Quadrilaterals

Quadrilateral is a two-dimensional figure characterized by having four sides, four vertices, and four angles. It can be broadly classified into two categories: concave and convex. Within the convex category, there are several specific types of quadrilaterals, including trapezoids, parallelograms, rectangles, rhombus, and squares.

The sum of the interior angles of a Quadrilateral is 360°. Let's learn what is a quadrilateral, its shapes, types, properties, formulas, and examples in detail.

What are Quadrilaterals?

A quadrilateral is defined as a four-sided polygon with four angles The quadrilateral is a type of polygon in which the sides are defined in a proper pattern.

A Quadrilateral is a shape with four sides, four corners, and four angles. No matter what type of quadrilateral it is, the total of all its inside angles always adds up to 360 degrees. Quadrilaterals come in various forms, each with distinct properties that define their angles, sides, and symmetry. Examples of quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapezoids, and kites. Each type has its special features and properties but they all are four-sided figures.

For example, in the diagram above , the quadrilateral defined as ABCD, ADCB, BCDA, CDAB, etc. It cannot be defined as ACBD or BDAC. Here, the quadrilateral's sides are AB, BC, CD, and DA, and the diagonals are AC and BD.

Properties of Quadrilateral

The properties of a quadrilateral are:

• It has 4 sides ( AB, BC, CD, and DA) .
• It has 4 vertices ( A, B, C, D) .
• It has 4 angles. ( ∠A, ∠B, ∠C, ∠D) .
• It has 2 diagonals. (AC and BD)
• The sum of its internal angles is 360°. ( ∠A + ∠B + ∠C + ∠D = 360°) .
• The sum of its exterior angles is 360°.

Convex and Concave Quadrilateral

Based on their properties, quadrilaterals are divided into two major types:

1. Convex quadrilaterals and
2. Concave quadrilaterals.

These concave and convex quadrilaterals can be further classified into their subdivisions.

Concave Quadrilateral

Quadrilaterals that have one interior angle greater than 180° and one diagonal lies outside the quadrilateral are called concave quadrilaterals.

One of the examples of a concave quadrilateral is a Dart. It is a quadrilateral with bilateral symmetry like a kite, but with a reflex interior angle.

Here, in the image given below, one of the interior angles of the quadrilateral is 210°, which is greater than 180°. Therefore,  the quadrilateral is a concave quadrilateral.

Concave Quadrilateral

Quadrilaterals that have all four interior angles less than 180° are called concave quadrilaterals.

There are various types of Concave Quadrilaterals, which are :

1. Trapezium
2. Kite
3. Parallelogram
4. Rectangle
5. Rhombus
6. Square

Common Types of Quadrilaterals

Quadrilaterals exhibit diverse shapes, ranging from the symmetrical squares and rectangles to the more complex and irregular parallelograms and trapezoids.

Square

A quadrilateral that has all sides equal and opposite sides parallel and all interior angles equal to 90° is called a Diagonals of squares bisect each other perpendicularly. Note that all squares are rhombus but not vice-versa. 

Properties of Square

The properties of a square are:

• All four sides of a square are equal to each other.
• The interior angles of a square are 90°.
• The diagonal of a square bisects each other at 90°.
• The opposite sides are parallel, and the adjacent sides are perpendicular in a square.

Where side is the length of any one of the sides.

Rectangle

Rectangle is a quadrilateral whose opposite sides are equal and parallel and all the interior angles equal to 90°.

Diagonals of a rectangle bisect each other.

Note that all the rectangles are parallelograms, but the reverse of this is not true.

Rectangle Properties

These are some of the important properties of rectangle:

• The opposite sides of a rectangle are parallel and equal. In the above example, AB and CD are parallel and equal, and AC and BD are parallel and equal.
• All 4 angles of a rectangle are equal and are equal to 90°. ∠A = ∠B = ∠C = ∠D = 90°.
• The diagonals of a rectangle bisect each other and the diagonals of a rectangle are equal, here, AD = BC.

Rhombus

Rhombus is a quadrilateral that has all sides equal and opposite sides parallel. Opposite angles of a rhombus are equal, and diagonals of the Rhombus bisect each other perpendicularly. .

Note all rhombus are parallelograms, but the reverse of this is not true.

Properties of Rhombus

Here are some of the key properties of a Rhombus:

• All 4 sides of a rhombus are equal. AB = BC = CD = AD.
• The opposite sides of a rhombus are parallel and equal. In the image above, AB is parallel to CD and AD is parallel to BC.
• The diagonals of a rhombus Bisect each other at 90°.

Where side is the length of any one of the sides.

Parallelogram

Parallelogram is a quadrilateral whose opposite sides are equal and parallel. Opposite angles of a Parallelogram are equal, and its diagonals bisect each other.

Properties of Parallelogram

The properties of a Parallelogram are:

• The opposite sides of a parallelogram are parallel and equal. In the above example, AB and CD are parallel and equal, and AC and BD are parallel and equal.
• The opposite angles in a parallelogram are equal. ∠A = ∠D and ∠B = ∠C.
• The diagonals of a parallelogram bisect each other.

Where, a and b are the adjacent sides of a parallelogram.

Trapezium

A trapezium is a quadrilateral that has one pair of opposite sides parallel. In a regular trapezium, non-parallel sides are equal, and its base angles are equal.

The area of trapezium is 1/2 × Sum of parallel sides × Distance between them.

Properties of Trapezium

Here are two important properties of a trapezium:

• The sides of the trapezium that are parallel to each other are known as the bases of trapezium. In the above image, AB and CD are the base of the trapezium.
• The sides of the trapezium that are non-parallel are called the legs. In the above image, AD and BC are the legs.

Where a, b, c, d are the side of trapezium and (a and b) are the parallel sides and the height (h) is the perpendicular distance between these parallel sides.

Kite

Kite has two pairs of equal adjacent sides and one pair of opposite angles equal. Diagonals of kites intersect perpendicularly.

The longest diagonal of the kite bisects the smaller one.

Properties of Kite

Let's discuss some of the properties of a kite.

• A kite has two pairs of equal adjacent sides. For example, AC = BC and AD = BD.
• The interior opposite angles that are obtuse are equal; here, ∠A = ∠B.
• The diagonals of a kite are perpendicular to each other; here, AB is perpendicular to CD.
• The longer diagonal of the kite bisects the shorter diagonal. Here, CD bisects AB.

where, a and b represent the lengths of the two pairs of equal sides of the kite.

Quadrilateral Theorems

• Sum of Interior Angles Theorem: In any quadrilateral, the sum of the measures of its interior angles equals 360 degrees.
• Opposite Angles Theorem: Within a quadrilateral, the sum of the measures of two opposite angles is 180 degrees.
• Consecutive Angles Theorem: Adjacent (consecutive) angles in a quadrilateral are supplementary, meaning their measures sum up to 180 degrees.
• Diagonals of Parallelograms Theorem: The diagonals of a parallelogram bisect each other, dividing each diagonal into two equal segments.
• Opposite Sides and Angles of Parallelograms Theorem: In a parallelogram, opposite sides are equal in length, and opposite angles are congruent.
• Diagonals of Rectangles and Rhombuses Theorem: In rectangles and rhombuses, the diagonals are equal in length. Additionally, the diagonals of a rectangle are congruent, while those of a rhombus bisect each other at right angles.
• Diagonals of Trapezoids Theorem: The diagonals of a trapezoid may have different lengths. However, the segment joining the midpoints of the non-parallel sides is parallel to the bases and is equal to half their sum.

Quadrilateral Lines of Symmetry

A quadrilateral has lines of symmetry that are imaginary lines that pass through the center of the quadrilateral and divide it into two similar halves. A line of symmetry can:

• Match two vertices on one side of the line with two vertices on the other.
• Pass through two vertices, and the other two vertices pair up when folded over the line.

A regular quadrilateral has four lines of symmetry. For example, a square has four lines of symmetry, including both its diagonals and the lines joining the midpoints of its opposite sides. A rectangle has two lines of symmetry, including the lines joining the midpoint of the opposite and parallel lines.

Quadrilateral Sides and Angles

The following table illustrates how the sides and angles of quadrilaterals make them different from one another:

Solved Examples on Quadrilaterals

Here are some solved examples on quadrilaterals for your help.

Question 1: The perimeter of quadrilateral ABCD is 46 units. AB = x + 7, BC = 2x + 3, CD = 3x - 8, and DA = 4x - 6. Find the length of the shortest side of the quadrilateral. 

Solution:

Perimeter = Sum of all sides

= 46 = 10x - 4 or [x = 5]

That gives, AB = 12 units, BC = 13 units, CD = 7 units, DC = 14 units

Hence, length of shortest side is 7 units (i.e. CD).

Question 2: Given a trapezoid ABCD (AB || DC) with median EF. AB = 3x - 5, CD = 2x -1 and EF = 2x + 1. Find the value of EF.

Solution:

We know that the Median of the trapezoid is half the sum of its bases.

= EF = (AB + CD) / 2

= 4x + 2 = 5x - 6  or [x = 8]

Therefore EF = 2x + 1 = 2(8) + 1 => EF = 17 units.

Question 3: In a Parallelogram, adjacent angles are in the ratio of 1:2. Find the measures of all angles of this Parallelogram.

Solution:

Let the adjacent angle be x and 2x.

We know that in of a Parallelogram adjacent angles are supplementary.

= x + 2x = 180° or [x = 60°]

Also, opposite angles are equal in a Parallelogram.

Therefore measures of each angles are 60°, 120°, 60°, 120°.

Quadrilateral Notes and Solution For Class 9

• Quadrilateral Class 9 Notes
• Quadrilateral Class 9 NCERT Solutions

Conclusion

Quadrilateral is a two-dimensional shape with four sides, corners, and angles, with a total interior angle sum of 360 degrees. There are two main types: concave, which has at least one angle greater than 180 degrees, and convex, where all angles are less than 180 degrees. Convex quadrilaterals include trapezoids, parallelograms, rectangles, rhombuses, squares, and kites. Each type has unique properties and formulas for calculating area and perimeter. For instance, the area of a rectangle is length times width, while a rhombus’s area is half the product of its diagonals. Symmetry and properties like equal sides or right angles vary among these shapes, making each useful for different applications in fields like architecture and design.