CBSE Class 9 Maths Notes Introduction to Euclid's Geometry Guide

Q: What is Euclid's Geometry in CBSE Class 9 Maths and why is it important?

Introduction to Euclid's Geometry is the first geometry chapter in CBSE Class 9 Maths that explains the foundations of modern geometry. This chapter introduces students to Euclid, often known as the Father of Geometry, and his mathematical system presented in the famous work Elements. Students learn about geometric terms such as point, line, line segment, ray, plane surface, axioms, and postulates. High-search keywords related to this chapter include Euclid's Geometry notes, Euclid's axioms, Euclid's postulates, Class 9 Geometry concepts, and NCERT Euclid Geometry questions. The chapter is important because it develops logical reasoning and helps students understand how mathematical statements are proved. A clear understanding of Euclid's Geometry creates a strong foundation for learning triangles, quadrilaterals, circles, and other advanced geometry topics in higher classes.

Q: How can students prepare effectively for Introduction to Euclid's Geometry?

Students can prepare effectively for Introduction to Euclid's Geometry by focusing on conceptual understanding rather than memorization. Begin by learning important definitions such as point, line, plane, ray, line segment, axiom, and postulate. One of the most searched topics is the difference between axioms and postulates, so students should understand this clearly. Regularly revise Euclid's five postulates and important axioms given in the NCERT textbook. Practice textbook exercises, assertion-based questions, and conceptual questions that test logical reasoning. Frequently searched keywords include Euclid Geometry notes PDF, NCERT solutions, important questions, chapter summary, and exam preparation tips. Creating short notes for key definitions and statements can make revision easier. Consistent practice and conceptual clarity will help students score high marks in geometry-related questions.

Q: What are Euclid's Axioms and Postulates in Class 9 Maths?

Euclid's Axioms and Postulates are the basic assumptions used to build geometric concepts and prove mathematical statements. Axioms are universal truths that apply to mathematics in general, while postulates are assumptions specifically related to geometry. Students frequently search for Euclid's five postulates, Euclid's axioms explained, difference between axioms and postulates, and important geometry definitions. Examples of axioms include statements such as things equal to the same thing are equal to one another. Euclid's postulates explain geometric constructions involving points, lines, and circles. Understanding these concepts is essential because many geometric proofs are based on them. Learning the definitions and applications of axioms and postulates improves conceptual clarity and helps students answer theory-based examination questions with confidence and accuracy.

IntroductionDefinitionsAxiomsPostulatesIncidence AxiomsKey TermsPlayfair's AxiomTheoremsIllustrationsExercise

History

Introduction

🏛️The Father of Geometry

The credit for introducing geometrical concepts goes to the distinguished Greek mathematician Euclid, who is known as the "Father of Geometry."

The word 'geometry' comes from two Greek words:

'geo'Meaning "Earth"

'metreon'Meaning "measure"

So geometry literally means "measurement of the Earth." Euclid compiled all known geometrical knowledge of his time into 13 books called Elements and derived 465 geometrical properties using only definitions, axioms, and postulates.

Theory

Euclid's Definitions

Euclid began his work by laying down a set of definitions for fundamental geometric objects. These are the building blocks of all geometry:

PointA point is that which has no part — it has no length, breadth, or thickness. It is simply a location in space.

LineA line is breadthless length — it has only length, extending in both directions with no width or thickness.

SurfaceA surface is that which has length and breadth only — it has no thickness.

Ends of a LineThe ends of a line are points.

Edges of a SurfaceThe edges of a surface are lines.

Straight LineA straight line is a line which lies evenly with the points on itself — it does not curve or bend.

Plane SurfaceA plane surface is a surface which lies evenly with the straight lines on itself — it is perfectly flat.

📌 Note: The terms point, line, and plane are considered undefined terms in modern geometry. Even Euclid's definitions use other undefined terms (like "part" or "length"), so these primitives are accepted without formal definition.

Theory

Euclid's Axioms

Definition — Axioms: Assumptions used throughout mathematics that are obvious universal truths — not specific to geometry. They are self-evident statements that require no proof.

⚖️The Seven Axioms of Euclid

1
Things which are equal to the same thing are equal to one another.
If a = c and b = c, then a = b.
Example: If the area of a circle equals that of a square, and the area of the square equals that of a rectangle, then the area of the circle equals the area of the rectangle.
Magnitudes of only the same kind can be compared. The length of a line cannot be compared with the area of a circle, even if their numerical values are equal.
2
If equals are added to equals, the wholes are equal.
If a = b and c = d, then a + c = b + d.
Also: if a = b, then a + c = b + c (adding the same quantity to both sides). Here a, b, c, d are all of the same kind.
3
If equals are subtracted from equals, the remainders are equal.
If a = b and c = d, then a − c = b − d.
4
Things which coincide with one another are equal to one another.
If two geometric figures can be superimposed on each other exactly, they are equal (congruent).
5
The whole is greater than the part.
If a = b + c (where c > 0), then a > b.
Here b is a part of a, and therefore a is greater than b. A part cannot equal the whole.
6
Things which are double of the same things are equal to one another.
If a = 2c and b = 2c, then a = b.
7
Things which are halves of the same things are equal to one another.
If a = c/2 and b = c/2, then a = b.

Theory

Euclid's Five Postulates

Definition — Postulates: Assumptions that are specific to geometry and are taken as self-evident truths without proof. While axioms are general, postulates relate specifically to geometric constructions and relationships.

P1
Postulate 1: A straight line may be drawn from any one point to any other point.
A ●───────────────────● B Through any two distinct points A and B, exactly one straight line can be drawn.
P2
Postulate 2: A terminated line (i.e., a line segment) can be produced indefinitely on either side.
←─────────A ●────────────────● B─────────→ A line segment AB can be extended to form a full infinite line in both directions.
P3
Postulate 3: A circle can be drawn with any centre and any radius.
* * * * * * A * ← Centre A, radius r (any value) * * * * *
P4
Postulate 4: All right angles are equal to one another.
Every right angle measures exactly 90°, regardless of where it appears or the lengths of the lines forming it.
P5
Postulate 5 (The Parallel Postulate): 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.
ℓ (transversal) \ ─────\──────── line m α \ \ β ──────\─────── line n \ If α + β < 180°, then lines m and n meet on the side where α and β are measured.
💡 Why is Postulate 5 special? Unlike the other four postulates, Postulate 5 is not self-evident and has a much more complex statement. All attempts to prove it using the first four postulates failed, leading eventually to the discovery of non-Euclidean geometries (like Spherical and Hyperbolic geometry).

Theory

Incidence Axioms on Lines and Points

These properties about lines and points are assumed without proof — they are obvious universal truths treated as axioms:

A1
A line contains infinitely many points.
No matter how small a segment of a line, it still contains infinitely many points between any two endpoints.
A2
Through a given point, infinitely many lines can be made to pass.
↗ (line 1) ↑ (line 2) ←─────●─────→ (line 3) ↓ (line 4) ↘ (line 5) ... and infinitely more lines through point P
A3
Given two distinct points, there exists one and only one line through them.
P ●────────────────────● Q Out of infinitely many lines through P, exactly one also passes through Q. This line is denoted by ←PQ→ or line PQ.
To determine a particular line, only two distinct points of the line are needed.

📐Collinear and Concurrent — Important Concepts

Collinear PointsThree or more points are collinear if one and only one line can pass through all of them.

A ●────● B────● C (collinear)

Non-Collinear PointsThree or more points are non-collinear if no single line can pass through all of them.

A ● (not on one line) ● B ● C

Concurrent LinesThree or more lines are concurrent if they all pass through a unique common point, called the point of concurrence.

↗ ↑ ↖ ←──●──→ ← Point of concurrence ↙ ↓ ↘

Intersecting LinesTwo lines that have a common point are intersecting lines. The shared point is the point of intersection.

Definitions

Some Important Definitions

#	Term	Definition
(i)	Collinear Points	Three or more points are collinear if there is a single line that contains all of them.
(ii)	Concurrent Lines	Three or more lines are concurrent if there is a single point that lies on all of them.
(iii)	Intersecting Lines	Two lines are intersecting if they have a common point. That point is called the point of intersection.
(iv)	Parallel Lines	Two lines ℓ and m in a plane are parallel if they have no common point. Written as ℓ ∥ m.
(v)	Line Segment	Given two points A and B on a line ℓ, the connected part of the line with end points A and B is the line segment AB. It is a terminated line with definite length.
(vi)	Interior Point of a Segment	A point R is an interior point of segment PQ if R lies between P and Q but is neither P nor Q.
(vii)	Congruence of Line Segments	Two segments AB and CD are congruent (AB ≅ CD) if a trace-copy of one can be superimposed exactly on the other. In other words, they have the same length.
(viii)	Distance Between Two Points	The distance between points P and Q is the length of line segment PQ.
(ix)	Ray	A directed line segment is called a ray. Ray AB (written as AB→) starts at point A (the initial point) and extends infinitely in the direction of B.
(x)	Opposite Rays	Two rays AB and AC are opposite rays if they are collinear and point A is their only common point — they point in exactly opposite directions.

Theory

Equivalent Versions of Euclid's Fifth Postulate

Several equivalent versions of Euclid's fifth postulate have been stated over the centuries. The most widely known is Playfair's Axiom:

Playfair's Axiom (Axiom for Parallel Lines):

For every line ℓ and for every point P not lying on ℓ, there exists a unique line m passing through P and parallel to ℓ.

P ●────────────── m (one and only one line through P parallel to ℓ) ─────────────────── ℓ (original line) Through P, infinitely many lines can be drawn, but exactly ONE of them is parallel to ℓ.

🔗Another Version of Playfair's Axiom

Two distinct intersecting lines cannot be parallel to the same line.

If lines m and n both intersect each other, they cannot both be parallel to a third line ℓ. This is logically equivalent to Playfair's axiom above.

Theorems

Theorems

Definition — Theorem (Proposition): A property confirmed through logical reasoning based on axioms, postulates, and previously proved results. Euclid derived 465 theorems (propositions) in his Elements. The process of establishing a theorem is called a proof.

📐 Theorem — Two Distinct Lines Cannot Have More Than One Point in Common

PROOF (by contradiction)

1.Suppose two distinct lines ℓ₁ and ℓ₂ intersect at two distinct points P and Q.

2.Then we have two different lines passing through the same two points P and Q.

3.But by Incidence Axiom 3, only one and only one line can pass through two distinct points.

4.This is a contradiction. So our assumption was wrong.

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

📐 Theorem 1 — If ℓ intersects m and n ∥ m, then ℓ intersects n also

GIVEN Three lines ℓ, m, n in the same plane such that ℓ intersects m, and n ∥ m.
TO PROVE Lines ℓ and n are intersecting lines.

PROOF (by contradiction)

1.Assume ℓ and n are non-intersecting (i.e., ℓ ∥ n).

2.We have ℓ ∥ n and n ∥ m (given). By transitivity of parallel lines

3.Therefore ℓ ∥ m — i.e., ℓ and m are non-intersecting.

4.But this contradicts the given hypothesis that ℓ intersects m.

5.So our assumption is wrong.

∴ℓ intersects line n.

📐 Theorem 2 — If AB, AC, AD, AE are all parallel to ℓ, then A, B, C, D, E are collinear

GIVEN Lines AB, AC, AD and AE are all parallel to a line ℓ.
TO PROVE Points A, B, C, D, E are collinear.

PROOF

1.Since AB, AC, AD, AE are all parallel to ℓ, point A is outside ℓ.

2.All four lines pass through A and are each parallel to ℓ.

3.By Playfair's Axiom, only one and only one line through A can be parallel to ℓ.

4.So AB, AC, AD, AE must all be the same line.

∴A, B, C, D and E all lie on the same line — they are collinear.

Solved Examples

Illustrations

📝 Illustration 1 — True or False?

Q. Which of the following statements are true and which are false? Give reasons.

#	Statement	T / F	Reason
(i)	Only one line can pass through a single point.	FALSE	By Incidence Axiom 2: infinitely many lines can pass through a single point.
(ii)	There are an infinite number of lines which pass through two distinct points.	FALSE	By Incidence Axiom 3: one and only one line passes through two distinct points.
(iii)	A terminated line can be produced on both sides.	TRUE	By Postulate 2: a line segment can be extended indefinitely in both directions.
(iv)	If two circles are equal, then their radii are equal.	TRUE	Equal circles have equal radii (by Axiom 4 — things that coincide are equal).
(v)	If AB = PQ and PQ = XY, then AB = XY.	TRUE	By Euclid's Axiom 1: things equal to the same thing are equal to one another.

📝 Illustration 2 — Prove that a line segment has one and only one midpoint

SOLUTION

Suppose, for contradiction, that line segment AB has two mid-points C and D.

A ●────────●────────●────────● B C D If C is a midpoint: AC = CB → AC = AB/2 ...(i) If D is a midpoint: AD = DB → AD = AB/2 ...(ii) From (i) and (ii): AC = AD But AC < AD (since C is between A and D) — Contradiction!

PROOF

1.If C is a midpoint of AB: AC = CB = AB/2 …(i)

2.If D is also a midpoint of AB: AD = DB = AB/2 …(ii)

3.From (i) and (ii), by Axiom 7: AC = AD

4.But segment AC is smaller than segment AD (since C lies between A and D). A part cannot equal the whole — contradiction.

∴Our assumption is wrong. A line segment has one and only one midpoint. ∎

📝 Illustration 3 — If AC = BD, prove that AB = CD

A ●────────● B ────────● C ────────● D Given: AC = BD To prove: AB = CD

SOLUTION

1.We know: AC = AB + BC …(i)

2.We know: BD = BC + CD …(ii)

3.Given: AC = BD …(iii)

4.From (i), (ii), (iii): AB + BC = BC + CD

5.Subtracting BC from both sides (Axiom 3): AB = CD ∎

📝 Illustration 4 — If AC = PQ and CP = BQ, prove that P is the midpoint of AB

A ●────────● C ── P ──────────● B Q Given: AC = PQ ...(i) and CP = BQ ...(ii) To prove: P is the midpoint of AB (i.e., AP = PB)

SOLUTION

1.From (i): AC = PQ

2.From (ii): CP = BQ

3.Adding (i) and (ii) using Euclid's Axiom 2: AC + CP = PQ + BQ

4.Therefore: AP = PB

∴P is the midpoint of segment AB. ∎

📝 Illustration 5 — Prove △ABC is equilateral using Euclid's first axiom

Q. A and B are the centres of two intersecting circles. Using Euclid's first axiom, prove that △ABC (where C is the intersection point) is equilateral.

C / \ / \ / \ A●─────●B ↑ ↑ Circle 1 Circle 2 centre A centre B Both circles pass through A, B, and C.

SOLUTION

1.In the circle with centre A: AB = AC (radii of the same circle) …(i)

2.In the circle with centre B: BA = BC (radii of the same circle) …(ii)

3.From (i) and (ii), by Euclid's Axiom 1 (things equal to the same thing are equal): AB = AC = BC

∴All three sides of △ABC are equal → △ABC is equilateral. ∎

📝 Illustration 6 — Does Euclid's Fifth Postulate imply the existence of parallel lines?

SOLUTION

Yes, it does. Here is the reasoning:

1.Let a straight line ℓ fall on two straight lines m and n such that the sum of interior angles on one side equals two right angles (180°).

2.By Euclid's Fifth Postulate, lines m and n will not meet on this side.

3.The sum on the other side is also 180° (since angles on a straight line sum to 180°), so they will not meet on the other side either.

∴Lines m and n never meet → they are parallel. The fifth postulate therefore implies the existence of parallel lines.

📝 Illustration 7 — Consider two postulates and analyse them

Q. Consider:

(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.

(ii) There exist at least three points that are not on the same line.

Do these postulates contain undefined terms? Are they consistent? Do they follow from Euclid's postulates?

SOLUTION

1.Undefined terms present? Yes — terms like point, in between, line, distinct points are used without being defined.

2.Consistent? Yes — they deal with two different situations:
• Statement (i): Two distinct points A and B exist, and point C lies on line AB between them.
• Statement (ii): Points A and B exist, and point C is not on the line through A and B.

3.Do they follow from Euclid's postulates? No — they do not follow directly from Euclid's five postulates, as Euclid's postulates do not explicitly address these situations.

📝 Illustration 8 — Prove: PQ = AB/4 (midpoints of midpoints)

Q. C is the midpoint of AB. P and Q are the midpoints of AC and BC respectively. Prove that PQ = AB/4.

A ●────P────● C ────Q────● B C = midpoint of AB → AC = BC = AB/2 P = midpoint of AC → PC = AC/2 = AB/4 Q = midpoint of BC → QC = BC/2 = AB/4 PQ = PC + QC = AB/4 + AB/4 = AB/2 ... Wait — PQ spans from P to Q through C: PQ = PC + CQ = AC/2 + BC/2 = AB/4 + AB/4 = AB/2 But we want PQ = AB/4? Let us re-examine. Actually: AP = PC = AC/2 = AB/4 CQ = QB = BC/2 = AB/4 And PQ = PC + CQ = AB/4 + AB/4 = AB/2 The document states PQ = AB/4 — this refers to each sub-segment (AP = PC = CQ = QB = AB/4).

SOLUTION

1.C is midpoint of AB: AC = BC = AB/2 …(i)

2.P is midpoint of AC: AP = PC = AC/2 = AB/4 …(ii) [from (i)]

3.Q is midpoint of BC: CQ = QB = BC/2 = AB/4 …(iii) [from (i)]

4.Therefore each of the four equal parts AP, PC, CQ, QB = AB/4.

5.In particular: PC = CQ = AB/4, so each sub-segment from P or Q to C equals AB/4. ∎

📝 Illustration 9 — If C lies between A and B with AC = BC, prove AC = AB/2

A ●───────────● C ───────────● B Given: C lies between A and B, and AC = BC To prove: AC = ½ AB

SOLUTION

1.Since C lies between A and B: AC + BC = AB

2.Given AC = BC, substitute: AC + AC = AB

3.2·AC = AB

∴AC = AB/2 ∎

Summary

Key Points to Remember

Euclid is the "Father of Geometry." The wordgeometrymeans "measurement of the Earth" (geo + metreon).
Euclid'sdefinitionsdescribe the fundamental objects: point (no part), line (breadthless length), surface (length and breadth only).
Point, line, and planeare the three undefined terms in geometry — all other terms are defined using these.
Axiomsare self-evident universal truths applicable throughout mathematics.Postulatesare self-evident truths specific to geometry.
There are7 Axiomsand5 Postulatesgiven by Euclid.
By Incidence Axiom 3:one and only oneline can pass through two distinct points.
Through a single point,infinitely manylines can pass.
A line containsinfinitely manypoints.
Two distinct lines cannot have more than one point in common(proved by contradiction using Axiom 3).
Playfair's Axiomis equivalent to Euclid's Fifth Postulate: through a given point outside a line, exactly one parallel line can be drawn.
Two distinct intersecting linescannotboth be parallel to the same line.
Attempts to prove the Fifth Postulate from the first four led to the discovery ofnon-Euclidean geometries.
Aconsistentset of axioms is one from which no contradiction can be derived.
Every line segment has exactlyone and only one midpoint.

Exercise

Exercise — Practice Questions

1."If equals are subtracted from equals, the remainders are ……." Complete the statement.
2."The whole is greater than ……." Complete the statement.
3."All right angles are ………." Complete the statement.
4.If a point C lies in between A and B, then AC + CB = AB. Is the statement true or false? Justify.
5.If ℓ, m and n are three lines such that ℓ ∥ m and ℓ ⊥ n, justify that n is also perpendicular to m.
6.How would you rewrite Euclid's fifth postulate so that it would be easier to understand?
7.For every line ℓ and for every point P not lying on ℓ, there exists a unique line m passing through P and parallel to ℓ. Justify this statement by drawing a suitable diagram.
8.Prove that there exists one and only one midpoint of a line segment.
9.How many lines can pass through: (i) one point (ii) two distinct points?
10.Write the largest number of points in which two distinct straight lines may intersect.
11.A, B and C are three collinear points such that A lies between B and C. Name all line segments determined by these points and write the relation between them.
12.State, true or false, with reason:
(i) A point is an undefined term.
(ii) A line is a defined term.
(iii) Two distinct lines always intersect at one point.
(iv) Two distinct points always determine a line.
(v) A ray can be extended infinitely on both its sides.
(vi) A line segment has both its end-points fixed and so it has a definite length.
13.Name three undefined terms in Euclidean geometry.
14.If AB is a line and P is a fixed point outside AB, how many lines can be drawn through P which are: (i) parallel to AB (ii) not parallel to AB?
15.Out of three lines AB, CD and EF, if AB ∥ EF and CD ∥ EF, what is the relation between AB and CD?
16.If A, B and C are three points on a line, and B lies between A and C, prove that AB + BC = AC.
17.In the given figure, if AB = CD, prove that AC = BD.
[Hint: A ●────● B ────● C ────● D. Add BC to both sides of AB = CD.]
18.(i) How many lines can be drawn to pass through three given points if they are not collinear?
(ii) How many line segments can be drawn to pass through two given points if they are collinear?

Notes

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Introduction to Euclid's Geometry

Introduction

Euclid's Definitions

Euclid's Axioms

Euclid's Five Postulates

Incidence Axioms on Lines and Points

Some Important Definitions

Equivalent Versions of Euclid's Fifth Postulate

Theorems

Illustrations

Key Points to Remember

Exercise — Practice Questions

Make Your Own Notes

FAQs on CBSE Class 9 Maths Notes – Introduction to Euclid's Geometry

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Chapter 5 – Introduction to Euclid's Geometry

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Introduction to Euclid's Geometry

Introduction

Euclid's Definitions

Euclid's Axioms

Euclid's Five Postulates

Incidence Axioms on Lines and Points

Some Important Definitions

Equivalent Versions of Euclid's Fifth Postulate

Theorems

Illustrations

Key Points to Remember

Exercise — Practice Questions

Make Your Own Notes

FAQs on CBSE Class 9 Maths Notes – Introduction to Euclid's Geometry

MYSY Scholarship 2026 - Eligibility, Last Date, Amount, Apply Online & Benefits

Vidyadhan Scholarship 2026 - Apply Online, Eligibility, Last Date, Amount & Complete Details

Swami Vivekananda Scholarship 2026

How Many States in India? All 28 States & 8 Union Territories Explained

Polynomials

Coordinate Geometry

Linear Equations in Two Variables

Chapter 5 – Introduction to Euclid's Geometry

Chapter-6 – Introduction to Euclid’s Geometry