Inscribed and central angles

Warm-up

Three quick questions from earlier lessons.

Discover

A central angle of $100^\circ$ and an inscribed angle stand on the same arc. Which is the inscribed angle?

$50^\circ$
$100^\circ$

Slide C along the arc. Watch the angle at C.

It stays the same: half of the central angle
It gets bigger as C rises

Key idea

Angles in a circle depend on the arc they stand on.

An inscribed angle is half the central angle that intercepts the same arc. Inscribed angles on the same arc are equal. An angle inscribed in a semicircle (standing on a diameter) is

90^\circ

.

Worked example

$AB$ is a diameter, $C$ on the circle, $\angle A = 35^\circ$ .

\angle C = 90^\circ

Inscribed in a semicircle.

\angle B = 180 - 90 - 35 = 55^\circ

Angle sum.

Your turn, guided

Your turn: an inscribed angle of $40^\circ$ . The central angle on the same arc.

80^\circ

Double.

Practice ★

A central angle measures $49^\circ$ . Find an inscribed angle that intercepts the same arc (degrees).
A central angle measures $57^\circ$ . Find an inscribed angle that intercepts the same arc (degrees).

An inscribed angle measures $45^\circ$ . Find the central angle on the same arc (degrees).

Watch out!

Practice ★★

Triangle $ABC$ is inscribed in a circle with $AB$ a diameter, and $\angle A = 67^\circ$ . Find $\angle B$ (degrees).
Triangle $ABC$ is inscribed in a circle with $AB$ a diameter, and $\angle A = 29^\circ$ . Find $\angle B$ (degrees).

Exit ticket

Three questions to check you have got it.

A central angle measures $49^\circ$ . Find an inscribed angle that intercepts the same arc (degrees).
A central angle measures $57^\circ$ . Find an inscribed angle that intercepts the same arc (degrees).
A central angle measures $165^\circ$ . Find an inscribed angle that intercepts the same arc (degrees).

Summary card

Inscribed angles

Inscribed = ½ central; on a diameter: 90°.

Watch out: halve, do not double.

About this lesson