Central Angles, Inscribed Angles, and Tangent Lines

Central Angle

A central angle has its vertex at the centre of the circle. Its measure equals the intercepted arc.

$\text{Central angle} = \text{intercepted arc}$

Inscribed Angle

An inscribed angle has its vertex on the circle. Its measure is half the intercepted arc.

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

Inscribed Angle Theorem

An inscribed angle is half the central angle that subtends the same arc:

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

Angle in a Semicircle

An inscribed angle that subtends a diameter is always 90°.

Tangent Line Properties

• A tangent line touches the circle at exactly one point.
• A tangent is perpendicular to the radius at the point of tangency.
• Two tangent segments from the same external point are equal in length.

Tangent-Radius Right Angle

If a line is tangent to a circle at point $P$, and $O$ is the centre, then $\angle OPT = 90°$ (where $T$ is along the tangent).

Tip 1: Draw the Radius to the Tangent Point

This creates a right angle, which often enables Pythagorean theorem or trig.

Tip 2: Inscribed = Half Central

This is the most-tested relationship.

Tip 3: Diameter → 90° Inscribed Angle

If the inscribed angle sits on a diameter, it's automatically 90°.