Space, shape and measurement: Explore, interpret and justify geometric relationships

Unit 3: Properties of cyclic quadrilaterals

Dylan Busa

Unit 3 outcomes

By the end of this unit you will be able to:

  • Define a cyclic quadrilateral.
  • Apply the theorem opposite angles of a cyclic quadrilateral are supplementary.
  • Apply the theorem exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
  • Apply the converses of equality between opposite angles, between angles in the same segment, and between exterior angles and interior opposite angles, to prove a quadrilateral is cyclic.

What you should know

Before you start this unit, make sure you can:

  1. State and use all the circle theorems covered in unit 2:
    1. A line drawn perpendicular to a chord from the centre of the circle bisects the chord.
    2. A line drawn from the circle centre to the mid-point of chord is perpendicular to the chord.
    3. The angle subtended by an arc or chord at the centre of a circle is twice the size of the angle subtended at the circumference.
    4. The diameter of a circle subtends a right angle at the circumference.
    5. If an angle subtended by a chord at a point on the circumference is a right angle, then the chord is a diameter.
    6. Angles subtended by the same arc or chord in the same segment of a circle (on the same side of the chord) are equal.

Introduction

A quadrilateral is any flat four-sided figure. Each of the four sides must be straight. A square is an example of a quadrilateral, as is a parallelogram. These are very special kinds of quadrilaterals with special properties.

Most quadrilaterals have no special characteristics other than that they have four straight sides and, therefore, four interior angles. We call these irregular quadrilaterals. Figure 1 shows various examples of quadrilaterals.

Figure 1: Examples of quadrilaterals

But some quadrilaterals are just the right shape that their four corners all lie on the circumference of the same circle. These are called cyclic quadrilaterals (see figure 2). Some special quadrilaterals such as squares, rectangles and parallelograms are always cyclic but many irregular quadrilaterals are cyclic as well.

Figure 2: Examples of cyclic quadrilaterals

 

Take note!

For a quadrilateral to be cyclic, all four vertices (corners) of the quadrilateral must lie on the circumference of the same circle.

Cyclic quadrilateral theorems

You do not need to be able to prove any of the cyclic quadrilateral theorems yourself. You can simply assume that they are true. The following sections explain the theorems that you need to be able to state and use. Note that they are numbered only for reference purposes and continue the numbering from unit 2. These theorems do not have official numbers.

Theorem 5

Let’s look at the theorem involving opposite angles of a cyclic quadrilateral.

Theorem 5: Opposite interior angles of a cyclic quadrilateral are supplementary

The opposite angles of a cyclic quadrilateral are supplementary (they add up to 180180).

If ABCDABCD is a cyclic quadrilateral, then ˆA+ˆC=180^A+^C=180 and ˆB+ˆD=180^B+^D=180.

Reason: opp s in cyclic quad

Example 3.1

Given the circle with centre OO with diameter GHGHand cyclic quadrilateral FGHIFGHI. GIGI is drawn and GˆHI=66G^HI=66. Determine the values of xx, yy and zz.

Solution

x=90(s in semi-circle)x=90(s in semi-circle)
y=1806690=24(s in Δ suppl)y=1806690=24(s in Δ suppl)
z=18066=114(s in cyclic quad)z=18066=114(s in cyclic quad)

Converse to theorem 5: Opposite interior angles are supplementary

If the opposite angles of a quadrilateral are supplementary (they add up to 180180), then the quadrilateral is cyclic.

If ˆA+ˆC=180^A+^C=180 or ˆB+ˆD=180^B+^D=180, then ABCD is a cyclic quadrilateral.

Reason: opp int s suppl

Theorem 6

The next theorem looks at exterior angles.

Theorem 6: Exterior angle of a cyclic quadrilateral equal to opposite interior angle

The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.

If ABCD is a cyclic quadrilateral, then ˆB2+ˆD.

Reason: ext =opp int in cyclic quad

Example 3.2

Given the circle with centre O and cyclic quadrilateral ABCD, and ˆO1=162. Determine the value the value of AˆBE.

Solution

ˆO2=360162=198( around a point)ˆD=99( at centre =2 at circumference)AˆBE=99(ext = opp int  in cyclic quad)

Converse to theorem 6: Exterior angle equal to opposite interior angle

If the exterior angle of a quadrilateral is equal to the opposite interior angle, then the quadrilateral is cyclic.

If AˆBE=ˆD, then ABCD is a cyclic quadrilateral.

Reason: ext =opp int

Proving a quadrilateral is a cyclic quadrilateral

So far, we have seen two ways in which we can prove that a quadrilateral is a cyclic quadrilateral.

  • If we can prove that the opposite interior angles of the quadrilateral are supplementary, then the quadrilateral is cyclic.
  • If we can prove that the exterior angle of the quadrilateral is equal to the opposite interior angle, then the quadrilateral is cyclic.

But there is a third way. Remember theorem 4 from unit 2? It stated that if the angles subtended by a chord of the circle are on the same side of the chord, then the angles are equal.

Figure 3: Angles in the same segment of a circle are equal

Because ˆQ and ˆP are both subtended by arc AB (or chord AB), then we know that ˆQ=ˆP. But now have a look at ABPQ. Can you see that all four vertices lie on the circumference of the same circle? Therefore, it is a cyclic quadrilateral.

Figure 4: All four vertices lie on the circumference of the same circle

This means that we can use the converse of theorem 4 to prove that a quadrilateral is cyclic. If the angles in the same segment of a circle are equal, then the quadrilateral made by the chord and the two angles must be a cyclic quadrilateral.

Take note!

There are three ways to prove that a quadrilateral is a cyclic quadrilateral:

Proof 1: Opp int s suppl

If ˆA+ˆC=180 or ˆB+ˆD=180, then ABCD is a cyclic quadrilateral.

Proof 2: ext =opp int


If AˆBE=ˆD, then ABCD is a cyclic quadrilateral.

Proof 3: s in same segment


If AˆCD=AˆBD, then ABCD is a cyclic quadrilateral.

Example 3.3

If COE is a diameter of circle centre O, prove that ABCD is a cyclic quadrilateral.

Solution

CˆDE=90(s in semi-circle)CˆDE=AˆBC=90
Therefore ABCD is a cyclic quadrilateral (ext =opp int ).

Exercise 3.1

  1. Find the value of the unknown angles:
    1. .
    2. .
    3. .
  2. In each case, determine if ABCD is a cyclic quadrilateral:
    1. .
    2. .

The full solutions are at the end of the unit.

Summary

In this unit you have learnt the following:

  • That a cyclic quadrilateral is any four-sided shape whose vertices all lie on the circumference of the same circle.
  • The opposite angles of a cyclic quadrilateral are supplementary.
  • The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
  • A quadrilateral can be proven to be a cyclic quadrilateral if you can show that:
    • the opposite angles are supplementary
    • the exterior angle is equal to the interior opposite angle
    • the angles subtended by one side of the quadrilateral are equal.

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