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NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid's Geometry

NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid's Geometry covers Euclid's definitions, axioms and postulates, and how they form the logical foundation of geometry.

Overview of Topics & Subtopics (Click to Jump)

1. Euclid's Definitions

Short Note:

• A point is that which has no part.

• A line is 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.

2. Axioms and Postulates

Short Note:

• Axioms (Common Notions): general statements accepted without proof, true in all branches of mathematics, e.g. 'things which are equal to the same thing are equal to one another'.

• Postulates: assumptions specific to geometry, also accepted without proof.

• Euclid's Five Postulates include: a straight line may be drawn from any one point to any other point; a terminated line can be produced indefinitely; all right angles are equal to one another; and the famous fifth (parallel) postulate.

3. Equivalent Versions of the Fifth Postulate

Short Note:

• Playfair's Axiom: For every line l and every point P not on l, there exists a unique line through P parallel to l. This is the most popular equivalent version used today.

NCERT Textbook Exercise Questions

Which of the following statements are true and which are false? (i) Only one line can pass through a single point.
Answer: False. Infinitely many lines can pass through a single point.
Give a definition for the term 'parallel lines' using undefined terms 'point' and 'line'.
Answer: Two lines in a plane are said to be parallel if they lie in the same plane, do not intersect however far extended in either direction, and the perpendicular distance between them remains constant.
In the given figure, if AB = PQ and PQ = XY, then does AB = XY? State the Euclid's axiom used.
Answer: Yes, AB = XY. This follows from Euclid's first axiom: things which are equal to the same thing are equal to one another.
If a point C lies between two points A and B such that AC = BC, prove that AC = ½ AB.
Answer: Since AC = BC and AC + BC = AB, substituting gives AC + AC = AB, so 2AC = AB, which gives AC = ½ AB.
How would you rewrite Euclid's fifth postulate so that it would be easier to understand?
Answer: Two distinct intersecting lines cannot both be parallel to the same line — or equivalently, through a point not on a given line, exactly one line can be drawn parallel to it (Playfair's Axiom).
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Class 9 › Mathematics › Introduction to Euclid's Geometry
Chapter notesRead the chapterPractice MCQsCheck what you rememberChapter test10 questions on one pageOnline testTest 1, 2 and 3, one question at a timePYQPrevious year questionsStudy materialPDF notes
Topics in this chapter: Euclid's DefinitionsAxioms and PostulatesEquivalent Versions of the Fifth Postulate