Question

In the given figure, O is the centre of the circle and arc ABC subtends an angle of ${130}^{\circ }$ at the centre. If AB is extended to P, find $\angle PBC.$

Answer 1

ANSWER:
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
⇒ ∠ ABC=230°2=115°
...(1)
Since ABP is a straight line, ∠ ABC + ∠ PBC = 180 ∘
⇒ ∠ PBC = 180 ∘ − 115 ∘
⇒ ∠ PBC = 65 ∘
...(2)
Hence, ∠ PBC = 65 ∘ .