Circles are like geometric playgrounds, full of cool angle relationships. Central angles boss it from the middle, while inscribed angles hang out on the edge, measuring half as much as their central buddies.

The inscribed angle theorem is a game-changer. It helps us figure out missing angles and shows how inscribed angles on the same arc are always twins. Plus, any angle in a semicircle is automatically a right angle!

Angles in Circles

Central and inscribed angles

• Central angles have vertex at center of circle formed by two radii intercept an arc on circumference
• Inscribed angles have vertex on circumference formed by two chords measure half of central angle subtending same arc
• Circumscribed angles have vertex outside circle formed by two tangent lines measure half difference of intercepted arcs

Inscribed angle theorem applications

• Inscribed Angle Theorem states measure of inscribed angle is half measure of central angle subtending same arc $m\angle ABC = \frac{1}{2} m\angle AOC$
• Can find measure of inscribed angle given central angle measure (if central angle is 120°, inscribed angle is 60°)
• Can determine measure of central angle given inscribed angle measure (if inscribed angle is 45°, central angle is 90°)
• Solve for unknown angles in problems involving inscribed and central angles (find missing angle measures in a circle with some given information)

Inscribed vs central angle relationships

• Inscribed angle measure always half measure of central angle subtending same arc $m\angle ABC = \frac{1}{2} m\angle AOC$
• All inscribed angles subtending same arc have equal measures (inscribed angles subtending same arc are congruent)
• Inscribed angles subtending semicircle are right angles measure 90° (angle inscribed in semicircle is a right angle)

Angles from chords and tangents

1. Angles formed by two chords intersecting inside circle measure half sum of measures of arcs intercepted by angle and its vertical angle $m\angle ABC = \frac{1}{2} (m\overset{\frown}{AC} + m\overset{\frown}{BD})$

2. Angles formed by two secant lines intersecting outside circle measure half difference of measures of intercepted arcs $m\angle ABC = \frac{1}{2} (m\overset{\frown}{AC} - m\overset{\frown}{BD})$

3. Angles formed by secant and tangent intersecting outside circle measure half difference of measures of intercepted arcs $m\angle ABC = \frac{1}{2} (m\overset{\frown}{AC} - m\overset{\frown}{BD})$

4. Angles formed by two tangent lines intersecting outside circle measure half difference of measures of intercepted arcs $m\angle ABC = \frac{1}{2} (m\overset{\frown}{AC} - m\overset{\frown}{BD})$