Angles in a semicircle and the same segment

A diameter subtends a right angle at the circumference: its central angle is \(180^\circ\), so the circumference angle is \(90^\circ\). The endpoints of the diameter must be the endpoints of the angle’s arms.

Example

If diameter \(AB=10\text{ cm}\) and chord \(AC=6\text{ cm}\), then \(\angle ACB=90^\circ\) (angle in a semicircle). Therefore \(BC=\sqrt{10^2-6^2}=8\text{ cm}\).

Angles subtended by the same chord at vertices on the same side of the chord are equal: they are angles in the same segment. If \(C\) and \(D\) lie on the same side of chord \(AB\), then \(\angle ACB=\angle ADB\). For example, if one is \(37^\circ\), the other is also \(37^\circ\).

Sharing a chord is not sufficient for equality: vertices on opposite sides give supplementary angles instead. Both diagrams illustrate theorem conditions and are not scale drawings for the numerical examples.