5.05 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.

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

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.

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:

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 180-92	Subtract 92 from both sides
\displaystyle m	\displaystyle =	\displaystyle 88\degree	Evaluate

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Example 2

Select all cyclic quadrilaterals:

A

B

C

D

E

F

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\degree	Add one pair of opposite angles
\displaystyle 59\degree+98\degree	\displaystyle =	\displaystyle 157\degree	Add the other pair

Option A is not a cyclic quadrilateral.

Option B:

\displaystyle 104\degree+76\degree	\displaystyle =	\displaystyle 180\degree	Add one pair of opposite angles
\displaystyle 97\degree+83\degree	\displaystyle =	\displaystyle 180\degree	Add 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.

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