Question

‘O’ is the centre of the circle and A, B, C

are points on it. Prove that

∠AOB = 2 (∠ABC + ∠BAC)
​

Answer 1

Step-by-step explanation:

so let solve

Aob =2(abc + bac)

2(x+x)

2(2x)

x=2/2

1

Answer 2

Given:

O is the center of the circle and A, B, C are the points on it.

To Prove:

∠AOB = 2(∠ABC + ∠BAC)

Solution:

We construct a line to meet CO.

Now, arc AC subtends ∠AOC at the center and ∝ABC in the remaining part of the circle.

Therefore, we see

∠AOC = 2∠ABC, since the angle subtended by an arc at the center of the circle, is double the angle subtended by it at a point in the remaining part of the circle. ...(I)

arc BC subtends ∠BOC at the center and ∠BAC in the remaining part of the circle, therefore,

∠BOC = 2∠BAC, because the angle subtended by an arc at the center of the circle, is double the angle subtended by it at a point in the remaining part of the circle ...(II)

Now we add equation (I) and equation (II),

⇒ ∠AOC + ∠BOC = 2∠ABC + 2∠ABC

⇒ ∠AOB = 2∠ABC + 2∠BAC

⇒ ∠AOB = 2(∠ABC + ∠BAC)

Hence proved.