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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.

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

Select all cyclic quadrilaterals:

  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

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