Central and Inscribed Angles

Geometry · Axiom Academy

LESSON Central and Inscribed Angles Understanding angles formed within and around circles — and the half-angle relationship that ties them together. 1. Two Important Circle Angles A central angle sits at the center, its sides are radii, and it equals the arc it cuts off. An inscribed angle sits on the circle, its sides are chords, and it is half that arc. Watch the inscribed vertex slide around the circle in the animation: the central angle stays fixed on its arc, and the inscribed angle stays locked at exactly half of it the whole way. Vertex at the center Sides are radii Equals the intercepted arc Vertex on the circle Sides are chords Half the intercepted arc An inscribed angle is half the central angle on the same arc 2. The Inscribed Angle Theorem Pin down one arc and read off both angles. The central angle equals the arc's measure; the inscribed angle is half of it. Play the animation: as the intercepted arc opens from 0° up to 120°, the central angle tracks the arc exactly while the inscribed angle stays locked at precisely half — the two readouts move in perfect lockstep. Run the theorem in both directions — from the central angle to the inscribed angle, then from the inscribed angle back to its arc. Play the animation to build each computation: the given angle appears, the ½× (or ×2) operation fires, and the answer counts out. Example 1 — Finding an Inscribed Angle Apply the Inscribed Angle Theorem:

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