# CBSE Class 9 Maths Revision Notes Chapter 5

## Class 9 Mathematics Revision Notes for Introduction to Euclid’s Geometry of Chapter 5

Extramarks’ Class 9 Mathematics Revision Notes for Introduction to Euclid’s Geometry of Chapter 5 can be used to gain a thorough understanding of Euclid’s Geometry. These notes provide a detailed overview to help students conceptualise, grasp and understand the applications of concepts. They will be able to score more with multiple revisions of these notes. Curated by subject matter experts, these notes can be easily accessed for quick revisions before examinations.

## Introduction to Euclid Geometry:

• A wide variety of applications of geometry have been observed since ancient times. Some of these applications were to demarcate the lands on riverbanks after being wiped out by floods, measurement of granaries, and borders laid down for different villages, Egyptian pyramids etc.
• The word geometry has its origin in the Greek Words ‘Geo’ – Earth and ‘Metrin’ – measurement.
• An Egyptian mathematician, Euclid studied the applications of geometry and systematically compiled these in a famous treatise ‘Elements’ in 300 BC.
• This book showed 28 definitions, five postulates and common conceptions which allowed people to understand and explore geometry. This is the Euclidean Geometric approach.
• A limited collection of assumptions is used to prove numerous assertions based on self-evident universal facts.

## Euclid’s Definitions:

Amongst the 23 definitions listed by Euclid,  a few of the important ones are listed below:

• A point is that which has no part.
• A line is a breadthless length.
• The ends of a line are points.
• A straight line is a line which lies evenly with the points on itself.
• A surface is that which has length and breadth only.
• The edges of a surface are lines.
• A plane surface is a surface which lies evenly with straight lines on itself.

Axioms and postulates are based on these definitions.

## Euclid’s Axioms:

Axioms have applications in all areas of geometry but are not directly related to it. Euclid’s axioms that stand true are:

1. Things that are equivalent to one another are also equivalent to that thing.
2. When equals are added together, the sums are also equal.
3. Equals can be subtracted from equals with equal results.
4. Things that coincide with one another are equal.
5. The whole is greater than the parts.
6. Things that are twice as much of a particular thing are equal to one another.
7. Things that are half the same are equivalent to one another.

Magnitude of the same kind is referred to in all these axioms.

• Axiom 1: If x=Z and y=Z, then x=y
• Axiom 2: If x=y, then x+Z=y+Z
• Axiom 3: If x=y, then x-Z=y-Z
• Axiom 4: This axiom justifies that everything equals itself, which is the principle of superposition.
• Axiom 5: If y is a part of x, then there is a quantity Z such that x=y+Z or x>y

Magnitudes of the same kind could be added, subtracted and compared.

## Euclid’s Postulates:

Assumptions unique to geometry were called postulates by Euclid. There are five such postulates.

### Postulate 1

• According to this postulate, “a straight line may be drawn from any one point to any other point.”
• This can also be stated as Axiom 5.1.
• Thus, given two distinct points, there is only one unique straight line that could pass through them.

### Postulate 2

• This postulate states that – “A terminated line can be produced indefinitely.”

### Postulate 3

• This postulate states that – “A circle can be drawn with any given centre and radius.”

### Postulate 4

• Postulate 4 states that – “All right angles are equal to one another.”

### Postulate 5

• According to postulate 5 – “If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.”

Postulates 1 through 4 are fundamental and hence are called ‘self-evident truths.’ But postulate 5 is a bit complicated and ought to be discussed. If any line LM intersects two lines, say PQ and RS at a place where the Sum of angles = 180, the lines LM and PQ will intersect at that point. It is important to note that even though in Mathematics, the terms axiom and postulate are used interchangeably, Euclid considers that they hold different meanings.

### System of Consistent Axioms:

A system of axioms is said to be consistent if it is impossible to deduce a statement from the axioms that contradict any of the given axioms.

### Proposition or Theorem:

These are statements that have been proven to hold true using Euclid’s axioms and postulates.

### Theorem:

Two distinct lines cannot have more than one point in common.

Proof:

Given, AB and CD are two lines.

To prove:

They intersect only at a single point or have no intersection.

Solution:

• Assume that the lines AB and CD cross at positions P and Q.
• The line AB must therefore pass through the points P and Q.
• Let us assume the line CD also runs through the P and Q points.
• This means that two lines are passing through two distinct points P and Q.
• However, only one line can cross through two separate places.
• This goes against our axiom that two separate lines can share more than one point.
• The lines AB and CD are unable to travel via points P and Q.
• These lines do not intersect.

### Equivalent Versions of Euclid’s Fifth Postulate:

• Version 1: For every line l and every point P not lying on l, there exists a unique line m passing through P and parallel to l.
• Two distinct intersecting lines could never be parallel to the same third line.

### Introduction to Euclid’s Geometry Class 9 Notes – Brief Overview of the Chapter

Extramarks offer a detailed explanation of ​​Class 9 Mathematics Chapter 5 – ‘Introduction to Euclid’s Geometry’ in the form of neatly organised notes. Concepts included in these notes are:

• Applications of Geometry before Euclid’s contribution
• Definitions, Axioms and Postulates as given by Euclid
• Geometrical concepts of a line, point, straight line and plane surfaces.

Revision sessions based on these notes can improve mathematical aptitude and allow easy scoring in the Class 9 examinations. Students can easily access these notes and either quickly revise the contents or gain an idea of the chapter. A conceptualisation of topics is extremely necessary to build confidence and avoid confusion during exam time.

### Introduction to Euclid’s Geometry Class 9 – Revision Notes

Class 9 Mathematics Chapter 5 Notes created by the subject matter experts of Extramarks provide students with an edge in their exam preparation strategy by listing out complicated mathematical concepts in simpler terms. A detailed description of the axioms, postulates and theorems given by Euclid form the core of these notes. Reading these notes will enable pupils to quickly cover quality points because they are well-structured and organised.

1. ### Euclid Geometry Class 9 Notes – Euclid’s Definition

Euclid explained certain mathematical concepts in his book ‘Elements’ in 300 BC.

Different geometrical terms listed in that book were used for preparing the axioms and postulates he wrote for the development of Geometry.

These geometrical concepts are :

1. Line
2. Straight line
3. Surface
4. Edges of a surface
5. A plane surface

A detailed reading of these concepts from Class 9 Mathematics Chapter 5 Notes by Extramarks will help students learn all these concepts and more.

1. ### Introduction to Euclid’s Geometry Notes – Euclid’s Axioms

Axioms are the assumptions based on which further explanations in mathematics can be made. The following axioms were given by Euclid for explaining certain theorems:

Axiom 1 : If a=b and b=c then a=c

Axiom 2 : If a=b is assumed, then a +c = b + c

Axiom 3 : If a=b, then a-c = b-c

Axiom 4 : The principle of superposition is that everything equals itself.

Axiom 5 : If a is a part of b, then there is a constant c for which, a=b +C or a>b

Subject matter experts have organised the Class 9 Mathematics Chapter 5 Notes in such a way that students can easily comprehend these topics. These notes will be useful for students to revise before exams and improve their marks in the Class 9 examinations. Every student’s self-assurance is based on how many revisions they do after learning a certain subject. They can boost their exam preparation and confidence with Extramarks’ Class 9 Mathematics Chapter 5 Notes.

1. ### Class 9 Mathematics Chapter 5 Notes – Euclid’s Postulates

The five postulates given by Euclid have been explained in Extramarks’ Class 9 Mathematics Chapter 5 Notes. These notes cover all topics in-depth for all-rounded exam preparation.

These postulates are :

Postulate 1 : A joining of two points together, forms a straight line, or a straight line connects two points together.

Postulate 2 : A terminated line can be extended indefinitely to infinity.

Postulate 3 : Any point can be used as a starting point for drawing a circle, and any value of the radius can be taken.

Postulate 4 : All right angles are equal, as suggested by Euclid.

Postulate 5 : This postulate needs arguments in favour of its accuracy and is unlike the other four postulates since those are self-evident universal truths.

Postulates and theories are formulated based on the premises known as axioms. Euclid’s geometry allows a way to establish that theorems hold by assuming the opposite of what the theorem deduces. Reasonable explanations must be provided for proving a theorem.

1. ### Class 9 Euclid Geometry Notes – Equivalent versions of Euclid’s fifth postulate

There exist two versions of the fifth postulate which are considered equivalent. These are :

• For any straight line, if there is a point ‘P’ which doesn’t lie on the straight line, there will be a unique line which passes through point ‘P’ and is parallel to the former straight line.
• Two distinct intersecting lines cannot have a single line parallel to them.

These equivalent postulates support the view that more than one intersection point cannot be present between two lines.

The Class 9 Mathematics Chapter 5 Notes provided by Extramarks explains Euclid’s axioms and postulates in a detailed and comprehensive manner, by including important pointers necessary for your exam preparation. A well-organised set of notes, with emphasis placed on exam-oriented questions for Class 9 exams, is a special feature of these notes. Curated especially for the benefit of students, these notes allow faster grasping of concepts by repetitive reading.

### 1. List Euclid’s definitions as given in ‘Elements’

Euclid’s definition as given in ‘Elements’ include:-
A point is that which has no part.
A line is a breadth-less length.
The ends of a line are points.
A straight line is a line which lies evenly with the points on itself.
A surface is that which has length and breadth only.
The edges of a surface are lines.
A plane surface is a surface which lies evenly with straight lines on itself.

### 2. Prove that AB and CD are two lines that intersect at one point or do not intersect.

Given : AB and CD are two lines.
To prove : They intersect at one point or do not intersect.

Assume that the lines AB and CD cross at positions P and Q. The line AB must therefore pass through the points P and Q.
Considering that the line CD also runs through the P and Q points, two lines are passing through two distinct points P and Q.
But, only one line can cross through two separate places.
This axiom goes against our belief that two separate lines can share more than one point.
Hence, the lines AB and CD are unable to travel via points P and Q.
They do not intersect.