Calculation of angles

To calculate angles in and around circles, we use properties such as the angle at the center theorem, the inscribed angle theorem, and properties of tangents and chords.

Important properties

• The angle at the center of a circle is twice any angle at the circumference subtended by the same arc: if $O$ is the center and $A$, $B$, $C$ are points on the circle, then $\angle AOB = 2 \angle ACB$.

• Angles in the same segment of a circle are equal.

• The angle in a semicircle is a right angle.

• The sum of opposite angles in a cyclic quadrilateral is 180°.

• The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.