Cyclic Quadrilaterals: Angles & Proofs | Year 11

01 / LEARN

Understand the idea.

Opposite angles use opposite arcs

A cyclic quadrilateral has four vertices on one circle. Label them in their order around the circle, then identify opposite vertices: in ABCD, the opposite pairs are A with C and B with D. Each opposite pair sums to $180^{\circ}$ .

The inscribed angles at opposite vertices intercept complementary arcs whose measures sum to $360^{\circ}$ . Each angle is half its arc measure, so their sum is $180^{\circ}$ . This reasoning does not impose the same sum on adjacent angles.

Converse and exterior angles

The converse is useful for proving cyclicity: a convex quadrilateral with a supplementary pair of opposite angles is cyclic. The total of all four angles being $360^{\circ}$ is insufficient, because every convex quadrilateral has that total.

Extending one side makes an exterior angle supplementary to its adjacent interior angle. In a cyclic quadrilateral, the opposite interior angle is supplementary to that same angle, so it equals the exterior angle. Keep the extended side and opposite vertex clear.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Solve an angle equation

ABCD is cyclic in that order. Angles B and D are $x + 20$ and $2 x + 10$ degrees. Find x and both angles.

Worked solution

1
Pair the opposite angles.
$(x + 20) + (2 x + 10) = 180$
B and D are opposite angles in the given cyclic order, so they are supplementary.
2
Solve the linear equation.
$\begin{aligned} 3 x + 30 & = 180 \\ 3 x & = 150 \\ x & = 50 \end{aligned}$
Collecting like terms and subtracting thirty isolates the unknown.
3
Substitute and check both angles.
$\begin{aligned} B & = 50 + 20 = 70^{\circ} \\ D & = 2 (50) + 10 = 110^{\circ} \\ B + D & = 180^{\circ} \end{aligned}$
The substituted values satisfy the circle theorem and are valid interior angles.

Answer

$x = 50$ , B $= 70^{\circ}$ , D $= 110^{\circ}$ .

EXAMPLE 02

A parallelogram on a circle

Prove that a cyclic parallelogram is a rectangle.

Worked solution

1
Use the parallelogram angle property.
$A = C, B = D$
Opposite interior angles of a parallelogram are equal.
2
Use the cyclic angle property.
$A + C = 180^{\circ}, B + D = 180^{\circ}$
Opposite angles of a cyclic quadrilateral are supplementary.
3
Determine every angle.
$A = B = C = D = 90^{\circ}$
Each equal pair summing to a straight angle consists of two right angles.
4
Apply the definition of rectangle.
The parallelogram has four right angles, so it is a rectangle.

Answer

It has four right angles, so it is a rectangle.

03 / YOUR TURN

Try it yourself.

Check your understanding

A convex quadrilateral has opposite angles $92^{\circ}$ and $88^{\circ}$ . Is it cyclic?

Show answer & reasoning

Yes.

$92^{\circ} + 88^{\circ} = 180^{\circ}$ . Since the stated quadrilateral is convex and these angles are opposite, the converse cyclic-quadrilateral theorem proves it is cyclic.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

$A B C D$ is cyclic in that order. Angle $A$ is $112^{\circ}$ . Find $C$ .
2 marks
Opposite angles of a cyclic quadrilateral are $x$ and $3 x$ . Find both.
2 marks
A side at vertex $C$ is extended to form an exterior angle of $128^{\circ}$ in cyclic $A B C D$ . Find the opposite interior angle $A$ .
2 marks
A convex quadrilateral has opposite angles $101^{\circ}$ and $79^{\circ}$ . Give a reason that its vertices lie on a circle.
2 marks
A student claims every parallelogram is cyclic because opposite sides are parallel. Give a counterexample.
3 marks
$A B C D$ is cyclic with $A = 80^{\circ}$ and $B = 95^{\circ}$ . Find $C$ and $D$ , then verify the angle sum.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

$A B C D$ is cyclic in that order. Angle $A$ is $112^{\circ}$ . Find $C$ .
2 marks
$180 - 112 = 68$ .
$68^{\circ}$ .
Opposite angles of a cyclic quadrilateral are $x$ and $3 x$ . Find both.
2 marks
$4 x = 180$ , so $x = 45$ .
$45^{\circ}$ and $135^{\circ}$ .
A side at vertex $C$ is extended to form an exterior angle of $128^{\circ}$ in cyclic $A B C D$ . Find the opposite interior angle $A$ .
2 marks
An exterior angle of a cyclic quadrilateral equals the opposite interior angle.
$128^{\circ}$ .
A convex quadrilateral has opposite angles $101^{\circ}$ and $79^{\circ}$ . Give a reason that its vertices lie on a circle.
2 marks
$101 + 79 = 180$ ; apply the converse cyclic-quadrilateral theorem.
Opposite angles are supplementary.
A student claims every parallelogram is cyclic because opposite sides are parallel. Give a counterexample.
3 marks
Its opposite angles sum to $120^{\circ}$ or $240^{\circ}$ , not $180^{\circ}$ .
A parallelogram with angles $60^{\circ}, 120^{\circ}, 60^{\circ}, 120^{\circ}$ .
$A B C D$ is cyclic with $A = 80^{\circ}$ and $B = 95^{\circ}$ . Find $C$ and $D$ , then verify the angle sum.
3 marks
$C = 180 - 80 = 100$ and $D = 180 - 95 = 85$ .
$80 + 95 + 100 + 85 = 360$ .
$C = 100^{\circ}$ , $D = 85^{\circ}$ ; total $360^{\circ}$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: geometric reasoning and real-number/proof material, regrouped into a school chapter; all exercises are original.

Official curriculum source ↗

Cyclic quadrilaterals

In this lesson

Understand the idea.

Opposite angles use opposite arcs

Converse and exterior angles

See the method in action.

Solve an angle equation

Pair the opposite angles.

Solve the linear equation.

Substitute and check both angles.

A parallelogram on a circle

Use the parallelogram angle property.

Use the cyclic angle property.

Determine every angle.

Apply the definition of rectangle.

Try it yourself.

Check your understanding

Keep practising