topic badge
AustraliaNSW
Stage 5.1-3

5.05 Cyclic quadrilaterals

Lesson

Cyclic quadrilaterals

If we place four points on a circle as shown below, we can connect them to form a quadrilateral inside the circle. We call this shape a cyclic quadrilateral.

A circle with 4 vertices on the circumference that are joined to form a cyclic quadrilateral.

Cyclic quadrilaterals have the special property that opposite angles of a cyclic quadrilateral are supplementary: they will add to 180\degree.

Two cyclic quadrilaterals with opposite angles labelled as supplementary. Ask your teacher for more information.

The converse of this is also true. Given four points that form a quadrilateral, if the quadrilateral has opposite angles that are supplementary then the quadrilateral is a cyclic quadrilateral.

The image shows a quadrilateral with opposite angles supplementary. Ask your teacher for more information.

This means that we can draw a circle that passes through all four points of the quadrilateral.

Examples

Example 1

Solve for m in the diagram below:

A cyclic quadrilateral with a pair of opposite interior angles of m and 92 degrees.

Show all working and reasoning.

Worked Solution
Create a strategy

Add the opposite interior angles of the cyclic quadrilateral and equate the sum to 180\degree.

Apply the idea
\displaystyle m+92\displaystyle =\displaystyle 180(Opposite angles of a cyclic quadrilateral)
\displaystyle m+92-92\displaystyle =\displaystyle 180-92Subtract 92 from both sides
\displaystyle m\displaystyle =\displaystyle 88\degreeEvaluate

Example 2

Select all cyclic quadrilaterals:

A
A cyclic quadrilateral with opposite angles of 74 and 129 degrees and opposite angles of 98 and 59 degrees on other sides.
B
A cyclic quadrilateral with opposite angles of 104 and 76 degrees and opposite angles of 97 and 83 degrees on other sides.
C
A cyclic quadrilateral with opposite angles of both 99 degrees and opposite angles of 77 and 85 degrees on other sides.
D
A cyclic quadrilateral with opposite angles of 139 and 41 degrees and opposite angles of 143 and 37 degrees on other sides.
E
A cyclic quadrilateral with opposite angles of 37 and 113 degrees and opposite angles of 95 and 115 degrees on other sides.
F
A cyclic quadrilaterals with opposite angles of 133 and 47 degrees and opposite angles of 132 and 48 degrees on other sides.
Worked Solution
Create a strategy

Add the opposite angles and check if they are supplementary.

Apply the idea

Option A:

\displaystyle 74\degree+129\degree\displaystyle =\displaystyle 203\degreeAdd one pair of opposite angles
\displaystyle 59\degree+98\degree\displaystyle =\displaystyle 157\degreeAdd the other pair

Option A is not a cyclic quadrilateral.

Option B:

\displaystyle 104\degree+76\degree\displaystyle =\displaystyle 180\degreeAdd one pair of opposite angles
\displaystyle 97\degree+83\degree\displaystyle =\displaystyle 180\degreeAdd the other pair

Option B is a cyclic quadrilateral.

Using the same process for the rest of the options, we will find that the quadrilaterals in options B, D and F are cyclic quadrilaterals.

Idea summary
A cyclic quadrilateral on a circle.

A cyclic quadrilateral is a quadrilateral that can be formed by connecting four points that lie on the same circle. The opposite angles of a cyclic quadrilateral are supplementary (add to 180\degree).

If a quadrilateral has opposite angles that are supplementary then it is a cyclic quadrilateral.

Outcomes

MA5.3-17MG

applies deductive reasoning to prove circle theorems and to solve related problems

What is Mathspace

About Mathspace