Class 9 Maths Chapter 5: Introduction to Euclid's Geometry

Introduction to Euclid's Geometry is Chapter 5 of NCERT Class 9 Maths. It explains how the Greek mathematician Euclid organised geometry into a logical system built on definitions, axioms and postulates, and how new results (theorems) are proved from them. These notes summarise every key statement in simple language with solved examples for CBSE exam preparation.

What you will learn

• Who Euclid was and why his book Elements is so important
• Why some terms like point, line and plane are left undefined
• Euclid's definitions of basic geometric ideas
• The difference between axioms and postulates, with Euclid's seven axioms and five postulates
• How a theorem is proved using axioms, postulates and earlier results

Key Concepts and Formulas

Geometry Before and After Euclid

The word geometry comes from Greek words meaning "earth measurement". Ancient civilisations in Egypt, Babylon and India (for example, the Sulbasutras used for building fire altars) used geometry for practical work like land measurement and construction. Around 300 BCE, Euclid, a teacher in Alexandria, Egypt, collected the known results and arranged them logically in a work called Elements, which has 13 books (chapters).

• Euclid started with a few basic assumptions and deduced everything else from them.
• This method of reasoning from accepted statements is called the deductive or axiomatic method.

Euclid's Definitions and Undefined Terms

Euclid tried to define every basic idea. Some of his definitions, in simple words, are given below.

• A point is that which has no part (no length, breadth or thickness).
• A line is length without breadth.
• The ends of a line are points.
• A straight line is one that lies evenly with the points on itself.
• A surface has only length and breadth; the edges of a surface are lines.
• A plane surface lies evenly with the straight lines on itself.
• These definitions use other words (like "part" and "breadth") that are themselves undefined, so today point, line and plane are accepted as undefined terms that we understand intuitively.

Axioms and Postulates

Euclid called the assumptions that apply to all of mathematics axioms (common notions), and the assumptions specific to geometry postulates. Today both words are used for statements accepted without proof. The axioms are listed in words first and then written in symbols, where a, b and c stand for any quantities.

• Axiom 1: Things equal to the same thing are equal to one another.
• Axiom 2: If equals are added to equals, the wholes are equal.
• Axiom 3: If equals are subtracted from equals, the remainders are equal.
• Axiom 4: Things that coincide with one another are equal.
• Axiom 5: The whole is greater than the part.
• Axiom 6: Things that are double of the same things are equal.
• Axiom 7: Things that are halves of the same things are equal.

Axiom 1: a = c and b = c ⇒ a = b
Axiom 2: a = b ⇒ a + c = b + c
Axiom 3: a = b ⇒ a − c = b − c
Axiom 5: if a = b + c (c > 0), then a > b
Axiom 6: a = 2c and b = 2c ⇒ a = b
Axiom 7: a = c/2 and b = c/2 ⇒ a = b

Euclid's Five Postulates

These five postulates describe what can be done in geometry, in our own words.

• Postulate 1: A straight line can be drawn from any one point to any other point.
• Postulate 2: A terminated line (line segment) can be extended indefinitely in either direction.
• Postulate 3: A circle can be drawn with any centre and any radius.
• Postulate 4: All right angles are equal to one another.
• Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side add up to less than two right angles, then the two lines, when extended far enough, meet on that side.
• Postulate 5 is called the parallel postulate; it is more complicated than the others and mathematicians studied it for centuries.

Postulate 5, with ∠1 and ∠2 the interior
angles on the same side of the transversal:
If ∠1 + ∠2 < 180°
then the two lines meet on that side

Theorems, Consistency and an Important Axiom

A theorem (or proposition) is a statement that is proved using definitions, axioms, postulates and previously proved results. A system of axioms is consistent if it is impossible to deduce a statement and its opposite from them.

• Axiom 5.1: Given two distinct points, there is a unique line that passes through them.
• Theorem 5.1: Two distinct lines cannot have more than one point in common.
• Proof idea: if two distinct lines met at two points P and Q, then two different lines would pass through P and Q, which contradicts Axiom 5.1. So they meet in at most one point.

Solved Examples

Example 1: If AB = PQ and PQ = XY, what can you say about AB and XY?

AB = PQ and XY = PQ
Things equal to the same thing are equal (Axiom 1)
∴ AB = XY

Example 2: Points A, B and C lie on a line with B between A and C. If AB = 4 cm and BC = 7 cm, find AC and compare it with AB.

AC = AB + BC
AC = 4 + 7 = 11 cm
Whole is greater than the part (Axiom 5)
∴ AC = 11 cm and AC > AB

Example 3: Riya and Kabir have equal savings. Each of them receives ₹50 as a gift. Which axiom tells us their savings are still equal?

Riya's savings = Kabir's savings
Riya's + ₹50 = Kabir's + ₹50
Equals added to equals give equal wholes
∴ Axiom 2

Example 4: If x + 6 = 15, which axiom lets us conclude x = 9?

x + 6 = 15
x + 6 − 6 = 15 − 6
Equals subtracted from equals (Axiom 3)
∴ x = 9, using Axiom 3

Example 5: If ∠P = 2∠X and ∠Q = 2∠X, how are ∠P and ∠Q related?

∠P = 2∠X and ∠Q = 2∠X
Doubles of the same thing are equal (Axiom 6)
∴ ∠P = ∠Q

Example 6: How many lines can pass through (a) one given point and (b) two distinct points M and N?

Answer: Through one point, infinitely many lines can be drawn. Through two distinct points M and N, exactly one line passes, by Axiom 5.1.

Common Mistakes to Avoid

• Do not mix up axioms and postulates when the question asks for Euclid's terms: axioms are general, postulates are specific to geometry.
• Always name the axiom or postulate you use; a correct statement without the reason usually loses marks in the CBSE board exam.
• Point, line and plane are undefined terms; do not try to give a formal definition in an answer.
• Write the fifth postulate carefully: the lines meet on the side where the interior angles add up to less than 180°.

Quick Revision

Axiom 1: a = c and b = c ⇒ a = b
Postulate 5: ∠1 + ∠2 < 180° ⇒ lines meet

• Euclid wrote Elements (13 books) around 300 BCE in Alexandria.
• Undefined terms: point, line, plane.
• Seven axioms are common notions; five postulates are specific to geometry.
• Through two distinct points there is exactly one line.
• Two distinct lines have at most one common point.

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