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.
Answers
Given : The bisectors of two adjacent angles A and B of a quadrilateral ABCD intersect at a point O such that ∠C + ∠D = k ∠AOB
To Find : value of k
Proof :
In quadrilateral ABCD ,
We know that , the sum of the angles of a quadrilateral is 360°.
∠A+∠B + ∠C + ∠D = 360°
∠A+∠B = 360° - [∠C + ∠D ] ……………..(1)
In ΔAOB, we have :
We know that , the sum of the angles of a triangle is 180°.
∠OAB + ∠OBA + ∠AOB = 180°
⇒ ½ ∠A + ½ ∠B + ∠AOB = 180°
[AO and BO are the bisectors of ∠Aand ∠B]
⇒ ½ [∠A + ∠B] + ∠AOB = 180°
⇒ ∠AOB = 180° - ½ [∠A + ∠B]
⇒ ∠AOB = 180° - ½ [360° - (∠C + ∠D)]
[From eq 1]
⇒ ∠AOB = 180° - ½ × 360° + ½ (∠C + ∠D)
⇒ ∠AOB = 180° - 180° + ½ (∠C + ∠D)
⇒ ∠AOB = ½(∠C + ∠D)
⇒ 2∠AOB = (∠C + ∠D)
⇒ (∠C + ∠D) = 2∠AOB…………(2)
We have , ∠C + ∠D = k ∠AOB
On comparing this with eq 2 , we get the value of k = 2
Hence the value of k is 2 .
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