Qstnid -C-15- In A Quadrilateral Abcd, Ao And Bo Are Bisectors Of Angle A And Angle B Respectively. Show That: ∠Aob = 1/2 (∠C + ∠D) - 16: Understanding Shapes (Including Polygons) Class 8 Maths - Icse-viii-mathematics

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ICSE-VIII-Mathematics

16: Understanding Shapes (Including Polygons) Class 8 Maths

with Solutions - page 8

Qstn# C-15 Prvs-Qstn

#15
In a quadrilateral ABCD, AO and BO are bisectors of angle A and angle B respectively. Show that:
∠AOB = 1/2 (∠C + ∠D)
Ans : 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.