Question

In the given figure, O is the centre of the circle and arc ABC subtends an angle of 130° at the centre. If AB is extended to P, find ∠PBC.

Answer 1

Reflex ∠AOC + ∠AOC = 360^∘
⇒ Reflex ∠AOC + 130^∘ + x = 360^∘
⇒ Reflex ∠AOC = 360^∘ − 130^∘
⇒ Reflex ∠AOC = 230^∘

We know that the angle subtended by an arc of a circle at the centre is double the angle subtended by it on the remaining part of the circle.

Here, arc AC subtends reflex ∠AOC at the centre and ∠ABC at B on the circle.

∴ ∠AOC = 2∠ABC
⇒ $\angle ABC=\frac{230°}{2}=115°$ ...(1)

Since ABP is a straight line, ∠ABC + ∠PBC = 180^∘
⇒ ∠PBC = 180^∘ − 115^∘
⇒ ∠PBC = 65^∘ ...(2)

Hence, ∠PBC = 65^∘.