A quadrilateral is a closed shape consisting of four straight line segments.

Polygons

## Parallelogram

Definition 2: Parallelogram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

### Example 1: Properties of a parallelogram

#### Question

$ABCD$ is a parallelogram with $AB∥DC$ and $AD∥BC$. Show that:

1. $AB=DC$ and $AD=BC$

2. $A ‸=C ‸$ and $B ‸=D ‸$

##### Prove $△ABC≡△CDA$

In $△ABC$ and $△CDA$,

(1)

$∴$ Opposite sides of a parallelogram have equal length.

We have already shown $A ‸ 2 =C ‸ 3$ and $A ‸ 1 =A ‸ 4$. Therefore,

$A ‸=A ‸ 1 +A ‸ 2 =C ‸ 3 +C ‸ 4 =C ‸.$(2)

Furthermore,

$B ‸=D ‸(△ABC≡△CDA)$(3)

Therefore opposite angles of a parallelogram are equal.

Summary of the properties of a parallelogram:

• Both pairs of opposite sides are parallel.

• Both pairs of opposite sides are equal in length.

• Both pairs of opposite angles are equal.

• Both diagonals bisect each other.

### Example 2: Proving a quadrilateral is a parallelogram

#### Question

Prove that if both pairs of opposite angles in a quadrilateral are equal, the quadrilateral is a parallelogram.

##### Find the relationship between $x ‸$ and $y ‸$

In $WXYZ$

$W ‸=Y ‸=y ‸(given)Z ‸=X ‸=x ‸(given)W ‸+X ‸+Y ‸+Z ‸=360 ° (sumofint.∠'sofquad.)∴2x ‸+2y ‸=360 ° ∴x ‸+y ‸=180 ° W ‸+Z ‸=x ‸+y ‸=180 °$(4)

But these are co-interior angles between lines $WX$ and $ZY$. Therefore $WX∥ZY$.

##### Find parallel lines

Similarly $W ‸+X ‸=180 °$. These are co-interior angles between lines $XY$ and $WZ$. Therefore $XY∥WZ$.

Both pairs of opposite sides of the quadrilateral are parallel, therefore $WXYZ$ is a parallelogram.

### Investigation 1: Proving a quadrilateral is a parallelogram

1. Prove that if both pairs of opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram.

2. Prove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

3. Prove that if one pair of opposite sides of a quadrilateral are both equal and parallel, then the quadrilateral is a parallelogram.

A quadrilateral is a parallelogram if:

• Both pairs of opposite sides are parallel.

• Both pairs of opposite sides are equal.

• Both pairs of opposite angles are equal.

• The diagonals bisect each other.

• One pair of opposite sides are both equal and parallel.

### Exercise 1:

Prove that the diagonals of parallelogram $MNRS$ bisect one another at P.

Hint: Use congruency.

To-do.

## Rectangle

Definition 3: Rectangle

A rectangle is a parallelogram that has all four angles equal to $90 °$.

A rectangle has all the properties of a parallelogram:

• Both pairs of opposite sides are parallel.

• Both pairs of opposite sides are equal in length.

• Both pairs of opposite angles are equal.

• Both diagonals bisect each other.

It also has the following special property:

### Example 3: Special property of a rectangle

#### Question

$PQRS$ is a rectangle. Prove that the diagonals are of equal length.

##### Prove $△PSR≡△QRS$

In $△PSR$ and $△QRS$

$PS=QR(equalopp.sidesofrectangle)SRisacommonsidePS ‸R=QR ‸S=90 ° (int.∠ofrectangle)∴△PSR≡△QRS(RHS)ThereforePR=QS$(5)

The diagonals of a rectangle are of equal length.

Summary of the properties of a rectangle:

• Both pairs of opposite sides are parallel.

• Both pairs of opposite sides are of equal length.

• Both pairs of opposite angles are equal.

• Both diagonals bisect each other.

• Diagonals are equal in length.

• All interior angles are equal to $90 °$.

### Exercise 2:

$ABCD$ is a quadrilateral. Diagonals $AC$ and $BD$ intersect at T. $AC=BD$, $AT=TC$, $DT=TB$. Prove that:

1. $ABCD$ is a parallelogram.

2. $ABCD$ is a rectangle.

To-do.

## Rhombus

Definition 4: Rhombus

A rhombus is a parallelogram with all four sides of equal length.

A rhombus has all the properties of a parallelogram:

• Both pairs of opposite sides are parallel.

• Both pairs of opposite sides are equal in length.

• Both pairs of opposite angles are equal.

• Both diagonals bisect each other.

It also has two special properties:

### Example 4: Special properties of a rhombus

#### Question

$XYZT$ is a rhombus. Prove that:

1. the diagonals bisect each other perpendicularly;

2. the diagonals bisect the interior angles.

##### Prove $△XTO≡△ZTO$
$XT=ZT(allsidesequalforrhombus)TOisacommonsideXO=OZ(diag.ofrhombusbisect)∴△XTO≡△ZTO(SSS)∴O ‸ 1 =O ‸ 4 ButO ‸ 1 +O ‸ 4 =180 ° (sum∠'sonstr.line)∴O ‸ 1 =O ‸ 4 =90 °$(6)

We can further conclude that $O ‸ 1 =O ‸ 2 =O ‸ 3 =O ‸ 4 =90 °$.

Therefore the diagonals bisect each other perpendicularly.

##### Use properties of congruent triangles to prove diagonals bisect interior angles
$X ‸ 2 =Z ‸ 1 (△XTO≡△ZTO)andX ‸ 2 =Z ‸ 2 (alt.int.∠'s,XT∥YZ)∴Z ‸ 1 =Z ‸ 2$(7)

Therefore diagonal $XZ$ bisects $Z ‸$. Similarly, we can show that $XZ$ also bisects $X ‸$; and that diagonal $TY$ bisects $T ‸$ and $Y ‸$.

We conclude that the diagonals of a rhombus bisect the interior angles.

To prove a parallelogram is a rhombus, we need to show any one of the following:

• All sides are equal in length.

• Diagonals intersect at right angles.

• Diagonals bisect interior angles.

Summary of the properties of a rhombus:

• Both pairs of opposite sides are parallel.

• Both pairs of opposite sides are equal in length.

• Both pairs of opposite angles are equal.

• Both diagonals bisect each other.

• All sides are equal in length.

• The diagonals bisect each other at $90 °$.

• The diagonals bisect both pairs of opposite angles.

## Square

Definition 5: Square

A square is a rhombus with all four interior angles equal to $90 °$.

OR

A square is a rectangle with all four sides equal in length.

A square has all the properties of a rhombus:

• Both pairs of opposite sides are parallel.

• Both pairs of opposite sides are equal in length.

• Both pairs of opposite angles are equal.

• Both diagonals bisect each other.

• All sides are equal in length.

• The diagonals bisect each other at $90 °$.

• The diagonals bisect both pairs of opposite angles.

It also has the following special properties:

• All interior angles equal $90 °$.

• Diagonals are equal in length.

• Diagonals bisect both pairs of interior opposite angles (i.e. all are $45 °$).

To prove a parallelogram is a square, we need to show either one of the following:

• It is a rhombus (all four sides of equal length) with interior angles equal to $90 °$.

• It is a rectangle (interior angles equal to $90 °$) with all four sides of equal length.

## Trapezium

Definition 6: Trapezium

A trapezium is a quadrilateral with one pair of opposite sides parallel.

## Kite

Definition 7: Kite

A kite is a quadrilateral with two pairs of adjacent sides equal.

### Example 5: Properties of a kite

#### Question

$ABCD$ is a kite with $AD=AB$ and $CD=CB$. Prove that:

1. $AD ‸C=AB ‸C$

2. Diagonal $AC$ bisects $A ‸$ and $C ‸$

##### Prove $△ADC≡△ABC$

In $△ADC$ and $△ABC$,

$AD=AB(given)CD=CB(given)ACisacommonside∴△ADC≡△ABC(SSS)∴AD ‸C=AB ‸C$(8)

Therefore one pair of opposite angles are equal in kite $ABCD$.

##### Use properties of congruent triangles to prove $AC$ bisects $A ‸$ and $C ‸$
$A ‸ 1 =A ‸ 2 (△ADC≡△ABC)andC ‸ 1 =C ‸ 2 (△ADC≡△ABC)$(9)

Therefore diagonal $AC$ bisects $A ‸$ and $C ‸$.

We conclude that the diagonal between the equal sides of a kite bisects the two interior angles and is an axis of symmetry.

Summary of the properties of a kite:

• Diagonal between equal sides bisects the other diagonal.

• One pair of opposite angles are equal (the angles between unequal sides).

• Diagonal between equal sides bisects the interior angles and is an axis of symmetry.

• Diagonals intersect at $90 °$.

### Investigation 2: Relationships between the different quadrilaterals

Heather has drawn the following diagram to illustrate her understanding of the relationships between the different quadrilaterals. The following diagram summarises the different types of special quadrilaterals.

1. Explain her possible reasoning for structuring the diagram as shown.

2. Design your own diagram to show the relationships between the different quadrilaterals and write a short explanation of your design.

### Exercise 3:

Use the sketch of quadrilateral $ABCD$ to prove the diagonals are perpendicular to each other.

Explain why quadrilateral $WXYZ$ is a kite. Write down all the properties of quadrilateral $WXYZ$.

To-do.

To-do.