High School Mathematics - 2 8.4 Theorems - Circles

Theorem: Angle is a semi circle is a right angle.

Given : Angle AOB is in semi circle.
To prove: Angle ACB = 90^{o}
Proof: Angle ACB = 1/2 angle AOB (angle subtended at the centre is twice angle at circumference)
Angle AOB = 180^{o} (straight line)
Hence ACB = 90^{o}

Theorem: Angles in the same segment of a circle are equal.