NCERT Solutions for Class 9 Introduction to Euclid's Geometry Maths Chapter 5 - FREE PDF Download
NCERT Solutions for Class 9 Maths chapter 5 Introduction to Euclid's Geometry by Vedantu introduces the foundational concepts and principles of geometry as developed by the ancient Greek mathematician Euclid. Known as the "father of geometry," . This chapter covers fundamental ideas such as definitions, axioms, and postulates, which are the building blocks of all geometric principles. Understanding these basics is crucial for understanding more advanced concepts later on. Vedantu’s solutions provide step-by-step explanations to help you understand these concepts thoroughly. The clear explanations make it easier to understand the material and apply it to solve problems.
Table of Content
Glance on Class 9 Maths Chapter 5 Introduction to Euclid's Geometry
NCERT Solutions for Class 9 Maths chapter 5 Introduction to Euclid's Geometry includes the topics introduction to geometry, Euclid’s definition, axioms. Postulates, Applications, Euclid's Theorem.
If a straight line intersects two straight lines and makes the interior angles on the same side less than two right angles, then the two lines will meet on that side when extended.
All the right angles (i.e. angles whose measure is 90°) are always congruent to each other i.e. they are equal irrespective of the length of the sides or their orientations.
This article contains chapter notes, important questions, exemplar solutions, exercises, and video links for Chapter 5 - Introduction to Euclid's Geometry, which you can download as PDFs.
There is one exercise (7 fully solved questions) in class 9th maths chapter 5 Introduction to Euclid's Geometry.
Access Exercise wise NCERT Solutions for Chapter 5 Maths Class 9
NCERT Solutions for Class 9 Maths Chapter 5, which is Introduction to Euclid’s Geometry, consists of 2 exercises. Practising these exercises will enable students to get a better idea of the concepts of Euclid’s Geometry. Given below are the details of the types of questions and their variations for each exercise:
Exercise 5.1: Exercise 5.1 has a variety of questions and their solutions. The kinds of questions in exercise 5 are as follows:
True or False
Definition of parallel lines, perpendicular lines, line segments, the radius of a circle, square, etc.
Matching Euclid’s postulates with two given postulates
Drawing figures to prove and explain equations on a line
Proof related to the midpoint of a line segment
Why is Euclid’s Axiom 5 considered a universal truth?
Access NCERT Solutions for Class 9 Maths Chapter 5– Introduction to Euclid’s Geometry
Exercise 5.1
1. Which of the following statements are true and which are false? Give reasons for your answers.
(a) Only one line can pass through a single point.
Ans: False.
Through a single point ‘P’ below, an infinite number of lines can pass. In the following figure, it can be seen that there are infinite numbers of lines passing through a single point P.
(b) There are an infinite number of lines which pass through two distinct points.
Ans: False.
Since through two distinct points, only one line can pass. In the following figure, it can be seen that there is only one single line that can pass through two distinct points P and Q.
(c) A terminated line can be produced indefinitely on both sides.
Ans: True.
A terminated line can be produced indefinitely on both sides. Let AB be a terminated line. It can be seen that it can be produced indefinitely on both sides.
(d) If two circles are equal, then their radii are equal.
Ans: True.
If two circles are equal, then their centre and circumference will coincide and hence, the radii will also be equal.
(e) In the following figure, if AB = PQ and PQ = XY, then AB = XY.
Ans: True.
It is given that AB and XY are two terminated lines (Line Segments) and both are equal to a third line PQ.
Euclid’s first axiom states that things which are equal to the same thing are equal to one another.
Therefore, the lines AB =PQ and PQ = XY, Hence AB = XY will be equal to each other.
2. Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?
(a) Parallel lines
(b) Perpendicular lines
(c) Line segment
(d) Radius of a circle
(d) Square.
Ans:
For the desired definition, we need the following terms:
Point
A small dot made by a sharp pencil on a sheet paper gives an idea about a point
A point has no dimension, it has only a position.
Line
A straight crease obtained by folding a paper, a straight string pulled at its two ends, the edge of a ruler are some close examples of a geometrical line.
The basic concept about a line is that it should be straight and that it should extend indefinitely in both the directions.
Plane
The surface of a smooth wall or the surface of a sheet of paper are close examples of plane.
Ray
A part of line l which has only one end- point A and contains the point B is called a ray AB
Angle
An angle is the union of two non- collinear rays with common initial point.
Circle
A circle is the set of all those points in a plane whose distance from a fixed point remains constant.
The fixed point is called the centre of the circle.
Quadrilateral.
A closed figure made of four line segments is called a quadrilateral
(a) Parallel Lines
If the perpendicular distance between two lines is always constant, then these are called parallel lines.
In other words, the lines which never intersect each other are called parallel lines.
To define parallel lines, we must know about point, lines, and distance between the lines and the point of intersection.
(b) Perpendicular Lines
If two lines intersect each other at \[90^\circ \] , then these are called perpendicular lines.
We are required to define line and the angle before defining perpendicular lines.
(c) Line Segment
A straight line drawn from any point to any other point is called as line segment.
To define a line segment, we must know about point and line segment
(d) Radius of a Circle
It is the distance between the centers of a circle to any point lying on the circle.
To define the radius of a circle, we must know about point and circle.
(e) Square
A square is a quadrilateral having all sides of equal length and all angles of same measure, i.e., \[90^\circ \].
To define square, we must know about quadrilateral, side, and angle.
3. Consider the two ‘postulates’ given below:
(i) Given any two distinct points A and B, there exists a third point C, which is between A and B.
(ii) There exists at least three points that are not on the same line. Do these postulates contain any undefined terms?
Are these postulates consistent?
Do they follow from Euclid’s postulates? Explain.
Ans:
There are various undefined terms in the given postulates.
The given postulates are consistent because they refer to two different situations.
Also, it is impossible to deduce any statement that contradicts any well-known axiom and postulate.
These postulates do not follow from Euclid’s postulates.
They follow from the axiom, “Given two distinct points, there is a unique line that passes through them”.
4. If a point C lies between two points A and B such that AC = BC, then prove that AC = 1 2 AB. Explain by drawing the figure.
Ans:
From the Figure, Given that,
AC = BC
Point C lies between two points A and B
To Prove:
AC = \[\dfrac{1}{2}\] AB
Proof:
Consider AC = BC
Adding AC to both the sides of the above Equation,
AC + AC = BC + AC …Equation (\[1\])
\[2\] AC = BC +AC
Here, (BC + AC) coincides with AB. It is known that things which coincide with one another are equal to one another.
∴ BC + AC = AB … Equation (\[2\])
It is also known that things which are equal to the same thing are equal to one another. Therefore, from Equations (\[1\]) and (\[2\]), we obtain
AC + AC = AB
\[2\] AC = AB
∴ AC = \[\dfrac{1}{2}\] AB
5. In the above question, point C is called a mid-point of line segment AB, prove that every line segment has one and only one mid-point.
Ans:
From the figure,
Given,
Let there be two mid-points, C and D.
C is the mid-point of AB.
To Prove:
Every line segment has one and only one mid-point.
Proof:
Let us assume, D be another mid- point of AB.
Therefore AD = DB ... Equation (\[1\])
But it is given that C is the mid- point of AB.
Therefore AC = CB ... Equation (\[2\])
Subtracting Equation (\[1\]) from Equation (\[2\]) we get
AC – AD = CB – DB
DC = – DC
\[2\] DC = \[0\]
DC = \[0\]
Therefore C and D coincide.
Thus, every line segment has one and only one mid- point.
6. In the following figure, if AC = BD, then prove that AB = CD.
Ans:
From the figure, it can be observed that
AC = AB + BC
BD = BC + CD
Given,
AC = BD
To Prove:
AB = CD
Proof:
AB + BC = BC + CD ….Equation (\[1\])
According to Euclid’s axiom, when equals are subtracted from equals, the remainders are also equal.
Subtracting BC from Equation (\[1\]), we obtain
AB + BC − BC = BC + CD − BC
AB = CD
Hence it is proved.
7. Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth?
Ans:
Axiom 5 states that the whole is greater than the part.
This axiom is known as a universal truth because it holds true in any field, and not just in the field of mathematics.
Let us take two cases – one in the field of mathematics, and one other than that.
I. Case I:
a. Let t represent a whole quantity and only a, b, c are parts of it.
b. t = a + b + c o Clearly, t will be greater than all its parts a, b, and c.
c. Therefore, it is rightly said that the whole is greater than the part.
II. Case II:
a. Let us consider the continent Asia. o
b. Then, let us consider a country India which belongs to Asia.
c. India is a part of Asia and it can also be observed that Asia is greater than India.
d. That is why we can say that the whole is greater than the part.
e. This is true for anything in any part of the world and is thus a universal truth.
Overview of Deleted Syllabus for CBSE Class 9 Maths Introduction to Euclid's Geometry
Class 9 Maths Chapter 5: Exercises Breakdown
Conclusion
NCERT Solutions for Maths Introduction to Euclid's Geometry Class 9 Chapter 5 by Vedantu is essential for building a strong foundation in geometry. This chapter introduces you to the foundation for understanding the basic principles of geometry. Euclid clearly defined geometric entities such as points, lines, and planes. In previous year exams, around 2 questions have been asked from this chapter, highlighting its significance in the overall curriculum. By thoroughly practising the problems and understanding the step-by-step solutions provided by Vedantu, you can confidently tackle algebraic expressions and identities.
Other Study Material for CBSE Class 9 Maths Chapter 5
Chapter-Specific NCERT Solutions for Class 9 Maths
Given below are the chapter-wise NCERT Solutions for Class 9 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.
Important Study Materials for Class 9 Maths
Important Study Materials for Class 9 Maths
FAQs on NCERT Solutions for Class 9 Maths Chapter 5 Introduction To Euclid'S Geometry
1. What are the main topics covered in NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid's Geometry as per the 2025–26 CBSE syllabus?
- Definitions of basic geometric terms such as point, line, plane, radius, and quadrilateral.
- Euclid’s five postulates and common notions (axioms), forming the basis for geometry.
- Comparing and contrasting axioms and postulates, including their practical applications.
- Proof-based questions involving midpoints, equality of line segments, and congruency of geometric figures.
- Understanding the logical structure of geometry, including why some postulates are considered ‘universal truths’.
2. How are Euclid's postulates and axioms different in Class 9 Maths Chapter 5 NCERT Solutions?
- Axioms are general universal truths used throughout mathematics, accepted without proof.
- Postulates are assumptions specific to geometry, also accepted without proof but confined to geometric concepts.
- NCERT emphasizes that while both are unproven assumptions, axioms have broader application, and postulates are geometry-specific.
3. What is the importance of understanding Euclid’s fifth postulate in the context of Chapter 5 NCERT Solutions?
Euclid’s fifth postulate is critical because it defines the nature of parallel lines and underpins much of plane geometry. Attempts to prove this postulate gave rise to non-Euclidean geometries, which are important for advanced mathematical understanding.
4. How does solving the NCERT Solutions for Class 9 Maths Chapter 5 help develop logical reasoning in students?
The solutions encourage step-by-step deduction, justification of each statement, and connecting geometric facts using axioms and postulates. This structured approach strengthens mathematical logic and the ability to build clear, valid proofs.
5. Why is the concept of congruent figures introduced through Euclid's axioms in NCERT Solutions for Chapter 5?
Introducing congruency through Euclid's axioms helps students recognize which properties (such as length or angle) are necessary for two figures to be considered equal. This foundational logic is integral for further topics like triangles and quadrilaterals.
6. What types of exam questions are most commonly asked from Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry?
- Statement and Reason questions (True/False with explanations).
- Definition-based questions on terms like point, line, parallel lines, etc.
- Proof or justification questions using Euclid’s postulates.
- Application of the ‘whole is greater than the part’ axiom.
7. Can practicing NCERT Solutions for Class 9 Maths Chapter 5 help score full marks? How?
Yes, regular practice using NCERT Solutions for Chapter 5 ensures familiarity with exam patterns, proof techniques, and key definitions, enhancing confidence and accuracy in both objective and proof-based questions, thus helping score full marks.
8. What is a common misconception about Euclid's postulates among Class 9 students, and how do NCERT Solutions clarify it?
A frequent misconception is that each postulate can be proved using previous ones. NCERT Solutions clarify that postulates are basic assumptions; the failure to prove the fifth postulate led to new geometries, emphasizing the limits of deductive chains in geometry.
9. What problem-solving approaches are recommended in NCERT Solutions for geometry proofs in Chapter 5?
- Begin by identifying which postulates or axioms apply to the given question.
- Draw accurate diagrams and clearly mark known and unknown quantities.
- Use logical sequencing: each step must logically follow from the previous using cited axioms or previously proved results.
10. How does Euclid's axiom ‘the whole is greater than the part’ appear in Class 9 Chapter 5 exam questions?
This axiom is applied in proof-based questions where students are asked to relate parts and wholes (e.g., proving the sum of parts is less than the total length of a segment), thus reinforcing the foundational logic.
11. What should students focus on when defining geometric terms in the NCERT Solutions for Class 9 Maths Chapter 5?
Students should use precise, textbook-aligned definitions, explicitly referencing prior concepts (e.g., to define a line segment, first understand a point and a line), ensuring clarity and logical order in their explanations.
12. What happens if two distinct midpoints are assumed for a line segment, according to NCERT Solutions for Chapter 5?
If two distinct midpoints are assumed, logical proof using Euclid’s axioms results in the conclusion that the two midpoints must coincide; therefore, a line segment can have only one unique midpoint.
13. How can students best use NCERT Solutions for Class 9 Maths Chapter 5 to prepare for CBSE board exams in 2025–26?
- Focus on thoroughly understanding and practicing every step in the official NCERT Solutions.
- Review all proof-based examples, especially those aligned with Euclid’s postulates and axioms.
- Understand exam-pattern questions and avoid memorizing without conceptual clarity.
14. Why is the study of Euclid’s geometry still relevant in the present CBSE Class 9 Maths curriculum?
The logical structure and language of Euclid’s geometry forms the basis for all modern geometric study. Mastery of these fundamentals is necessary for approaching advanced topics and logical reasoning in mathematics.
15. Which concepts from Chapter 5 are prerequisites for success in later geometry chapters as per the NCERT Class 9 Maths solutions?
Understanding points, lines, planes, postulates, axioms, and proof logic are essential for subsequent chapters like ‘Lines and Angles’, ‘Triangles’, and ‘Quadrilaterals’ in the Class 9 Maths curriculum.
