## Quadrilateral Properties | Trapezium, parallelogram, Rhombus

What is *Quadrilaterals*

A* Quadrilaterals* is defined as a simple closed figure bounded by four lines in plane. If any four points are not colliner, then we obtain a closed figure.

There above all types of figures are know as **quadrilaterals.** We can easily draw many more quadrilaterals and we can identify many around us.

**Terminology related to Quadrilaterals :**

**Sides and adjacent sides :**

The four line segments AB, BC, CD & DA are known as sides of **quadrilateral**. Two sides of quadrilaterals which have a common end point are called the adjacent sides.

Thus ( AB, BC) ; ( BC , CD ) ; ( CD , DA ) ; ( DA , AB ) are four pairs of adjacent sides of the quadrilateral of ABCD.

#### Vertices of the quadrilateral :

The points A, B, C & D are the Vertices of the *quadrilateral* ABCD.

#### Angles and adjacent sides :

The four angles ∠DAB , ∠ABC, ∠BCD & ∠CDA are known as angles of quadrilateral. These angles can be also denoted as ∠A , ∠B, ∠C & ∠D respectively.

Two angles of the a quadrilateral having a common side are called its adjacent angles. Thus (∠A , ∠B ) ; ( ∠B, ∠C ) ; (∠C , ∠D ) ; ( ∠D, ∠A) are four pairs of adjacent angles of the quadrilateral of ABCD.

Two angles of a quadrilateral which are not adjacent angles are the opposite angles . Thus (∠A , ∠C ) ; ( ∠B, ∠D ) are two pairs of opposite angles of the quadrilateral of ABCD.

#### Diagonals of the quadrilateral:

The line segment joining the opposite vertices of a quadrilateral is called a Diagonals of the quadrilateral.

Here AC and BD are two diagonals of the quadrilateral ABCD.

#### Perimeter of the quadrilateral:

Quadrilateral perimeter mean, The sum of length of all the four sides of quadrilateral.

Here Perimeter of the quadrilateral ABCD = AB + BC + CD + DA.

### Types of quadrilaterals with formulas :

#### Properties and formulas of Trapezium (Trapezoid) :

The quadrilateral having one pair of opposite sides parallel to each other is called a* trapezium**.*

Here AD || BC , Height from base AD to base BC is ” h” and length of AD = a and BC = b

**Area of the trapezoid** = 1/2 x sum of parallel sides x height

**Area of Trapezium (Trapezoid) ABCD** = **(1/2 ) (a + b) h.**

- If the non -parallel sides are equal then diagonals will be too.
- Median of trapezium = 1/2 x sum of parallel sides ( median is the line equidistant from the parallel sides). Here EF is
**median of trapezium**ABCD . EF = (a + b) /2. (here AE = EB & DF= FC).

**Isosceles trapezium :**

The quadrilateral having one pair of opposite sides parallel to each other and other pair of sides are equal then it is called an *Isosceles trapezium*.

#### Parallelogram Properties and formulas:

If both pairs of opposite sides of the quadrilateral are parallel, such a quadrilateral is called *parallelogram.*

**Area of parallelogram** = Base x height = **bh**

**Perimeter of parallelogram** = **2 ( b+ c )**

*Area of parallelogra*m = product of any two adjacent sides x sine of the included angle = b x c x sin B.

- The opposite sides of a parallelogram are
**equal in length**. Thus AB = DC and AD = BC. - Opposite angles of a parallelogram are
*congruent*(equal measure). - The diagonals of a parallelogram bisect each other. Here BE = ED and AE = EC.
- Each diagonal of a parallelogram divides it into two triangles of the same area. Here area of ΔABC = Area of ΔACD and area of ΔABD = area of ΔBCD.
- Bisectors of the angles of parallelogram form a
*rectangle*. - A parallelogram inscribed in circle is a
**rectangle**. - A parallelogram circumradius about a circle is a
**rhombus.** - The sum of the squares of the diagonals is equal to the sum of the square of the four sides.
- Here AC
^{2}+ BD^{2}= AB^{2}+ BC^{2}+ CD^{2}+ DA^{2}= 2 b^{2}+ 2 c^{2}

#### Properties and formulas of Rectangle:

A parallelogram having any one of its angle as right angle ( 90°) is a* rectangle.*

**Area of the rectangle = bh**

**Perimeter of the rectangle = 2 (b + h)**

**Length of diagonal ( l ) = √ b ^{2} + h^{2}**

- The diagonals of a rectangle are equal and bisect each other.
- The opposite sides of a rectangle are parallel and opposite sides of a rectangle are congruent. Here AD || BC and AB || DC.
- All four angles of a rectangle are right angles. Here ∠A =∠B = ∠C = ∠D = 90°.
- All
**rectangles**are parallelograms but the reverse is not possible.

#### Properties and formulas of Rhombus:

A parallelogram having equal adjacent sides is called a *rhombus*. Here height, AB = BC = CD = DA = b & AB || DC , AD || BC and are the diagonals

**Area of the Rhombus** = 1/2 **x** product of diagonals **x** sine of the angle between them.

**Area of the Rhombus ABCD** = (1/2) d1 d2 ( Diagonals are right angle so sin 90° = 1)

*Here Area of the Rhombus ABCD* =** bh**

**Perimeter of rhombus** =** 4b**

- All the sides of a rhombus are congruent. Here AB = BC = CD = DA = b.
- Opposite sides of a rhombus are parallel. Here AB || DC & AD || BC.
- The diagonals of a rhombus bisect each other at right angles. Here ∠AED = ∠AEB = ∠BEC = ∠DEC = 90°.
- Opposite internal angles of a rhombus are congruent (equal in size). Here ∠ABC = ∠ADC and ∠BAD = ∠DCB.
- The sum of any two consecutive internal angles of a rhombus equal to 180°. Here ∠A + ∠B = ∠B + ∠C = ∠C +∠D = ∠D + ∠A = 180°.
- All
**rhombuses**are parallelograms but the reverse is not possible. - A
*rhombus*may or may not be a square but all square but all squares are rhombuses.

#### Properties and formulas of Square :

A parallelogram having equal adjacent sides and all angles of right angle (90°) is called square.

Here length of the side for square ABCD = a

Length of diagonal = **d = √2 a**

*Area of the square* =** b ^{2}**

**Area of the square** ABCD =** (1/2) d ^{2}**

**Perimeter of the square** **= 4b**

- All four sides of a square are
**congruent**. - Opposite sides of a square are parallel. Here AB || DC & AD || BC.
- The diagonals of a square are equal. Here AE = BD
- The diagonals of a square bisect each other at right angles. Here ∠AED = ∠AEB = ∠BEC = ∠DEC = 90°.
- All angles of a square are 90° . Here ∠A = ∠B = ∠C = ∠D = 90°.
- All squares are belongs to a special kind of rectangles where all the sides have equal length.
- Side is the diameter of the inscribed circle.
- Diagonal is the diameter of the circumradius circle.
**Here Diameter = √2 a & circumradius = a / √2**

#### Area of Kite:

The quadrilateral having two pairs of equal adjacent sides is called *kite*.

**Here BC = DC = a & AB = AD = b**

** d1** is the length of a diagonal.

* d2 *is the length of the other diagonal.

*Area of kite* = **(1/2) d1 d2.**

### Some Important points in quadrilaterals

- Parallelograms, trapeziums, rhombuses, rectangles and squares are all
__quadrilaterals__. - Parallelograms, rhombuses, rectangles and squares are all trapeziums.
- Rhombuses, rectangles and squares are all Parallelograms.
- All squares are Rhombuses but the converse is not true.
- all squares , rhombuses are kites.

Classifications of Triangles with properties | Triangle Area Formulas

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