Theorem 10.9 : Angles in the same segment of the circle are equal Class 9

Given: A circle with center ‘O’. Having points P and Q subtends angles ∠PAQ and ∠PBQ at points A and B.

To Prove: ∠PAQ = ∠PBQ

Proof: From Theorem 10.8: Angle subtended by an arc at the center is double the angle subtended by it at any other point on the circle.

Therefore, ∠POQ = 2∠PAQ ….(i)

and, ∠POQ = 2∠PBQ ….(ii)

from eq(i) and (ii), we get

2∠PBQ = 2∠PAQ

∠PBQ = ∠PAQ 

Hence,Proved.