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In a Quadrilateral Abcd, Ao and Bo Are Bisectors of Angle a and Angle B Respectively. Show That: - Mathematics

Sum

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

∠AOB = `1/2` (∠C + ∠D)

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Solution

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.

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APPEARS IN

Selina Concise Mathematics Class 8 ICSE
Chapter 16 Understanding Shapes
Exercise 16 (C) | Q 15 | Page 188
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