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प्रश्न
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.
उत्तर
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.
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