# Types and Properties of Quadrilaterals (Quadrangles)

# Quadrilaterals Overview

A quadrilateral is a polygon with four sides (edges) or four vertices or four corners it can also be called as a **TETRAGON**

## Properties of Quadrilaterals

- Four Sides (
**Tetragon**) - Four Angles (
**Quadrangle**) - Four Vertices
- Interior angles’ sum to 360°
- Exterior angles’ sum to 360°

## Important Terms Related to Quadrilaterals

### Diagonals

Diagonals are line segments that join two opposite vertices of a quadrilateral (corners).

- A quadrilateral can have only two diagonals

In the figure below line segment AD represents a diagonal of the Quadrilateral ABCD.

### Adjacent Sides

Two sides of a quadrilateral are adjacent, if they share a common vertex.

In the quadrilateral shown above the 4 pair of adjacent sides are

- AD and DC
- DC and BC
- BC and AB
- AB and AD

### Adjacent Angles

Two angles are of a quadrilateral adjacent, if they share a common side.

In the quadrilateral shown above the 4 pair of adjacent angles are

- ∠1 and ∠3
- ∠3 and ∠4
- ∠2 and ∠4
- ∠1 and ∠2

## Types of Quadrilaterals

### Convex Quadrilaterals

All interior angles are strictly less than 180 degrees

#### 1. Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel and equal sides

##### Properties of Parallelogram

**Theorem**: Opposite sides are equal.**Theorem**: Opposite angels are equal.**Theorem**: Diagonals bisect each other.**Theorem**: The diagonals of a parallelogram bisect each other to form two pairs of congruent triangles.- In a parallelogram if one angle is right angle, then all angles are right angle(i.e., It is a square).
**Theorem**: Parallelograms on the same base and between the same parallels are equal in area.

#### 2. Rhombus (Rhomb)

A rhombus is a parallelogram whose all four sides are congruent.

##### Properties of Rhombus

- All four sides are equal in length
- Opposite sides are parallel(i.e., It’s a parallelogram).
- Opposite angles are equal
- Consecutive angles are supplementary (i.e., Sum of consecutive angle = 180°).
- Diagonals are perpendicular bisectors of one another.
- In case, if one angle is right angle, then all angles are right angle(i.e., It will be a square).
- Four congruent triangles are formed by diagonals.

#### 3. Rectangle [Equiangular Quadrangle]

A rectangle is a plane figure with four straight sides and four right angles.

##### Properties of Rectangle

- Opposite sides that are equal and parallel (i.e., It’s a parallelogram).
- Adjacent sides are of unequal length
- Consecutive angles are supplementary (180 degree)
- Diagonals are equal and bisect each other.

#### 4. Square {Regular Quadrilateral}

A square is a rhombus whose all angles are right angles

##### Properties of Square

- The diagonals are equal and bisect each other and at right angle
- Opposite sides are parallel
- All four angles are equal
- All four sides are equal.

#### 5. Kite

A kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.

##### Properties of Kites

- Two disjoint pairs of adjacent sides are equal (by definition)
- One diagonal is the perpendicular bisector of the other diagonal.
- One diagonal is a line of symmetry (it divides the quadrilateral into two congruent triangles that are mirror images of each other).
- One diagonal bisects a pair of opposite angles

#### 6. Trapezoid (Trapezium)

A quadrilateral with at least one pair of parallel sides is known as a trapezoid

##### Properties of Kites

- The bases are parallel by definition
- Each lower base angle is supplementary to the upper base angle on the same side.

### Concave Quadrilaterals

At least one angle with a measure in between 180 degrees and 360 degrees

### Cyclic Quadrilaterals

A cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle

#### Properties of Cyclic Quadrilaterals

- The opposite angles of a cyclic quadrilateral sum to 180°
- i.e. a + c = 180°
- and b + d = 180°
- The exterior angle of a cyclic quadrilateral is equal to the interior

opposite angle - i.e. e = c

## Pictorial Representation of Classification of Quadrilaterals

## Angle Sum Property of a Quadrilateral

The sum of the angles of a quadrilateral is 360°

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