Year 10Extension

Circle Theorems

Circle theorems are Year 11 content in the Australian curriculum. Offered here as preparation for senior mathematics.

Every circle theorem is a rule about angles made by chords, radii and tangents. Learn the picture that goes with each name.

Method

Angle at the centre is twice the angle at the circumference

\angle_{\text{centre}} = 2\angle_{\text{circumference}}

Same arc, both angles.

Angle in a semicircle is a right angle

A special case of the one above.

Angles in the same segment are equal

Standing on the same arc.

A tangent meets a radius at 90 degrees

Opposite angles of a cyclic quadrilateral add to 180

e.g.

\text{The angle at the centre standing on arc } AB \text{ is } 100^\circ. \text{ Find the angle at the circumference.}

Identify which arc both angles stand on. $\text{arc } AB$
Name the theorem that connects them. $[\text{angle at centre} = 2 \times \text{angle at circumference}]$
Write the relationship. $100 = 2x$
Solve. $x = 50^\circ$
State the answer with the reason. $50^\circ \ [\text{angle at the centre}]$

Name the theorem beside every answer.

\text{Angle at the centre is } 140^\circ. \text{ Angle at circumference?}

Answer

70^\circ

\text{A triangle has one side as the diameter. The opposite angle?}

Answer

90^\circ

\text{A cyclic quadrilateral has one angle } 75^\circ. \text{ Find the opposite one.}

Answer

105^\circ

\text{A tangent meets a radius. What is the angle?}

Answer

90^\circ

\text{Two angles stand on the same arc. What do you know?}

Answer

\text{They are equal.}

\text{Why is the semicircle angle } 90^\circ?

Answer

\text{The centre angle is } 180^\circ, \text{ and half of that is } 90^\circ.