CBSE • Class 9 • Mathematics • Chapter 5 (Introduction to Euclid's Geometry) • Exercise 5.1

Exercise 5.1: Introduction to Euclid's Geometry — NCERT Solutions

Definitions, axioms and postulates — distinguishing assumptions from proven facts.

Aligned to the latest NCERT 2024-25 edition • 7 questions in this exercise • Free plan, no credit card

What this exercise covers

Axiom vs theoremEuclid's postulatesLogical structure

5 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 5.1 Q1 • 3 marks

Which of the following statements are true and which are false? Give reasons. (i) Only one line can pass through a single point. (ii) There are an infinite number of lines which pass through two distinct points. (iii) A terminated line can be produced indefinitely on both sides.

Hint (Socratic — try this first)▾

Think about how many lines you can draw through one dot versus through two fixed dots.

Step-by-step solution▾

(i) False. Through a single point we can draw infinitely many lines in different directions. So it is not true that only one line passes through a single point. (ii) False. By Euclid's first postulate, exactly one line can be drawn through two distinct points, not infinitely many. (iii) True. A terminated line (a line segment) can be extended endlessly in both directions, according to Euclid's second postulate.

Common mistake:

Students often confuse (i) and (ii) by assuming one line always passes through one point, forgetting that a single point allows infinitely many lines.

Exercise 5.1 Q2 • 3 marks

Give a definition for each of the following terms. Are there other terms that need to be defined first? (i) parallel lines (ii) perpendicular lines (iii) line segment.

Hint (Socratic — try this first)▾

Before defining these, ask which basic words (like 'point', 'line', 'plane') must already be understood?

Step-by-step solution▾

90^\circ

Common mistake:

Students give the definitions but forget to mention the prior undefined terms that these definitions depend on.

Exercise 5.1 Q3 • 3 marks

State whether each statement is an axiom, a postulate, or a theorem: (i) The whole is greater than the part. (ii) A straight line may be drawn from any one point to any other point. (iii) The sum of angles in a triangle is 180°.

Hint (Socratic — try this first)▾

Which statements are self-evident universal truths, which are geometric assumptions, and which must be proved?

Step-by-step solution▾

(i) Axiom.'The whole is greater than the part' is one of Euclid's common notions (axioms) that applies to all mathematics, not just geometry. (ii) Postulate. This is Euclid's first postulate, an assumption specific to geometry. (iii) Theorem. The angle-sum property of a triangle is not assumed — it is proved using postulates and axioms, so it is a theorem. Key idea: Axioms/common notions are universal self-evident truths, postulates are geometric assumptions, and theorems are proved results.

Common mistake:

Students commonly label the triangle angle-sum as a postulate, not realising that anything requiring proof is a theorem.

Exercise 5.1 Q4 • 2 marks

Explain the difference between an axiom and a postulate as used by Euclid, giving one example of each.

Hint (Socratic — try this first)▾

Which type of assumption did Euclid treat as common to all sciences, and which one applied only to geometry?

Step-by-step solution▾

Axiom (common notion): A basic self-evident truth assumed to be true throughout mathematics, not restricted to geometry. Example:'Things which are equal to the same thing are equal to one another.' Postulate: An assumption specific to geometry that Euclid stated without proof to build the subject. Example:'A straight line can be drawn joining any two points.' (First postulate.) Difference: Axioms are general assumptions used everywhere in mathematics, while postulates are geometry-specific assumptions.

Common mistake:

Students treat 'axiom' and 'postulate' as identical, missing that postulates are geometry-specific while axioms are universal.

Exercise 5.1 Q5 • 2 marks

How many lines can pass through (a) one given point, and (b) two given distinct points? Justify each answer.

Hint (Socratic — try this first)▾

Try drawing many lines through a single dot, then attempt the same through two fixed dots.

Step-by-step solution▾

(a) Through one given point: Infinitely many lines can pass through a single point, since a line can be drawn in any direction through that point. (b) Through two distinct points: Exactly one line can pass through two distinct points. This is guaranteed by Euclid's first postulate, which states that a unique straight line joins any two points.

Common mistake:

Students sometimes reverse the two cases, saying one line through one point and many through two points.

Re-read the chapter summary first. Open Introduction to Euclid's Geometry and refresh the key concepts: Axiom, Postulate, Theorem, Euclid's postulates.
Try each problem yourself before opening the solution. Ten focused minutes per problem usually beats reading two finished solutions.
Use Guru AI for the questions you get stuck on. The Socratic AI tutor walks you through it with hints instead of dictating the answer.
Mark the questions you got wrong and revisit them after 24 hours — the spacing is what locks the method into long-term memory.

Exercise 5.1 — Definitions, axioms and postulates — distinguishing assumptions from proven facts.
Exercise 5.2 — Betweenness, uniqueness of a line through two points and simple deductive steps.

Exercise 5.2 →

Betweenness, uniqueness of a line through two points and simple deductive steps.

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