Cyclic Quadrilaterals

A cyclic quadrilateral has all four corners on a circle: its opposite angles add up to 180°, and its exterior angle equals the interior opposite angle.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 17 real past-paper questions below

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What a Cyclic Quadrilateral Is

A quadrilateral is cyclic when all four corners sit on the same circle. Name it in order round the circle, like ABCD.

Rule

Opposite angles of a cyclic quadrilateral add up to $18 0^{\circ}$ . So A pairs with C, and B pairs with D.

Common mistake

A shape with a corner at the centre O is not cyclic. O is inside the circle, not on it.

Opposite Angles Add to 180°

The 2-mark question: find one angle, then give the reason. One mark each.

Worked example

1
$∠ B A D = 7 6^{\circ}$ — This is the angle you are given, at corner A.
2
$A and C are opposite$ — Go round the shape A, B, C, D. C is two corners on from A, so it is the opposite corner, not a neighbour.
3
$∠ B C D = 18 0^{\circ} - 7 6^{\circ}$ — Opposite angles of a cyclic quadrilateral add up to 180°, so take the given angle away from 180°.
4
$∠ B C D = 10 4^{\circ}$ — Do the subtraction. Check: 76 + 104 = 180.

Rule

Reason: opposite angles of a cyclic quadrilateral add up to 180°.

Common mistake

Taking the angle next to it instead of the opposite one. Neighbours in a cyclic quadrilateral do not have to add to $18 0^{\circ}$ .

The Exterior Angle

Sometimes one side is made longer, past the corner. The angle outside the shape is the exterior angle.

Worked example

1
$∠ B C E = 8 1^{\circ}$ — DC is made longer past C to a point E. The 81° sits outside the shape, between CE and CB.
2
$∠ B C D = 18 0^{\circ} - 8 1^{\circ} = 9 9^{\circ}$ — DCE is a straight line, and angles on a straight line add up to 180°. This gives the inside angle at C.
3
$A is opposite C$ — Going round A, B, C, D, corner A is across from corner C.
4
$∠ B A D = 18 0^{\circ} - 9 9^{\circ} = 8 1^{\circ}$ — Opposite angles of a cyclic quadrilateral add up to 180°. The answer matches the exterior angle.

Key idea

The exterior angle equals the interior opposite angle. Both are $8 1^{\circ}$ . Still write both reasons in the exam.

Angles in Algebra

Exams often write the angles with a letter. Use the opposite pairs to make equations.

A cyclic quadrilateral has angles, in order round the circle, $p^{\circ}$ , $7 n^{\circ}$ , $(3 n + 19)^{\circ}$ and $5 n^{\circ}$ . Find p.

Worked example

1
$7 n + 5 n = 180$ — 7n and 5n are opposite, so they add up to 180°. Start with the pair that has only n in it.
2
$12 n = 180$ — Collect the n terms: 7n + 5n = 12n.
3
$n = 15$ — Divide both sides by 12.
4
$3 (15) + 19 = 64$ — Put n = 15 into the angle opposite p.
5
$p = 180 - 64 = 116$ — p and 64° are opposite, so they add up to 180°.
6
$116 + 105 + 64 + 75 = 360$ — Check: 7n = 105 and 5n = 75. All four angles add up to 360°, as in any quadrilateral.

Key idea

Write the equation down first. The equation 7n + 5n = 180 earns a mark for working even if you slip later.

Sometimes you get a ratio instead. Here p and m are opposite, $p : m = 4 : 5$ , and y is opposite $(m - 15)^{\circ}$ . Find y.

Worked example

1
$4 + 5 = 9 parts$ — The opposite pair shares 180° in the ratio 4 : 5. Add the ratio numbers to get the total parts.
2
$180 \div 9 = 20$ — Share 180, not 360: only the opposite pair adds up to 180°. One part is 20°.
3
$m = 5 \times 20 = 100$ — m has 5 parts.
4
$m - 15 = 85$ — Work out the angle opposite y.
5
$y = 180 - 85 = 95$ — y and 85° are opposite, so they add up to 180°.

Is It Cyclic?

Some questions ask you to show that a quadrilateral is cyclic. The rule also works backwards.

Rule

If a pair of opposite angles adds up to 180°, the four corners lie on a circle, so the quadrilateral is cyclic. If they do not add up to 180°, it is not cyclic.

Quadrilateral ABCD has $A = (x + 10)^{\circ}$ , $B = 2 x^{\circ}$ , $C = (3 x - 50)^{\circ}$ and $D = (2 x - 40)^{\circ}$ . Show that ABCD is cyclic.

Worked example

1
$(x + 10) + 2 x + (3 x - 50) + (2 x - 40) = 360$ — You do not know yet that it is cyclic, so use the fact true for every quadrilateral: the angles add up to 360°.
2
$x + 2 x + 3 x + 2 x = 8 x$ — Collect the x terms first.
3
$10 + (- 50) + (- 40) = - 80$ — Now collect the numbers. Keep each minus sign with its number.
4
$8 x - 80 = 360$ — Put the two parts together.
5
$8 x = 440$ — Add 80 to both sides.
6
$x = 55$ — Divide both sides by 8.
7
$A = 55 + 10 = 65, C = 3 (55) - 50 = 115$ — Work out one pair of opposite angles, A and C.
8
$65 + 115 = 180$ — Opposite angles add up to 180°, so ABCD is a cyclic quadrilateral. Write this reason in words.

Common mistake

Using "opposite angles add up to 180°" to find x. That is the thing you must show, so you cannot use it. Find x from 360° first.

Which Method? Read the Question

First check that all four corners are on the circle.

You see	You do
Four corners on the circle	opposite angles add to 180°
A side extended past a corner	straight line, then opposite angle
Angles in algebra	opposite pair adds to 180, solve
"Show that it is cyclic"	360° to find x, then check an opposite pair is 180°
A corner at the centre O	Angle at the Centre
Several facts in one diagram	Combining Circle Theorems

Key idea

If the question says "give a reason", write the words in full, not just "cyclic quad".

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