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CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals PDF Download

Here, we have provided CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals. Students can view these CBSE Class 9 Maths Notes Chapter 8 before exams for better understanding of the chapter.

Ananya Gupta22 May, 2024

CBSE Class 9 Maths Notes Chapter 8: In Class 9 Math, Chapter 8 talks about Quadrilaterals, which are shapes with four straight sides. This chapter helps you understand different kinds of quadrilaterals, like parallelograms, rectangles, squares, rhombuses, and trapeziums.

You'll learn about their special features and how to recognize them. We'll also explore things like opposite sides and angles, diagonals, and special properties like symmetry and congruence. By understanding this chapter well, you'll be better prepared for more advanced math topics later on.

CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals Overview

These notes on Chapter 8, Quadrilaterals, are written by subject experts of Physics Wallah in simple language to help students understand the concepts easily. In this chapter, you will learn about different types of quadrilaterals, such as parallelograms, rectangles, squares, rhombuses, and trapeziums. The notes explain their properties, like the relationships between sides and angles, and how to identify each type of quadrilateral. By studying these notes, students can grasp the fundamentals of quadrilaterals, which will be useful for more advanced math topics in the future.

CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals PDF

You can access the CBSE Class 9 Maths Notes for Chapter 8 Quadrilaterals in PDF format using the provided link. These notes provide comprehensive explanations and examples to help you grasp the concepts of triangles effectively.

CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals PDF

CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals

Quadrilaterals

Quadrilaterals are a specific type of polygon characterized by having exactly four sides. These geometric shapes are formed by the union of four line segments. Common examples of quadrilaterals include squares, rectangles, parallelograms, and trapezoids. Each quadrilateral has four vertices (corners) and four angles. The sum of the interior angles of a quadrilateral is always 360 degrees. Understanding the properties and types of quadrilaterals is fundamental in geometry, as they form the basis for more complex shapes and are widely used in various mathematical applications.

Examples of Quadrilaterals

Parallelograms

A parallelogram is a specific type of quadrilateral defined by its unique properties. One defining characteristic of a parallelogram is that its opposite sides are both parallel and equal in length. This means that if you extend the opposite sides indefinitely, they will never intersect. Parallelograms encompass various other quadrilaterals, including rectangles, rhombuses, and squares, each possessing additional properties beyond those of a standard parallelogram. In contrast, a trapezium is another type of quadrilateral, but it differs from a parallelogram in that only one pair of its opposite sides are parallel. Therefore, it does not qualify as a parallelogram. Despite this distinction, trapeziums share similarities with parallelograms, such as having opposite sides of equal length. Understanding the properties and distinctions of parallelograms and trapeziums is crucial in geometry, as they form the basis for many geometric concepts and mathematical calculations.

In the diagram,

Opposite Sides

$A B ∥ D C$ and $A D ∥ B C$

$A B = D C$ and $A D = B C$

Opposite Angles are Equal.

From figure,

$\hat{A} = \hat{C}$ and $\hat{B} = \hat{D}$

Diagonals of a Parallelogram Bisect Each Other

In the diagram,

$O D = O B$ and $O A = O C$

Each diagonal divides the parallelogram into two congruent triangles

In the diagram,

$△ A B C ≅ △ C D A$

$△ A B D ≅ △ C D B$

Opposite Sides of a Quadrilateral

Two sides of a quadrilateral, which have no common point, are called opposite sides.

In the diagram, $A B$ and $D C$ is one pair of opposite sides.
$D C$ and $B C$ is the other pair of opposite side

Understanding Quadrilateral Properties

Quadrilaterals are four-sided polygons with distinct properties that define their angles and sides. One fundamental aspect of quadrilaterals is the concept of consecutive sides, opposite angles, and consecutive angles. Consecutive sides are two sides of a quadrilateral that share a common endpoint. For example, in a quadrilateral ABCD, AB and BC form one pair of consecutive sides, while BC and CD, CD and DA, and DA and AB represent the other three pairs of consecutive sides. Opposite angles in a quadrilateral are a pair of angles that do not share a side in their intersection. In the quadrilateral ABCD, angles A and C form one pair of opposite angles, while angles B and D constitute another pair of opposite angles. Consecutive angles, on the other hand, are two angles of a quadrilateral that include a side in their intersection. For instance, angles A and B represent one pair of consecutive angles, while angles B and C, C and D, and D and A form the other three pairs of consecutive angles. Understanding these properties helps in identifying and analyzing quadrilaterals, facilitating geometric calculations and problem-solving in various mathematical contexts.

Theorem 1 Statement

The diagonals of a parallelogram bisect each other.
If two sides of a triangle are unequal, the longer side has the greater angle opposite to it.
$A B C D$ is a parallelogram in which diagonals $A C$ and
$B D$
intersect each other at $O$ .

To prove:

The diagonals $A C$ and

B D

bisect eac $\hat{B}$ and $\hat{D}$ areh other that is,

$A O = O C$ and

$B O = D O$ .

Proof:

$A B ∥ C D$ (By definition of parallelogram)

$A C$ is a transversal.

$∴ O \hat{A} B = O \hat{C} D . . . . . (i)$ (Alternate angles are equal in a parallelogram)

Also,

$A B = D C$ (Opposite sides are equal in a parallelogram)

Now in $Δ A O B$ and $\Delta COD

$A B = D C$ (Opposite sides of parallelogram are equal)

$O \hat{A} B = O \hat{C} D$ (Proved by $(i)$ )

$A \hat{O} B = C \hat{O} D$ (Vertically opposite angles are equal)

Therefore,

△ A O B ≅ △ C O D

( $A A S$ Congruency condition)

Therefore,

$A O = O C$ and $B O = O D$ (corresponding parts of congruent triangles are congruent) that is the diagonals of a parallelogram bisect each other.

Sufficient Conditions for a Quadrilateral to be a Parallelogram

Identifying a parallelogram involves understanding its defining properties. These properties provide sufficient conditions for determining whether a quadrilateral is a parallelogram. One key property is that if a quadrilateral is a parallelogram, then its opposite sides are equal. Conversely, if both pairs of opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram. Another condition is that if the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. This means that the point where the diagonals intersect divides each diagonal into two equal parts. Additionally, if either pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram. This condition emphasizes the significance of parallelism in identifying parallelograms. Understanding these conditions enables us to efficiently recognize and classify parallelograms, aiding in geometric analysis and problem-solving.

Theorem $2$

Statement:

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Given:

$A B C D$ is a quadrilateral in which diagonals $A C$ and $B D$ intersect at $O$ such that $A O = O C$ and $B O = O D$ .

To prove:

$A B C D$ is a parallelogram.

Proof:

In triangles $A O B$ and $C O D$ ,

$A O = C O$ (Given)

$B O = O D$ (Given)

$A \hat{O} B = C \hat{O} D$ (Vertically opposite angles are equal)

Therefore,

Δ A O B ≅ Δ C O D

( $S A S$ Congruency condition)

Therefore,

$O \hat{A} B = O \hat{C} D$ ( $c p c t$ )

Since these are alternate angles made by the transversal $A C$ intersecting $A B$ and $C D$

Therefore,

$A B ∥ C D$

Similarly,

$A D ∥ B C$

Hence, $A B C D$ is a parallelogram.

Theorem $3$ :

Statement:

A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel.

Given:

$A B C D$ is a quadrilateral in which $A B ∥ C D$ and $A B = C D$ .

To prove:

$A B C D$ is a parallelogram.

Construction:

Join $A C$ .

Proof:

In triangles $A B C$ and $A D C$ ,

$A B = C D$ (Given)

$B \hat{A} C = A \hat{C} D$ (Alternate angles are equal)

$A C = A C$ (Common side)

Therefore,

Δ A B C ≅ Δ C D A

( $S A S$ Congruency condition)

$B \hat{C} A = D \hat{A} C$ (Corresponding parts of corresponding triangles)

Since these are alternate angles,

$A B ∥ C D$

Thus, in the quadrilateral $A B C D$ , $A B ∥ C D$ and $A D ∥ B C$

Therefore, $A B C D$ is a parallelogram.

Special Parallelograms

Parallelograms encompass a diverse set of quadrilaterals, including rectangles, rhombuses, and squares. Each of these special parallelograms has distinct properties and characteristics.

Rectangle:

A rectangle is a parallelogram with all interior angles measuring 90 degrees, making it a right angle. Consequently, opposite sides of a rectangle are equal in length.

Rhombus:

A rhombus is a parallelogram with all sides of equal length. This means that opposite sides are equal and parallel. However, the angles of a rhombus are not necessarily 90 degrees, except in the case of a square.

Square:

A square is a special case of both a rectangle and a rhombus. It possesses all the properties of a rectangle, including right angles, and all the sides are equal in length like a rhombus.

Relationships Between Special Parallelograms:

In terms of relationships, every rectangle and rhombus is inherently a parallelogram. Therefore, they are depicted as subsets of a parallelogram. Furthermore, because a square possesses the characteristics of both a rectangle and a rhombus, it is represented by the overlapping shaded region in the diagram. Understanding the distinctions and relationships between these special parallelograms is crucial for geometry and problem-solving applications.

Rectangle

A rectangle is a parallelogram with one of its angles as a right angle.

In the above figure,

Let, $\hat{A} = 90^{\circ}$

Since,

$A D ∥ B C$ ,

$\hat{A} + \hat{B} = 180^{\circ}$

(Sum of interior angles on the same side of transversal $A B$ )

Therefore,

$\hat{B} = 90^{\circ}$

Here,

$A B ∥ C D$ and $\hat{A} = 90^{\circ}$ (Given)

Therefore,

$\hat{A} + \hat{D} = 180^{\circ}$

$∴ \hat{D} = 90^{\circ}$

$∴ \hat{C} = 90^{\circ}$

Corollary: Each of the four angles of a rectangle is a right angle.

Rhombus

A rhombus is a parallelogram with a pair of its consecutive sides equal.

$A B C D$ is a rhombus in which $A B = B C$ .

Since a rhombus is a parallelogram,

$A B = D C$ and $B C = A D$

Thus, $A B = B C = C D = A D$

Corollary: All the four sides of a rhombus are equal (congruent).

Square

A square is a rectangle with a pair of its consecutive sides equal.

Since square is a rectangle, each angle of a rectangle is a right angle and $A B = D C$ , $B C = C D$ .

Thus,

$A B = B C = C D = A D$

Each of the four angles of a square is a right angle and each of the four sides is of the same length.

Theorem $4$

Statement:

The diagonals of a rectangle are equal in length.

Given:

$A B C D$ is a rectangle.

$A C$ and $B D$ are diagonals.

To prove:

$A C = B D$

Proof:

Let, $\hat{A} = 90^{\circ}$ (By definition of rectangle)

$\hat{A} + \hat{B} = 180^{\circ}$ (Consecutive interior angle)

$\hat{A} = \hat{B} = 90^{\circ}$

Now in triangles, $A B D$ and $A B C$ ,

$A B = A B$ (Common side)

$\hat{A} = \hat{B} = 90^{\circ}$ (Each angle is a right angle)

$A D = B C$ (Opposite sides of parallelogram)

Therefore,

Δ A B D ≅ Δ B A C

Therefore,

$B D = A C$ (Corresponding parts of corresponding triangles)

Hence the theorem is proved.

Converse of Theorem $4$ :

Statement:

If two diagonals of a parallelogram are equal, it is a rectangle.

Given:

$A B C D$ is a parallelogram in which $A C = B D$ .

To prove:

Parallelogram $A B C D$ is a rectangle.

Proof:

In triangles $A B C$ and $D B C$ ,

$A B = D C$ (Opposite sides of parallelogram)

$B C = B C$ (Common side)

$A C = B D$ (Given)

Therefore,

Δ A B C ≅ Δ D C B

(

S S S

congruency condition)

Therefore,

$A \hat{B} C = D \hat{C} B$ (Corresponding parts of corresponding triangles)

But these angles are consecutive interior angles on the same side of transversal $B C$ and $A B ∥ D C$ .

Therefore,

$A \hat{B} C + D \hat{C} B = 180^{\circ}$

But,

$A \hat{B} C = D \hat{C} B$

Therefore,

$A \hat{B} C = D \hat{C} B = 90^{\circ}$

Therefore, by definition of rectangle, parallelogram $A B C D$ is a rectangle.

Hence the theorem is proved.

Theorem $5$ :

Statement:

The diagonals of a rhombus are perpendicular to each other.

Given:

$A B C D$ is a rhombus.

Diagonal $A C$ and $B D$ intersect at $O$ .

To prove:

$A C$ and $B D$ bisect each other at right angles.

Proof:

A rhombus is a parallelogram such that

$AB = DC = AD = BC . . . . . . (i)$

Also the diagonals of a parallelogram bisect each other.

Hence,

$B O = D O$ and $A O = OC . . . . . . (ii)$

Now, compare triangles $A O B$ and $A O D$ ,

$A B = A D$ (From $(i)$ above)

$B O = D O$ (From $(ii)$ above)

$AO = AO$ (Common side)

Therefore,

Δ A O B ≅ Δ A O D

(

S S S

congruency condition)

Therefore,

A O ˆ B = A O ˆ D

(Corresponding parts of corresponding parts)

B D

is a straight line segment.

Therefore,

A O ˆ B + A O ˆ D = 180 \circ

But,

A O ˆ B = A O ˆ D

(Proved)

Therefore,

A O ˆ B = A O ˆ D = 180 \circ 2

A O ˆ B = A O ˆ D = 90 \circ

That is, the diagonals bisect at right angles.

Hence the theorem is proved.

Converse of Theorem $5$ :

Statement:

If the diagonals of a parallelogram are perpendicular then it is a rhombus.

Given:

$A B C D$ is a parallelogram in which $A C$ and $B D$ are perpendicular to each other.

To prove:

$A B C D$ is a rhombus.

Proof:

Let $A C$ and $B D$ intersect at right angles at $O$ .

A O ˆ B = 90 \circ

In triangles $A O D$ and $C O D$ ,

$A O = O C$ (Diagonals bisect each other)

$O D = O D$ (Common side)

A O ˆ D = C O ˆ D = 90 \circ

(Given)

Therefore,

Δ A O D ≅ Δ C O D

(

S A S

congruency condition)

$A D = D C$

That is, the adjacent sides are equal.

Therefore, by definition, $A B C D$ is a rhombus.

Hence the theorem is proved.

Benefits of CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals

Conceptual Understanding: These notes provide a comprehensive explanation of the properties and characteristics of quadrilaterals, helping students build a strong conceptual foundation in geometry.
Clarity in Definitions: By clearly defining terms such as parallelograms, rectangles, rhombuses, and squares, the notes ensure that students understand the distinctions between different types of quadrilaterals.
Problem-Solving Skills: Through worked examples and exercises, students can enhance their problem-solving abilities in geometry. Practice questions included in the notes enable students to apply the concepts they've learned to solve a variety of problems.
Preparation for Exams: CBSE Class 9 Maths exams often include questions related to quadrilaterals. These notes serve as a valuable resource for exam preparation, ensuring that students are well-equipped to tackle questions on this topic.

CBSE Maths Notes For Class 9

Chapter 1 – Number System

Chapter 2 – Polynomials

Chapter 3 – Coordinate Geometry

Chapter 4 – Linear Equations in Two Variables

Chapter 5 – Introduction to Euclid’s Geometry

Chapter 6 – Lines and Angles

Chapter 7 – Triangles

Chapter 8 – Quadrilaterals

Chapter 9 - Areas of Parallelograms and Triangles

CBSE Class 9 Maths Notes Chapter 10 - Circles Notes

CBSE Class 9 Maths Notes Chapter 11 - Constructions Notes

CBSE Class 9 Maths Notes Chapter 12 - Herons Formula Notes

CBSE Class 9 Maths Notes Chapter 13 - Surface Areas and Volumes Notes

CBSE Class 9 Maths Notes Chapter 14 - Statistics Notes

CBSE Class 9 Maths Notes Chapter 15 - Probability Notes

CBSE Class 9 Maths Notes Chapter 8 FAQs

What are quadrilaterals?

Quadrilaterals are polygons with four sides. They come in various shapes and sizes, such as squares, rectangles, parallelograms, rhombuses, trapezoids, and kites.

What are the properties of a parallelogram?

Parallelograms have opposite sides that are equal and parallel. Their opposite angles are also equal.

What are the conditions for a quadrilateral to be a parallelogram?

A quadrilateral is a parallelogram if its opposite sides are equal and parallel, or if its diagonals bisect each other, or if one pair of opposite sides are equal and parallel.

How can we calculate the area and perimeter of quadrilaterals?

The area of a quadrilateral can be calculated using various formulas depending on the type of quadrilateral. Perimeter is calculated by adding the lengths of all sides. These calculations are essential in fields such as construction, architecture, and engineering.

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CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals PDF Download

CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals Overview

CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals PDF

CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals

Quadrilaterals

Examples of Quadrilaterals

Parallelograms

Diagonals of a Parallelogram Bisect Each Other

Each diagonal divides the parallelogram into two congruent triangles

Opposite Sides of a Quadrilateral

Understanding Quadrilateral Properties

Theorem 1 Statement

Sufficient Conditions for a Quadrilateral to be a Parallelogram

Theorem 2 2

Special Parallelograms

Rectangle

Rhombus

Square

Theorem 4 4

Benefits of CBSE Class 9 Maths Notes Chapter 8 Quadrilaterals

CBSE Class 9 Maths Notes Chapter 8 FAQs

What are quadrilaterals?

What are the properties of a parallelogram?

What are the conditions for a quadrilateral to be a parallelogram?

How can we calculate the area and perimeter of quadrilaterals?

Theorem $2$

Theorem $4$