Cyclic Quadrilaterals

A cyclic quadrilateral has all four corners on a circle. Its angle properties come from the angle at the centre theorem, and proving a quadrilateral is cyclic is a skill you will use in many Grade 12 riders.

Opposite angles are supplementary

Theorem. The opposite angles of a cyclic quadrilateral are supplementary. (Reason: opp $∠$ s of cyclic quad)

Proof. Let $A B C D$ be cyclic with centre $O$ . Join $O B$ and $O D$ .

$\hat{O}_{1} = 2 \hat{A}$ and reflex $\hat{O}_{2} = 2 \hat{C}$ ( $∠$ at centre $= 2 \times ∠$ at circumference)
$\hat{O}_{1} + \hat{O}_{2} = 360°$ ( $∠$ s round a pt)
So $2 \hat{A} + 2 \hat{C} = 360°$ , and $\hat{A} + \hat{C} = 180°$

Example. If $\hat{B} = 2 \hat{D}$ in cyclic quadrilateral $A B C D$ , then $3 \hat{D} = 180°$ , so $\hat{D} = 60°$ and $\hat{B} = 120°$ .

Exterior angle equals interior opposite angle

Theorem. An exterior angle of a cyclic quadrilateral equals the interior opposite angle. (Reason: ext $∠$ of cyclic quad)

Example. $P QR S$ is cyclic and $P S$ is produced to $T$ . If $\hat{R S T} = 3 x$ and $\hat{P QR} = 2 x + 20°$ , then $3 x = 2 x + 20°$ , so $x = 20°$ .

Proving a quadrilateral is cyclic

Show one of these:

A pair of opposite angles is supplementary. (converse opp $∠$ s of cyclic quad)
An exterior angle equals the interior opposite angle. (converse ext $∠$ of cyclic quad)
A side subtends equal angles at the other two vertices. (converse $∠$ s in the same seg)

Example. A parallelogram is cyclic only if it is a rectangle. Its opposite angles are equal (parallelogram) and must also add up to 180° (cyclic), so each is 90°.

Never use cyclic quadrilateral properties until you have shown the quadrilateral is cyclic.

Key terms

Cyclic quadrilateral: A quadrilateral with all four vertices on one circle.
Supplementary angles: Two angles that add up to 180°.
Converse: A theorem with the conclusion and the condition swapped.

Common mistakes

Adding adjacent angles instead of opposite angles.
Using the exterior angle with the wrong interior angle.
Assuming a quadrilateral is cyclic because it is drawn in a circle without all four vertices on the circle.
Giving 'cyclic quad' as the reason when proving that a quadrilateral is cyclic.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

$A B C D$ is a cyclic quadrilateral with $\hat{B} = 2 \hat{D}$ . Calculate $\hat{B}$ and $\hat{D}$ .

Worked solution

Hidden - try the question first

Solution

$\hat{B} + \hat{D} = 180°$ (opp $∠$ s of cyclic quad)
$3 \hat{D} = 180° \Rightarrow \hat{D} = 60°$
$\hat{B} = 120°$

Question 2

5 marks total

routine

In the diagram, $O$ is the centre of the circle and $A B C D$ is a cyclic quadrilateral. Prove the theorem that the opposite angles of a cyclic quadrilateral are supplementary, that is, $\hat{A} + \hat{C} = 180°$ .

Worked solution

Hidden - try the question first

Solution

Construction: join $O B$ and $O D$
$\hat{O}_{1} = 2 \hat{A}$ ( $∠$ at centre $= 2 \times ∠$ at circumference)
$\hat{O}_{2} = 2 \hat{C}$ ( $∠$ at centre $= 2 \times ∠$ at circumference)
$\hat{O}_{1} + \hat{O}_{2} = 360°$ ( $∠$ s round a pt)
$\hat{A} + \hat{C} = 180°$

Question 3

6 marks total

complex

$P QR S$ is a cyclic quadrilateral. $P S$ is produced to $T$ . $\hat{R S T} = 3 x$ and $\hat{P QR} = 2 x + 20°$ .

(a) Calculate $x$ . (3)

(b) Calculate $\hat{P S R}$ . (2)

Worked solution

Hidden - try the question first

Solution

(a) $O B$ (ext $∠$ of cyclic quad)
$3 x = 2 x + 20°$
$x = 20°$
(b) $\hat{R S T} = 60°$
$\hat{P S R} = 180° - 60° = 120°$ ( $∠$ s on a str line)
(c) $\hat{QR S} = 180° - 95° = 85°$ (opp $∠$ s of cyclic quad)

Question 4

8 marks total

problem-solving

(a) In diagram (a), $A B C D$ is a parallelogram whose vertices lie on a circle. Prove that $A B C D$ is a rectangle. (4)

(b) In diagram (b), $N M$ is produced to $P$ . In quadrilateral $K L M N$ , $\hat{K} = 85°$ and $\hat{L M P} = 85°$ . Prove that $K L M N$ is cyclic. (2)

Worked solution

Hidden - try the question first

Solution

(a) $O B$ is a parallelogram, so $\hat{A} = \hat{C}$ (opp $∠$ s of $∥$ m)
$A B C D$ is cyclic, so $\hat{A} + \hat{C} = 180°$ (opp $∠$ s of cyclic quad)
$2 \hat{A} = 180° \Rightarrow \hat{A} = 90°$
A parallelogram with a right angle is a rectangle
(b) $\hat{L M P} = \hat{K} = 85°$ : the exterior angle at $M$ equals the interior opposite angle
$K L M N$ is cyclic (converse ext $∠$ of cyclic quad)
(c) A rhombus is a parallelogram, so a cyclic rhombus has all its angles equal to $90°$ (from (a))
A rhombus with a right angle is a square

FAQs

Is the cyclic quadrilateral proof examinable?

Yes. The proof that opposite angles are supplementary is examinable in Grade 11 and Grade 12. The converse is used as a result, not proved.

How do I prove that four points lie on a circle?

Show that the quadrilateral they form is cyclic, using one of the three converses.

Which shapes are always cyclic?

Rectangles (including squares) and isosceles trapeziums. Kites with two right angles are also cyclic.

Opposite angles are supplementary

Exterior angle equals interior opposite angle

Proving a quadrilateral is cyclic

Key terms

Common mistakes

Practice questions

FAQs

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