Circle Theorems: Angles in Same Segment

Definition of Angles in Same Segment

Angles in the same segment of a circle are angles that are subtended by the same chord and lie on the circumference of the circle. The phrase same segment means these angles are on the same arc created by that chord.

In other words, if you have a chord AB in a circle, any angles formed at points on the circumference that lie on the same side of the chord AB will be angles in the same segment.

Example: If chord AB subtends angles at points C and D on the circumference, and both points lie on the same arc AB, then these angles are equal: $\angle ACB = \angle ADB$.

Properties of Angles in Same Segment

• Equal Angles: Angles in the same segment are equal in size. This means if two angles are subtended by the same chord and lie on the same arc, they have the same measure.
• Chord and Arc Relationship: The chord divides the circle into two segments (arcs), and angles in the same segment correspond to the same arc.
• Opposite Sides of Chord: Angles in the same segment lie on the same side of the chord, not opposite sides.

This property is fundamental for solving many circle geometry problems involving unknown angles.

Applications and Examples

This theorem is often used to identify equal angles in circle diagrams and to calculate unknown angles when given partial information about the circle.

For instance, if chord AB subtends angles at points C and D on the circumference, and both points lie on the same arc AB, then:

$\angle ACB = \angle ADB$

This equality allows you to find missing angles or prove that two angles are equal, which is useful in many geometric proofs and problems.

For example, consider a circle with chord AB and points C and D on the circumference such that both lie in the same segment formed by AB. If $\angle ACB = 40^\circ$, then $\angle ADB$ is also $40^\circ$.