,

Sum of angles of a quadrilateral = 360°

For Quadrilateral ABCD,

∠A+ ∠B + ∠C + ∠D = 360°

Putting the value of ∠A+ ∠B from equation 1

360° – 2 ∠AOB + ∠C + ∠D = 360°

2 ∠AOB = ∠C + ∠D

Hence, Proved.