Cyclic quadrilaterals | Year 11

01 / LEARN

The idea, step by step

Opposite angles use opposite arcs

A cyclic quadrilateral has all four vertices on one circle. For vertices A, B, C, D in that order, A and C are opposite, as are B and D. Each opposite pair sums to $180^{\circ}$ . Adjacent angles do not generally have a fixed sum.

The reason is the inscribed-angle theorem: opposite angles intercept the two arcs between the remaining vertices, whose measures total a full turn. Always state the vertex order so the opposite pair is unambiguous.

Converse and exterior angles

For a convex quadrilateral, if a pair of opposite angles sums to $180^{\circ}$ , then it is cyclic. This converse is useful when the circle is not already given. Merely finding four angles that sum to $360^{\circ}$ proves nothing about cyclicity, because every quadrilateral has that sum.

An exterior angle formed by extending one side of a cyclic quadrilateral equals the opposite interior angle: the exterior angle is supplementary to its adjacent interior angle, and the opposite angle is supplementary to the same angle.

A common trap

Using adjacent angles as the supplementary pair.

Instead: Label vertices around the circle and pair A with C, B with D.

02 / WORKED EXAMPLES

See the thinking

Example 1

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.

B and D are opposite, so $(x + 20) + (2 x + 10) = 180$ .
Solve $3 x + 30 = 180$ , giving $x = 50$ .
Substitute: B is $70^{\circ}$ and D is $110^{\circ}$ .

x = 50

, B

= 70^{\circ}

, D

= 110^{\circ}

Example 2

A parallelogram on a circle

Prove that a cyclic parallelogram is a rectangle.

Opposite angles in a parallelogram are equal.
Opposite angles in a cyclic quadrilateral sum to $180^{\circ}$ .
Equal angles summing to $180^{\circ}$ must each be $90^{\circ}$ . The other opposite pair is also right-angled.

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

Check your understanding

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

Show answer

Yes.

Their sum is $180^{\circ}$ , so the converse theorem applies.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

ABCD is cyclic in that order. Angle A is $112^{\circ}$ . Find C.
2 marks
Opposite angles of a cyclic quadrilateral are x and 3x. Find both.
2 marks
A side at vertex C is extended to form an exterior angle of $128^{\circ}$ in cyclic ABCD. 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
ABCD 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.

ABCD 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 3x. 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 ABCD. 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 $60^{\circ}, 120^{\circ}, 60^{\circ}, 120^{\circ}$ .
ABCD 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 ↗

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