Theorems 10.7 to 10.12: Angles in a Circle

Theorem 10.7

Statement

The measure of a central angle of a minor arc of a circle, is double that of the angle subtended by the corresponding major arc.

Class Activity

Prove theorem 10.7 by taking central angle $\angle COD = 180^{\circ}$ and $\angle COD > 180^{\circ}$ (reflex angle).

Theorem 10.8

Statement

Any two angles in the same segment of a circle are equal.

Class Activity

Result

Any number of angles in the same segment of a circle are equal.

Theorem 10.9

Statement

The angle in a semi- circle is a right angle.

Class Activity

Corrolary

If an arc of a circle subtends a right angle at any point of the remaining part of the circle, then the arc is a semi- circle.