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In this tutorial on basic geometry concepts, we cover the types and properties of quadrilaterals: Parallelogram, rectangle, square, rhombus, trapezium.

#### Definition:

A quadrilateral is a simple closed figure with four sides.

There are five types of quadrilaterals.

• Parallelogram
• Rectangle
• Square
• Rhombus
• Trapezium

One common property of all quadrilaterals is that the sum of all their angles equals 360°.

Let us look into the properties of different quadrilaterals.

## Parallelogram #### Properties of a parallelogram

• Opposite sides are parallel and congruent.
• Opposite angles are congruent.
• Diagonals bisect each other and each diagonal divides the parallelogram into two congruent triangles.
• If one of the angles of a parallelogram is a right angle then all other angles are right and it becomes a rectangle.

#### Important formulas of parallelograms

• Area = L * H
• Perimeter = 2(L+B)

## Rectangles #### Properties of a Rectangle

• Opposite sides are parallel and congruent.
• All angles are right.
• The diagonals are congruent and bisect each other (divide each other equally).
• Opposite angles formed at the point where diagonals meet are congruent.
• A rectangle is a special type of parallelogram whose angles are right.

#### Important formulas for rectangles

• If the length is L and breadth is B, then

Length of the diagonal of a rectangle = √(L2 + B2)

• Area = L * B
• Perimeter = 2(L+B)

## Squares #### Properties of a square

• All sides and angles are congruent.
• Opposite sides are parallel to each other.
• The diagonals are congruent.
• The diagonals are perpendicular to and bisect each other.
• A square is a special type of parallelogram whose all angles and sides are equal.
• Also, a parallelogram becomes a square when the diagonals are equal and right bisectors of each other.

#### Important formulas for Squares

• If ‘L’ is the length of the side of a square then length of the diagonal = L √2.
• Area = L2.
• Perimeter = 4L

## Rhombus #### Properties of a Rhombus

• All sides are congruent.
• Opposite angles are congruent.
• The diagonals are perpendicular to and bisect each other.
• Adjacent angles are supplementary (For eg., ∠A + ∠B = 180°).
• A rhombus is a parallelogram whose diagonals are perpendicular to each other.

#### Important formulas for a Rhombus

If a and b are the lengths of the diagonals of a rhombus,

• Area = (a* b) / 2
• Perimeter = 4L

## Trapezium #### Properties of a Trapezium

• The bases of the trapezium are parallel to each other (MN ⫽ OP).
• No sides, angles and diagonals are congruent.

#### Important Formulas for a Trapezium

• Area = (1/2) h (L+L2)
• Perimeter = L + L1 + L2 + L3

## Summary of properties

Summarizing what we have learnt so far for easy reference and remembrance:

 S.No. Property Parallelogram Rectangle Rhombus Square 1 All sides are congruent ✕ ✕ ✓ ✓ 2 Opposite sides are parallel and congruent ✓ ✓ ✓ ✓ 3 All angles are congruent ✕ ✓ ✕ ✓ 4 Opposite angles are congruent ✓ ✓ ✓ ✓ 5 Diagonals are congruent ✕ ✓ ✕ ✓ 6 Diagonals are perpendicular ✕ ✕ ✓ ✓ 7 Diagonals bisect each other ✓ ✓ ✓ ✓ 8 Adjacent angles are supplementary ✓ ✓ ✓ ✓

Properties of Lines and Angles
Properties and formulas of Circles
Types of Triangles and Properties

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1. Sophie says:

Do opposite sides in a quadrilateral have to be equal in order for it to have diagonals that are perpendicular?

• Maryam says:

Yes when sides on opposite are equal then only the diagnols perpendicularly bisect each other

• K.K.Mukherjee says:

It is not correct.The diagonals are perpendicular if and only if all the sides are equal as in the case of a rhombus or a square.
The diagonals will be perpendicular also if the pairs of adjacent sides are equal as in the case of a kite

However, in a kite the diagonals are perpendicular but the diagonals are not congruent. In a rhombus, rectangle or square for it to have diagonals that is perpendicular and congruent all of its sides must be congruent. In the case of kite, of all of its sides are congruent it will become either a rhombus or a square.

• Sid says:

No all sides of a quadrilateral should be congruent for the diagonals to be perpendicular

• Aaeesha says:

No
Diagonal will be perpendicular only when all sides are equal not opposite sides.

2. Piyush says:

Are the measure of a pair of parallel sides in a trapezium equal?

• Harish says:

No they are not.
If they were, it would become a parallelogram and a trapezium is not a parallelogram

• Krish says:

No they are never same. The are always different .
If they will be equal it will not be a trapezium

• Aaeesha says:

Not necessarily equal
Sometimes it’s equal then the trapezium is called isotrapezium.

3. anonymous says:

are the diagonals of the rectangles are congruent?
how to proof it?

• Anil says:

Since the opposite sides of a rectangle are congruent, it’s diagonals are also congruent. This can be done by applying Pythagoras theorem..

• Gurleen says:

Take a triangle with common base from that dignols and made that triangle congruent by sas then both side will be equal by cpct

4. Grreshma says:

If ab =33 bc =27 CD = 33 DA=21 area of quadrilateral ?

• Simron says:

Area of the quadrilateral = Area of rectangle + Area of Right angled triangle
Length of the rectangle = 33cm
Breadth of the rectangle = 21cm
Area of rectangle = Length x Breadth
= 33 x 21
= 693 sq. cm
Now in right angled triangle,
Hypotenuse = 33cm
Base = 6cm
Therefore, the height of the triangle = Square root of (square of hypotenuse – square of base) [ By Pythagoras Theorem ]
= Square root of(33 x 33 – 6 x 6)
= Square root of 1053
= 32.44cm
Area of Right angled Triangle = 1/2 x base x height
= 1/2 x (27-21) x 32.44
= 1/2 x 6 x 32.44
= 97.32 sq. cm

So, Area of the quadrilateral = Area of rectangle + Area of Right angled triangle
= 693 + 97.32
= 790.32 sq. cm

• Saurabh says:

Why are you asking because it is quite easy?

You know that when the heights of quadrilateral are given which will perpendicular to any one of diagonal then fomula for area is given by
=1/2×(h1+h2)×diagonal

5. Kevin says:

What are the names of the three different quadrilaterals with two congruent
diagonals?

6. Ajaz says:

Explain the sum of properties of quadrilateral is 360 degree.

7. VUSI says:

Are diagonals the bisectors of angles in a quadrilateral or parallelogram?

• Mangalam says:

Parallelogram

8. Kunal says:

Is parallelogram a trapezium?

• S Doraswany Naidu says:

Never a parallelogram be a trapezium.
A parallelogram can form a rectangle.

9. Bodhisatya says:

I have an query on trapezium
If a angle of a trapezium is 90°then what is it called

• Harshit says:

Rectangle OR square

10. nabaah says:

squares are my favourite type of quadrilaterals