Euclid's Geometry

Class 9 Maths • Chapter 05 • Introduction

🧠 Why study Euclid’s Geometry?

Geometry is not about drawing figures — it is about logical thinking. Euclid showed that complex results can be built from simple assumptions.

CBSE Focus: This chapter trains your reasoning skills.

1. Who was Euclid?

Euclid was a Greek mathematician, often referred to as the "Father of Geometry". He collected all the known work of his time and arranged it in his famous treatise called 'Elements'.

He divided the 'Elements' into 13 chapters, each called a book.

Key Definitions

Euclid began with 23 definitions in Book 1. Here are the most important ones:

1. A point is that which has no part.
2. A line is breadthless length.
3. The ends of a line are points.
4. A surface is that which has length and breadth only.

2. Axioms and Postulates

Euclid assumed certain properties which were not to be proved. These assumptions are divided into two types:

Term	Description	Example
Axioms	Applicable to all branches of mathematics (Algebra, Geometry, etc.)	"The whole is greater than the part."
Postulates	Applicable specifically to Geometry.	"A straight line may be drawn from any one point to another."

Very Important Comparison

Term	Meaning	Example
Definition	Explains what something is	A point has no dimension
Axiom	Universal truth	Whole > part
Postulate	Geometry-specific assumption	Line can be drawn joining two points

Keep sorting until you’ve classified all statements correctly 🎯

Axiom or Postulate?

Read the statement and classify it.

"Things equal to the same thing are equal to one another."

3. Euclid's 5 Postulates

These are the foundation of Euclidean Geometry.

Postulate 1

A straight line may be drawn from any one point to any other point.

Postulate 2

A terminated line (line segment) can be produced indefinitely.

Postulate 3

A circle can be drawn with any center and any radius.

Postulate 4

All right angles are equal to one another.

Postulate 5 (The Famous One)

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 (180°), then the two straight lines, if produced indefinitely, meet on that side.

Think first: What happens if ∠1 + ∠2 = 180°?

Reveal Answer

The lines will never meet — they will be parallel.

Visual Representation of Postulate 5

4. Euclid's Axioms (Common Notions)

Things which are equal to the same thing are equal to one another.
If equals are added to equals, the wholes are equal.
If equals are subtracted from equals, the remainders are equal.
Things which coincide with one another are equal to one another.
The whole is greater than the part.
Things which are double of the same things are equal to one another.
Things which are halves of the same things are equal to one another.

🎯 Exam Smart Zone

Definitions are NOT axioms
Postulates apply only to geometry
Postulate 5 explains parallel lines
Axioms are used across all maths chapters

Tip: Write exact statements — wording matters.

Concept Mastery Quiz

1. How many dimensions does a solid have?

A) 1

B) 2

C) 3

2. "The whole is greater than the part" is an:

A) Axiom

B) Postulate

C) Definition

3. Two distinct lines cannot have more than ____ point in common.

A) 1

B) 2

C) Infinite

4. Euclid belongs to which country?

A) India

B) Greece

C) Egypt

5. A point has how many dimensions?

A) 0

B) 1

C) 2