Inscribed Angles

Central Angle

An angle with its vertex at the center of the circle.

Measure: Equal to the measure of its intercepted arc.

Inscribed Angle

An angle with its vertex on the circle and sides containing chords.

Inscribed Angle Theorem

An inscribed angle is half the measure of its intercepted arc.

$\text{Inscribed angle} = \frac{1}{2} \times \text{intercepted arc}$

Corollaries

Inscribed angles that intercept the same arc are congruent.

An angle inscribed in a semicircle is a right angle (90°).

• Because it intercepts a 180° arc
• $\frac{180°}{2} = 90°$

Inscribed Quadrilateral

If a quadrilateral is inscribed in a circle:

Opposite angles are supplementary (sum to 180°)

Arc-Angle Relationships

• Central angle = arc measure
• Inscribed angle = $\frac{1}{2}$ arc measure
• Tangent-chord angle = $\frac{1}{2}$ arc measure