21.2 Circle theorems — Learn

Prerequisites: angles on a straight line total $180^\circ$; angles in a triangle total $180^\circ$; identify a radius, chord and circumference.

Objectives: apply the angle-at-the-centre, semicircle, same-segment and cyclic-quadrilateral theorems.

Angles at the centre and circumference

An angle at the centre has its vertex at the circle's centre. An angle at the circumference has its vertex on the circle. When both angles stand on the same arc, the centre angle is twice the circumference angle.

Example 1. The angle at the circumference is $38^\circ$. Work out the angle at the centre standing on the same arc.

The centre angle is twice the circumference angle.

Substitute: $2\times38^\circ=76^\circ$.

So the centre angle is $76^\circ$.

An angle in a semicircle is always $90^\circ$. This follows because a straight angle at the centre is $180^\circ$, and the circumference angle is half of it.

Angles in the same segment are equal. They stand on the same chord and lie in the same part of the circle.

Example 2. Two angles stand on the same chord in the same segment. One is $51^\circ$. Find the other.

Angles in the same segment are equal.

Therefore the other angle is $51^\circ$.

Cyclic quadrilaterals

A cyclic quadrilateral has all four vertices on one circle. Its opposite angles add to $180^\circ$.

Example 3. Opposite angles are $x^\circ$ and $112^\circ$. Find $x$.

Opposite angles in a cyclic quadrilateral total $180^\circ$.

Write the equation: $x+112=180$.

Subtract $112$: $x=68^\circ$.

Watch out. Do not double the circumference angle when the angles stand on different arcs. Check that both angles stand on the same arc before using the centre theorem.

Guided investigation

Predict: if the circumference angle changes, how will the centre angle change? Record a rule before moving the control.

Test: change the circumference angle and compare the two displayed angles.

Explain: write why the centre angle stays twice the circumference angle when both angles stand on the same arc.

Progress check

For the chapter problem, the two circumference angles $38^\circ$ and $x^\circ$ stand on the same chord in the same segment. Use that fact to predict $x$.

You should now be able to

• calculate an angle at the centre from an angle at the circumference;
• use the $90^\circ$ angle in a semicircle;
• use equal angles in the same segment;
• calculate opposite angles in a cyclic quadrilateral.