Question

Prove that an angle in a semicircle is a right angle.

Answer 1

Given: AB is a diameter of a circle C(O,r) and ∠ACB is an angle in the semicircle.
To Prove: ∠ACB = 90°



Proof: We know that the angle subtended by an arc at the centre of a circle is
twice the angle formed by it at any point on the remaining part of the circle.
∴ ∠AOB = 2∠ACB [Since angle subtended by arc AB = 2 × angle formed by it at C]
⇒ 2∠ACB = 180° [Since ∠AOB is a straight angle]
⇒ ∠ACB = 90°