Question

If the bisectors of two adjacent angles A and B of a quadrilateral ABCD intersect at a point O such that ∠C + ∠D = k ∠AOB, then find the value of k.

Answer 1

The quadrilateral can be drawn as follows:

We have AO and BO as the bisectors of angles ∠A and ∠B respectively.

In ,ΔAOB We have,

∠AOB + ∠1+ ∠2 = 180°

∠AOB = 180°-(∠1 + ∠2)

∠AOB = 180° -`(1/2∠A +1/2∠B)` 

 `∠AOB = 180°- 1/2 (∠A+ ∠B)` …… (I)

By angle sum property of a quadrilateral, we have:

∠A+ ∠B + ∠C + ∠D = 360°

∠A+∠B = 360°-( ∠C+ ∠D) 

Putting in equation (I):

`∠AOB = 180°- 1/2[360° - ( ∠C+ ∠D)]`

`∠AOB= 180° - 180° +(∠C+ ∠D)/2`

`∠AOB = 1/2 (∠C + ∠D)` 

 (∠C + ∠D) =2∠AOB …… (II)

On comparing equation (II) with

(∠C + ∠D) = k ∠AOB 

We get k = 2.

Hence, the value for k is 2.