Overview

Circle angle facts convert arc information into angle measures. These are foundational for cyclic geometry.

Key Ideas

• An inscribed angle is half the central angle subtending the same arc.
• If $AB$ is a diameter, then $\angle ACB = 90^\circ$ (Thales).
• All inscribed angles subtending the same arc are equal.

Core Skills

Match Angles to Arcs

Identify which arc an inscribed or central angle intercepts. Angle measures are determined by that arc alone.

Use Thales for Right Angles

If a chord is a diameter, the angle opposite it on the circle is a right angle. This is often the fastest way to create a right triangle.

Equal Inscribed Angles

If two inscribed angles subtend the same arc, set them equal immediately to unlock angle chasing.

Worked Example

A central angle is $60^\circ$. Find the inscribed angle on the same arc.

The inscribed angle is $30^\circ$.

More Examples

Example 1: Same Arc

Two inscribed angles subtend arc $AB$. If one is $35^\circ$, find the other.

They are equal, so it is $35^\circ$.

Example 2: Diameter

If $AB$ is a diameter of a circle and $C$ is on the circle, find $\angle ACB$.

$\angle ACB = 90^\circ$ by Thales.

Strategy Checklist

• Mark the intercepted arc for each angle.
• Use central angles as twice the inscribed angles on the same arc.
• Look for a diameter to create a right triangle quickly.

Common Pitfalls

• Using the inscribed angle rule with different arcs.
• Forgetting that Thales requires a diameter.
• Using arc length rather than arc measure in degrees.

Practice Problems