Course 12.2Lesson 1 of 3

Cyclic quadrilaterals and their angles

A cyclic quadrilateral has all four corners on one circle. The sum of each pair of opposite angles is $180^\circ$ . We prove this with the angle at the centre, find unknown angles, and decide which four-sided figures can be cyclic.

about 25 min1 simulation1 practice question1 Python program5 questions to check yourself

This lesson is free to read. Enroll free to keep your place and take the tests.

Lesson 1 of 3Show all lessons

Try the code and the simulations beside the lesson.

Four lamps round a pond

A park has a round pond. Four lamp posts stand on its edge. A path joins each post to the next one, so the four paths make a four-sided figure.

Every corner of this figure touches the circle. A four-sided figure with all four corners on one circle is called a cyclic quadrilateral. The circle is its circumcircle. Points that lie on one circle are called concyclic points.

The theorem

In the cyclic quadrilateral $ABCD$ below, the opposite angles are $\angle A$ and $\angle C$ , and also $\angle B$ and $\angle D$ .

A circle with its centre O marked and a four-sided figure ABCD drawn inside it, with all four corners on the circle. A is at the top left, D at the top right, C at the right and B at the bottom left. The angles are marked: angle A is 95 degrees, angle B is 70 degrees, angle C is 85 degrees and angle D is 110 degrees. A line below says that A plus C is 95 plus 85, which is 180 degrees, and B plus D is 70 plus 110, which is 180 degrees.

In the picture, $\angle A = 95^\circ$ and $\angle C = 85^\circ$ . They add to $180^\circ$ . So do $70^\circ$ and $110^\circ$ .

Theorem. The sum of each pair of opposite angles of a cyclic quadrilateral is $180^\circ$ .

Two angles that add to $180^\circ$ are called supplementary. So the opposite angles of a cyclic quadrilateral are supplementary.

Why it is true

We use a fact from the first course of this chapter. The angle that an arc makes at the centre is double the angle it makes at any point on the remaining part of the circle. This holds even when the arc is more than half the circle.

Let $O$ be the centre of the circle through $A$ , $B$ , $C$ and $D$ .

The arc $ABC$ makes an angle at the centre. The corner $D$ is on the remaining part of the circle. So that angle at the centre is $2 \times \angle ADC$ .
The arc $ADC$ also makes an angle at the centre. The corner $B$ is on the remaining part. So that angle is $2 \times \angle ABC$ .
The two arcs together make the whole circle. Their angles at the centre add to $360^\circ$ .

So $2 \times \angle ADC + 2 \times \angle ABC = 360^\circ$ . Divide by $2$ :

\displaystyle \angle ADC + \angle ABC = 180^\circ.

The same argument with the arcs $BAD$ and $BCD$ gives $\angle A + \angle C = 180^\circ$ .

Using the theorem

Example 1. $ABCD$ is a cyclic quadrilateral with $\angle A = 95^\circ$ and $\angle B = 70^\circ$ . Find $\angle C$ and $\angle D$ .

$\angle C$ is opposite to $\angle A$ . So $\angle C = 180^\circ - 95^\circ = 85^\circ$ .
$\angle D$ is opposite to $\angle B$ . So $\angle D = 180^\circ - 70^\circ = 110^\circ$ .

Check. The four angles of a quadrilateral add to $360^\circ$ , and $95 + 70 + 85 + 110 = 360$ .

Example 2. Two opposite angles of a cyclic quadrilateral are $(4x + 5)^\circ$ and $(2x + 25)^\circ$ . Find $x$ and the two angles.

They are opposite angles, so they add to $180^\circ$ : $(4x + 5) + (2x + 25) = 180$ .
Collect the terms: $6x + 30 = 180$ , so $6x = 150$ and $x = 25$ .
The angles are $4 \times 25 + 5 = 105^\circ$ and $2 \times 25 + 25 = 75^\circ$ .

Check. $105 + 75 = 180$ .

The other pair comes free

The four angles of any quadrilateral add to $360^\circ$ . If $\angle A + \angle C = 180^\circ$ , then $\angle B + \angle D = 360^\circ - 180^\circ = 180^\circ$ . One pair settles the other.

Which figures can be cyclic

Use the theorem as a test. A figure can be cyclic only if both pairs of opposite angles add to $180^\circ$ .

A rectangle is cyclic. All four angles are $90^\circ$ , and $90 + 90 = 180$ . A square is cyclic too.
An isosceles trapezium is cyclic. The third lesson shows why.
A parallelogram that is not a rectangle is not cyclic. Its opposite angles are equal, so they add to $180^\circ$ only when each is $90^\circ$ .
A rhombus that is not a square is not cyclic, for the same reason.

Practise with new numbers

Try it in Python

The program places four points on a circle, measures the four angles of the quadrilateral from their positions, and adds the opposite angles.

Common mistakes

Adding neighbouring angles. The theorem is about opposite angles. In the picture, $\angle A + \angle B = 95 + 70 = 165$ , not $180$ .
Using $360^\circ$ for a pair. A pair of opposite angles adds to $180^\circ$ . All four angles together add to $360^\circ$ .
Forgetting the circle. The theorem needs all four corners on one circle. A quadrilateral with a corner off the circle is not cyclic.
Naming the wrong angle. In the quadrilateral $ABCD$ , the angle $\angle C$ is opposite to $\angle A$ , not to $\angle B$ .

Remember these

A cyclic quadrilateral has all four corners on one circle.
Opposite angles of a cyclic quadrilateral add to $180^\circ$ .
All four angles of a quadrilateral add to $360^\circ$ .
A rectangle and an isosceles trapezium are cyclic. A parallelogram that is not a rectangle is not.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

In a cyclic quadrilateral $PQRS$ , $\angle P = 64^\circ$ and $\angle Q = 117^\circ$ . Find $\angle R$ and $\angle S$ .

Two opposite angles of a cyclic quadrilateral are $(6x)^\circ$ and $(4x + 40)^\circ$ . Find $x$ .

Can a rhombus that is not a square be cyclic?

A cyclic quadrilateral has $\angle A = 88^\circ$ . What do $\angle B$ and $\angle D$ add to?

Why is the quadrilateral in the picture with angles $95^\circ$ , $70^\circ$ , $85^\circ$ and $110^\circ$ possible, but one with $95^\circ$ , $70^\circ$ , $85^\circ$ and $100^\circ$ is not cyclic?