+-=x�+-=x+�-=x�+ Mathematics Handout

QUADRILATERAL

+-=x�+-=x+�-=x�+ A. Definition of quadrilateral 

There are many quadrilateral figures such as square, rhombus,

parallelogram and many more. Even there is composite figure. We can see these

quadrilateral figures in our daily life, like the shape of an object. The definition

of quadrilateral is a 2-dimensional figure/shape with 4 sides and 4 angles. All of

the sides are straight and the total of the angles is always 360°.

B. Properties of each quadrilateral Quadrilateral is the same as Android’s slogan ‘Be Together Not The Same’,

which means that each quadrilateral figure has the different properties and

formula. Some of the quadrilateral figure has many properties but some of them

has just like 2-5 properties. So, to know more, I will explain one by one about

the properties of quadrilateral figures that I know.

● Square: I am sure that you are familiar with this quadrilateral figure.

Square has some properties that other quadrilateral figure don’t have. So

we can say that square is a unique quadrilateral figure. Now we will find

out about the properties of square.

○ The unique properties of square is that all sides has the same

length.

○ The other unique properties in square is that the intersecting

diagonals in square form 4 right angles.

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○ All angles in square are right angle.

○ The diagonal in square intersect and bisect each other.

○ The opposite sides are parallel.

○ The diagonal in square are equal in length.

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○ Each of the 2 diagonals of square is an axis of symmetry line.

○ Square fill its frame in 8 ways.

So, that’s all about square.

● Rectangle: I think that this quadrilateral figure is the second best known

quadrilateral figure after square. Rectangle has some properties that is

the same with square. Now we will find out about the properties of

rectangle.

○ All angles in rectangle are right angle.

○ The opposite sides are equal in length.

○ The opposite sides are parallel.

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○ The diagonal in rectangle intersect and bisect each other

○ Each of the 2 diagonal of rectangle is an axis of symmetry line

○ Rectangle fill its frame in 4 ways

So, that’s all about rectangle.

● Parallelogram: This quadrilateral figure is almost the same as rectangle.

Parallelogram has some properties that is the same as rectangle. Now we

will find out about the properties of parallelogram.

○ The opposite sides are equal in length.

○ The opposite sides are parallel.

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○ The opposite interior angles are equal.

○ The diagonals in parallelogram bisect each other.

○ Each of the 2 diagonals of parallelogram separates it to 2 congruent

triangles.

So, that’s all about parallelogram

● Kite: I am sure that everyone especially kids know this quadrilateral figure.

Kite is one of the toy that kids like to play. Now we will find out about the

properties of kite.

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○ There are 2 pair of sides with equal length.

○ One of the diagonal is an axis of symmetry.

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○ One of the diagonal bisects the other diagonal and its perpendicular

to that other diagonal.

○ There is a pair of opposite interior angles which are equal.

○ The intersection of the diagonals form 90°.

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So, that’s all about kite.

● Trapezium: There are 3 types of trapezium which are arbitrary, isosceles

and right angle. Now we will find out about the properties of each

trapezium.

○ Arbitrary

■ All of the sides are not equal in length.

■ There is one pair of sides that is parallel.

○ Isosceles

■ There is one pair of opposite sides that is equal in length.

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■ There is one pair of sides that is parallel.

○ Right angle

■ This trapezium has one right interior angle

■ There is one pair of sides that is parallel.

So, that’s all about trapezium.

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● Rhombus: This quadrilateral figure is similar to parallelogram and kite but

all the side has the same length. Now we will find out about the properties

of rhombus.

○ All of the sides has the equal length.

○ One of the diagonal bisects the other diagonal and its perpendicular

to that other diagonal.

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○ There are two pair of opposite interior angles which are equal.

○ The intersection of the diagonals form 90°.

So, that’s all about rhombus.

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C. Proving area of each quadrilateral   To prove the area of quadrilateral figure, we can divide the figure to

several shape. After divide it, we count the area of each shape than we add all of

it. So, to know more, I will explain one by one about how to prove the area of

each quadrilateral figure.

● Square: to prove the area of square, we can divide it to several shape.

There are many ways to prove it, but I will tell only three ways.

○ The first way:

In this way, let us say the top side is ‘a’ and the right side is

‘b’.

So, it means that the lower side is also ‘a’ and the left side is

‘b’. We divide the ABCD square to two rectangles so it means

that side ‘a’ is divided to two. After that, we calculate the

area of each rectangle than we add all of it.

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○ The second way:

 

In this way, let us say the top side is ‘a’ and the right side is

‘b’.

So, it means that the lower side is also ‘a’ and the left side is

‘b’. We divide the ABCD square to two triangles. After that,

we

calculate the area of each triangle than we add all of it.

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○ The third way:

In this way, let us say the top side is ‘a’ and the right side is

‘b’.

So, it means that the lower side is also ‘a’ and the left side is

‘b’. We divide the ABCD square to three triangles. So, it

means

that side ‘a’ is divided to two. After that, we calculate the

area

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So, that’s how to prove square area.

● Rectangle: to prove the area of rectangle, we can divide it to several

shape. There are two ways to prove it.

○ The first way:

In this way, let us say the top side is ‘a’ and the right side is

‘b’.

So, it means that the lower side is also ‘a’ and the left side is

‘b’. We divide the ABCD rectangle to two rectangles so it

means that side ‘a’ is divided to two. After that, we calculate

the area of each rectangle than we add all of it.

○ The second way:

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In this way, let us say the top side is ‘a’ and the right side is

‘b’.

So, it means that the lower side is also ‘a’ and the left side is

‘b’. We divide the ABCD rectangle to two triangles. After that,

we calculate the area of each triangle than we add all of it.

○ The third way:

+-=x�+-=x+�-=x�+ In this way, let us say the top side is ‘a’ and the right side is

‘b’.

So, it means that the lower side is also ‘a’ and the left side is

‘b’. We divide the ABCD rectangle to three triangles. So, it

means that side ‘a’ is divided to two. After that, we calculate

the area of each triangle than we add all of it.

So that’s how to prove rectangle area.

● Parallelogram: to prove the area of parallelogram, we can divide it to

several shape. There are two ways to prove it.

○ The first way:

In this way, let us say the lower side is ‘b’ and the height is ‘h’.

So, it means that the top side is also ‘b’. We divide the

ABCD

parallelogram to two triangles. After that, we calculate the

area of each triangle than we add all of it.

○ The second way:

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In this way, let us say the lower side is ‘b’ and the height is ‘h’.

So, it means that the top side is also ‘b’. We divide the

ABCD

parallelogram to two parts. After that, we combine the two

parts so it will become one rectangle. After that we calculate

the area.

So, that’s how to prove parallelogram area.

● Kite: to prove the area of kite, we can divide it to several shape. There is

one way to prove it.

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In this way, let us say the diagonal 1 is ‘a’ and the diagonal 2 is

‘b’. We divide the ABCD kite to two triangles. After that, we

calculate the area of each triangle than we add all of it.

So that’s about how to prove kite area.

● Trapezium: to prove the area of trapezium, we can divide it to several

shape. There is one way to prove it.

○ The first way:

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In this way, let us say the top side is ‘a’, the lower side is ‘b’

and the height is ‘h’ We divide the ABCD trapezium to two

parts. After that, we calculate the area of each the part than

we add all of it.

So, that’s about how to prove trapezium area.

● Rhombus: to prove the area of rhombus, we can divide it to several shape.

There are three ways to prove it.

○ The first way:

+-=x�+-=x+�-=x�+ In this way, let us say the diagonal 1 is ‘a’ and the diagonal 2 is

‘b’. We divide the ABCD rhombus to two triangles. After

that, we calculate the area of each triangle than we add all of

it.

○ The second way:

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In this way, let us say the diagonal 1 is ‘a’ and the diagonal 2 is

‘b’. We divide the ABCD rhombus to two triangles. After

that, we calculate the area of each triangle than we add all of

it.

○ The third way:

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In this way, let us say the diagonal 1 is ‘a’ and the diagonal 2 is

‘b’. We divide the ABCD rhombus to four triangles. After

that, we calculate the area of each triangle than we add all of

it.

So, that’s about how to prove rhombus area.

So that’s all the information that I can give. Thank you for reading.