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.
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
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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.
FAQs on NCERT Solutions Class 9 Maths Chapter 5 Introduction to Euclid's Geometry
1. What are the Main differences between a Postulate and a Theorem? How do the Axioms differ from the Postulates?
Postulate | Theorem |
A postulate is a statement that is accepted true without any proof. | A theorem is a statement which can be proven using postulates, previously proved statements and deductive reasoning. |
It is often called axioms. | It is also called propositions. |
Example: Euclid’s postulates, i.e. A straight line may be drawn by joining anyone point to any other point. | Example: Pythagoras theorem, i.e. In any right-angled triangle, the square of its hypotenuse is equal to the sum of squares of the other two sides. |
The postulates and axioms are often used interchangeably. They differ from each other by the fact that postulates are specific to geometry while, axioms are used throughout mathematics.
2. How many Postulates did Euclid Propose and What are these Postulates?
Euclid proposed five postulates. These five postulates of Euclid are given below:
A straight line may be drawn by joining anyone point to any other point.
A terminated line can be produced indefinitely.
A circle can be drawn of any radius with any centre.
All right angles are equal to one another
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.
3. What are the Other Two Equivalent Versions of Euclid’s Fifth Postulate? What was the Outcome of Attempts to Prove Euclid’s Fifth Postulate?
The other two equivalent versions of Euclid’s fifth postulate are:
For every line l and for every point P not lying on the line, there exists a unique line m which passes through the point P and is parallel to the line l.
Two distinct intersecting lines cannot be parallel to the same given line.
The outcome of attempts to prove Euclid’s fifth postulate using the first four postulates failed. But this led to the discovery of several other geometries, called non-Euclidean geometries. The non-Euclidean geometry is also called spherical geometry. In spherical geometry, lines are not straight. They are parts of great circles.
Euclidean geometry is only valid for the figures in the plane. It fails for the curved surfaces.
4. What is Euclid's Geometry in Class 9?
Euclid’s Geometry, also known as Euclidean Geometry is Chapter 5 of Class 9 Maths and is the study of plane and solid figures based on various axioms and theorems given by the Greek mathematician Euclid. Euclidean geometry deals with the relationship between all these figures. Students can refer to the NCERT Solutions For Class 9 Maths Chapter 5 “Introduction to Euclid’s Geometry” from Vedantu, to help in a better and clear understanding of these concepts.
5. What do we mean by Euclidean Geometry?
The term Geometry comes from the Greek words ‘Geo’, meaning the ‘Earth’, and ‘Metre’, meaning ‘to measure’. The need for the concept of Geometry dates back in time when Greek mathematician Euclid proposed some postulates and axioms that described the relationship and properties of the solid and plane figures. This form of geometry is hence known as Euclidean Geometry. Euclidean geometry is based on Euclid’s ‘Elements,’ which is a mathematical and geometrical book consisting of 13 parts explaining the axioms and postulates.
6. What is the Euclid Axiom?
A proposed statement that declares a certain assumed fact and is generally accepted without any proof is known as an axiom. The Euclidean axiom is also known as the postulates of Euclidean Geometry are the five postulates given by Euclid in the field of Plane Geometry. The axioms are common to entire mathematics whereas the Postulates refer to the assumptions specific to geometry.
7. Are solutions available?
Yes, the solutions for Chapter 5 of Class 9 Maths are easily available on Vedantu. Here, you will find the best possible explanations for this chapter and all your NCERT Solutions for your exercises are provided in PDF format too, which is totally free to download and you can study from it even in the offline mode. For this:
Visit Vedantu
You can select your preferred chapter
Click on the download PDF option.
Once you’re redirected to a page you will be able to download it.
8. Can I score full marks in Chapter 5 Class 9?
Maths is a scoring subject. All you have to do is be consistent and practice regularly. If you face difficulty in solving these problems, then you can refer to Vedantu for help at free of cost. Practising the NCERT Solutions for Chapter 5 of Class 9 Maths from Vedantu will bring perfection and help you score full marks in your examination. With the help of the Vedantu website and the Vedantu app, you will be able to ace your exams.
9. What is covered in Chapter 5 of Euclid Geometry Class 9 Maths?
Chapter 5 of Euclid Geometry Class 9 Maths, titled "Introduction to Euclid's Geometry," covers the fundamental principles of geometry as established by the ancient Greek mathematician Euclid. The chapter includes definitions, axioms, and postulates that form the basis of geometric reasoning and proofs.
10. What type of questions can I expect from Chapter 5 of Euclid's Geometry Class 9 in exams?
Questions from Euclid's Geometry Class 9 Chapter 5 typically involve:
Proving geometric properties using Euclid's postulates and axioms.
Understanding and applying definitions and basic terms.
Logical reasoning based on Euclidean principles.
Solving problems that require constructing figures based on given postulates.
11. Why is studying geometry important in Euclid Geometry Class 9 Maths Chapter 5 Solutions PDF?
Studying Euclid's geometry is important because it forms the foundation of all geometric reasoning and proof techniques. It enhances logical thinking, and problem-solving skills, and provides a historical perspective on the development of mathematical thought.
12. What are Euclid’s five postulates in Class 9 Chapter 5 Maths?
Euclid's five postulates in Class 9 Chapter 5 Maths are:
A straight line segment can be drawn joining any two points.
A straight line can be extended indefinitely in both directions.
A circle can be drawn with any centre and any radius.
All right angles are equal to one another.
If a straight line intersects two straight lines and makes the interior angles on the same side less than two right angles, the two lines will meet on that side when extended.
13. What problems do the solutions address in Chapter 5 Maths Class 9?
The solutions typically cover in Class 9th Maths Chapter 5 problems related to:
Identifying points, lines, and planes in diagrams.
Applying definitions of geometric terms to solve problems.
Recognizing congruent and similar shapes based on their properties.
(Optional) Understanding and applying axioms and postulates in basic geometric proofs.