The alternate segment theorem

The angle between a tangent and a chord equals the angle subtended by that chord at a point on the circumference in the alternate segment. Identify the tangent, its point of contact, and the chord before matching angles.

Example

AT is tangent at A and AB is a chord. The angle between AT and AB is \(45^\circ\), so \(\angle ACB=45^\circ\). C lies in the segment opposite this tangent–chord angle.

The theorem uses an angle at the circumference, not the centre. The radius to A is perpendicular to the tangent, so it can connect tangent–chord reasoning to a triangle containing the centre.