AustraliaNSW
Stage 5.1-3

Lesson

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^\circ$180°.

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 there exists a circle that all four points of the quadrilateral lie on.

#### Worked Example

Find the value of $x$x.

Think: The quadrilateral lies on a circle so it must a cyclic quadrilateral. We also know that a cyclic quadrilateral has opposite angles that are supplementary.

Do: Set up an equation and solve for $x$x

 $x+102^\circ$x+102° $=$= $180^\circ$180° Opposite angles of a cyclic quadrilateral are supplementary $x$x $=$= $180^\circ-102^\circ$180°−102° $x$x $=$= $78^\circ$78°

Summary

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^\circ$180°).

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

#### Practice questions

##### Question 1

In the adjacent figure, $ABCD$ABCD is inscribed inside a circle.

1. The sum of any two opposite angles in the quadrilateral $ABCD$ABCD is $\editable{}$°

2. Can a parallelogram without right angles be inscribed inside a circle?

Yes

A

No

B

Yes

A

No

B
##### Question 2

1. Select all correct options.

A

B

C

D

E

F

A

B

C

D

E

F
##### Question 3

Solve for $m$m in the diagram below:

1. Show all working and reasoning.

### Outcomes

#### MA5.3-17MG

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