Angle Properties in a Circle

1. Central Angle: A central angle is an angle whose vertex is at the center of the circle. The measure of a central angle is equal to the measure of the arc it intercepts.

Example: In the circle below, angle AOB is a central angle. The measure of angle AOB is equal to the measure of arc AB.

A / \ / \ / \ O-------B

1. Inscribed Angle: An inscribed angle is an angle whose vertex is on the circle and whose sides intersect the circle at two different points. The measure of an inscribed angle is half the measure of the intercepted arc.

Example: In the circle below, angle ACB is an inscribed angle. The measure of angle ACB is half the measure of arc AB.

A / \ / \ / \ O-------B \ \ C

1. Tangent-Chord Angle: When a tangent and a chord intersect at a point on the circle, the measure of the angle formed is half the measure of the intercepted arc.

Example: In the circle below, angle ACD is formed by the tangent AD and the chord CD. The measure of angle ACD is half the measure of arc CD.

A / \ / \ / \ O-------B \ \ C | D

1. Intersecting Chords Angle: When two chords intersect inside a circle, the measure of each angle formed is half the sum of the measures of the intercepted arcs.

Example: In the circle below, angle ABD is formed by the intersecting chords AB and CD. The measure of angle ABD is half the sum of the measures of arcs AD and BC.

A / \ / \ / \ O-------B \ / \ / C | D

1. Cyclic Quadrilateral Angle: In a cyclic quadrilateral (a quadrilateral whose vertices lie on a circle), the opposite angles are supplementary, meaning their measures add up to 180 degrees.

Example: In the circle below, quadrilateral ABCD is a cyclic quadrilateral. Angle ABC and angle ADC are opposite angles, and their measures add up to 180 degrees.

A / \ / \ / \ O-------B \ / \ / C | D

Remember, these angle properties in a circle can be used to solve various geometry problems involving circles and their components.