NCERT Solutions for Class 9 Maths Chapter 5 - Introduction To Euclids Geometry

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?

• Exercise 5.2: Exercise 5.2 has 2 questions - one asks to rewrite Euclid’s fifth postulate in an easy-to-understand manner and another asks whether Euclid’s fifth postulate implies the existence of parallel lines. Both these questions are conceptual questions, answering which will improve a student’s understanding of the theory.

Access NCERT Solutions for Class-9 Maths Chapter 5– 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.

Exercise 5.2

1. How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?

Ans:

• Two lines are said to be parallel if they are equidistant from one other and they do not have any point of intersection. 

• In order to understand it easily, let us take any line l and a point P not on l. Then, by Playfair’s axiom (equivalent to the fifth postulate), there is a unique line m through P which is parallel to l. 

• The distance of a point from a line is the length of the perpendicular from the point to the line. 

• Let AB be the distance of any point on m from l and CD be the distance of any point on l from m. 

• It can be observed that AB = CD. In this way, the distance will be the same for any point on m from l and any point on l from m. 

• Therefore, these two lines are everywhere equidistant from one another.

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2. Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.

Ans:

• If a straight line l falls on two straight lines m and n such that the sum of the interior angles on one side of l is two right angles, then by Euclid's fifth postulate the lines will not meet on this side of l.  

• Next, we know that the sum of the interior angles on the other side of line l will also be two right angles. Therefore, they will not meet on the other side also. 

• So, the lines m and n never meet and are, therefore, parallel.

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

You can opt for Chapter 5 - Euclid's Geometry NCERT Solutions for Class 9 Maths PDF for Upcoming Exams and also You can Find the Solutions of All the Maths Chapters below.

NCERT Solutions for Class 9 Maths

• Chapter 1 - Number System

• Chapter 3 - Coordinate Geometry

• Chapter 4 - Linear Equations in Two Variables

• Chapter 5 - Introductions to Euclids Geometry

• Chapter 6 - Lines and Angles

• Chapter 7 - Triangles

• Chapter 8 - Quadrilaterals

• Chapter 9 - Areas of Parallelogram and Triangles

• Chapter 10 - Circles

• Chapter 11 - Constructions

• Chapter 12 - Herons formula

• Chapter 13 - Surface area and Volumes

• Chapter 14 - Statistics

• Chapter 15 - Probability

Introduction to Euclidean Geometry 

Chapter 5 Euclid's Geometry Class 9 is divided into five sections and four exercises. The first section is the introduction with no exercise. The Second and Third section explains the two important properties whereas the Fourth and Fifth sections revisit the topics taught in class 9.

List of Exercises and topics covered in Introduction To Euclid's Geometry Class 9:

What is Euclidean Geometry?

The word ‘geometry’ comes from the Greek words ‘geo’, meaning the ‘earth’, and ‘metre’, meaning ‘to measure’. Geometry appears to have originated from the need for measuring land. Euclidean geometry is the study of plane and solid figures on the basis of axioms and postulates employed by the Greek mathematician named Euclid. It deals with the properties and relationship between all the things. The studies of space and solids were summarised into statements called definitions. Euclid began his exposition by listing all the 23 definitions in Book 1 of the ‘Elements’.

Elements

The study of solids led to the study of surfaces, curves, lines and points. A solid object has shape, size and position. A solid always has a surface which separates one part from the other. The boundaries of these surfaces are either curves or straight lines. And these straight lines are made up of points. All these gave rise to the concept of dimensions:

• Solids - Three Dimensional.

• Surface - Two Dimensional.

• Curve/ Line - One Dimensional.

• Point - No Dimensions.

Euclid’s Elements is a mathematical and geometrical book consisting of 13 parts written by ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt. Later the ‘Elements’ was divided into thirteen books which popularized geometry all over the world.  Given below are the few important points mentioned in the book:

1. A point is that which has no part. 

2. A line is a breadthless length. 

3. The ends of a line are points. 

4. A straight line is a line which lies evenly with the points on itself. 

5. A surface is that which has length and breadth only. 

6. The edges of a surface are lines. 

7. A plane surface is a surface which lies evenly with the straight lines on itself.

Axioms and Postulates

On studying the various definitions, Euclid concluded them into certain assumed properties which weren’t proved.  These were considered as universal truth and were divided into two parts: Axioms (common to entire mathematics) and Postulates (assumptions specific to geometry).

Euclidean Geometry Axioms

Elements is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. He gave five postulates for plane geometry known as Euclid’s Postulates and the geometry is known as Euclidean geometry.

1. Things are equal to one another if those things are equal to the same thing.

2. The wholes are equal if equals are added to equals.

3. The remainders are equal if equals are subtracted from equals.

4. Things are equal to one another if they coincide with one another.

5. The whole will be greater than the part.

Euclid’s Five Postulates

It can be seen that the definition of a few terms needs extra specification. Now let us discuss these Postulates in detail.

Euclid’s Postulate 1

“A straight line can be drawn from any one point to another point.”

This postulate states that at least one straight line passes through two distinct points but he did not mention that there cannot be more than one such line. Although throughout his work he has assumed there exists only a unique line passing through two points.

Euclid’s Postulate 2

“A terminated line can be further produced indefinitely.”

In simple words what we call a line segment was defined as a terminated line by Euclid. Therefore this postulate means that we can extend a terminated line or a line segment in either direction to form a line. In the figure given below, the line segment AB can be extended as shown to form a line.

Euclid’s Postulate 3

“A circle can be drawn with any centre and any radius.”

Any circle can be drawn from the end or start point of a circle and the diameter of the circle will be the length of the line segment.

Euclid’s Postulate 4

“All right angles are equal to one another.”

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.

Euclid’s Postulate 5

“If a straight line falling on two other 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 the side on which the sum of angles is less than two right angles.”

Historical Facts

Need for Geometry in Ancient Civilizations

Geometry was studied in various forms in every ancient civilisation, be it in Egypt, Babylonia, China, India, Greece, the Incas, etc. The people of these civilisations faced several practical problems which required the development of geometry in various ways. For example, whenever the river Nile overflowed, it wiped out the boundaries between the adjoining fields of different landowners. After such flooding, these boundaries had to be redrawn. For this purpose, the Egyptians developed a number of geometric techniques and rules for calculating simple areas and also for doing simple constructions. The knowledge of geometry was also used by them for computing volumes of granaries, and for constructing canals and pyramids. They also knew the correct formula to find the volume of a truncated pyramid.

Pyramid, Harappa And Mohenjo - Daro

You know that a pyramid is a solid figure, the base of which is a triangle, or square, or some other polygon, and its side faces are triangles converging to a point at the top. In the Indian subcontinent, the excavations at Harappa and Mohenjo-Daro, etc. show that the Indus Valley Civilisation (about 3000 BCE) made extensive use of geometry. It was a highly organised society. The cities were highly developed and very well planned. For example, the roads were parallel to each other and there was an underground drainage system. The houses had many rooms of different types. This shows that the town dwellers were skilled in mensuration and practical arithmetic. The bricks used for constructions were kiln fired and the ratio length: breadth: thickness, of the bricks, was found to be 4: 2: 1.

Other Ancient Uses of Geometry

In ancient India, the Sulbasutras (800 BCE to 500 BCE) were the manuals of geometrical constructions. The geometry of the Vedic period originated with the construction of altars (or vedis) and fireplaces for performing Vedic rites. The location of the sacred fires had to be in accordance with the clearly laid down instructions about their shapes and areas if they were to be effective instruments. Square and circular altars were used for household rituals, while altars whose shapes were combinations of rectangles, triangles and trapeziums were required for public worship. The Sri Yantra (given in the Atharvaveda) consists of nine interwoven isosceles triangles. These triangles are arranged in such a way that they produce 43 subsidiary triangles. Though accurate geometric methods were used for the constructions of altars, the principles behind them were not discussed.

Documents and Proofs of Geometry

These examples show that geometry was being developed and applied everywhere in the world. But this was happening in an unsystematic manner. What is interesting about these developments of geometry in the ancient world is that they were passed on from one generation to the next, either orally or through palm leaf messages, or by other ways. Also, we find that in some civilisations like Babylonia, geometry remained a very practical oriented discipline, as was the case in India and Rome. The geometry developed by Egyptians mainly consisted of the statements of results. There were no general rules of the procedure. In fact, Babylonians and Egyptians used geometry mostly for practical purposes and did very little to develop it as a systematic science. But in civilisations like Greece, the emphasis was on the reasoning behind why certain constructions work. The Greeks were interested in establishing the truth of the statements they discovered using deductive reasoning.

Thales, Pythagoras And Euclid

A Greek mathematician, Thales is credited with giving the first known proof. This proof was of the statement that a circle is bisected (i.e., cut into two equal parts) by its diameter. One of Thales’ most famous pupils was Pythagoras (572 BCE), whom you have heard about. Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent. This process continued until 300 BCE. At that time Euclid, a teacher of mathematics at Alexandria in Egypt, collected all the known work and arranged it in his famous treatise, called ‘Elements’. He divided the ‘Elements’ into thirteen chapters, each called a book. These books influenced the whole world’s understanding of geometry for generations to come. In this chapter, we shall discuss Euclid’s approach to geometry and shall try to link it with the present-day geometry.

What So Special About Vedantu’s NCERT Solutions? 

Vedantu is an ardent believer of smart work and harbour experienced teaching professionals who are adept at learning and possess a greater passion for imparting the same. Vedantu makes the learning experience fun by offering solutions in a step by step explanation of numerical problems to help you improve your understanding of the concept related to the topics. NCERT Solution of Class 9 Chapter 5 is engineered by the experts of Vedantu to serve it as an excellent material for practise and make the learning process more convenient. 

NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry - Free PDF

The main strength of the Vedantu’s NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry lies in the following points: 

1. It is written keeping in mind the age group of the students.

2. The solutions are in simple language and emphasis on basic facts, terms, principles and applications on various concepts. 

3. Complicated solutions are broken down into simple parts and well spaced to save the students from the unnecessary strain on their minds.

4. It gives a gist of the entire chapter and concept in the form of solutions.

5. The answers are treated systematically and presented in a coherent and interesting manner.

6. The content is kept concise, brief and self-explanatory.

7. Some answers are incorporated with necessary images to facilitate the understanding of the concept.

8. The solutions are in accordance with the latest syllabus and exam specifications.

Vedantu tried its best to render you real help by providing the NCERT Solutions of Introduction To Euclid's Geometry Class 9th. It aimed to deliver sufficient problems and solutions to practice and build a strong foundation on the chapter.

Deeper Into The Exercises - Types of Questions.

Each of the topics is followed by compact exercises. The exercises aim to test your knowledge and depth of understanding of the different axioms and postulates that are introduced in this chapter. Regardless, it must be noted that the numerical problems of this chapter are mostly based on postulates and other associated concepts. To further help you improve your understanding of these topics and related concepts, numbers of solved examples of numerical problems are also offered. Moreover, a thorough step by step explanation is provided for each solved example. It can help understand which methods are to be used to approach different types of questions for solving them accurately.

NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid Geometry - PDF Download

NCERT Solutions for Class 9 Maths Chapter 5 Introduction To Euclid's Geometry Free PDF available on Vedantu are solved by experts. Download Free Study Material for Class 9 to score more marks. The Vedantu team has verified how many exercises and types of questions are there in Euclid Geometry Class 9.

Section 5.2 - Exercise 5.1

The first exercise of this chapter consists of 7 questions and is covered in details in NCERT Solutions for Maths Class 9 Chapter 5. Most of the questions of this exercise are based on the Postulates of Euclid’s Geometry which plays important in. There are basically three types of questions found from this section:

Type 1: Validation of statements on the basis of axioms and postulates.

Type 2: Definition

Type 3: Analysis of postulates.

Type 4: Proving sums.

These types of questions involve a lot of steps to reach the solution and hence comes with a risk of making a lot of silly mistakes. Make sure that you have a clear understanding of Euclid’s Geometry, its axioms and postulates to minimise the room for silly errors. Also, a better understanding of the steps involved would help them clear their lingering doubts easily. Get all your doubts clear and strengthen your knowledge of the different concepts covered in the chapter by referring to our NCERT Solutions for Class 9th Maths Chapter 5. Each numerical problem has been explained step by step to make it easy for you to understand them and grasp the logic behind the same. Additionally, you will also find many helpful tips and alternative techniques to solve similar problems accurately and with more confidence.

Section 5.3 - Exercise 5.2

The second exercise in chapter 5 class 9 maths consists of 2 questions and is mostly based on the Equivalent Versions of Euclid’s Fifth Postulate. Once you grasp the concept, you will be able to answer all the questions. Given below are the types questions found related to the topic:

Type 1: Rewriting of Euclid’s fifth postulate.

Type 2: Questions on Euclid’s fifth postulate.

Refresh your knowledge of the prime factorisation method and revise the fundamentals of the chapter. Doing so, you will gain more confidence as to how to approach important questions for an exam without making any mistakes. It will also prove useful in helping you solve similar types of numerical problems efficiently and in less time. Study the shortcut techniques from up close by taking a quick look at the NCERT Solutions for Class 9 Maths Chapter 5 pdf offered online in its PDF format.You can ace your upcoming board examination quite easily and with many conveniences by incorporating NCERT Solutions for Class 9 Maths Chapter 5 pdf into your revision plan. Download Vedantu’s study solutions from it’s learning portal with just a click and improve your learning experience without much ado.

You need to solve the problem based on rational numbers without from Chapter 5 Maths Class 9 using the long division method. You are also required to state if they have a non-terminating repeating decimal expansion or a terminating decimal expansion. Solve the exercise and match your answer with our chapter - based solutions online, to gauge your understanding of the topics more effectively.

Revising NCERT Solutions for Class 9 Chapter 5 Maths persistently will go a long way to help you ace their preparation for your upcoming board examination and will prove useful in scoring well in them. It will help you effectively solve this type of numerical problem accurately and in less time.

Summary

• Even though Euclid provided definitions for a point, line, and plane, these definitions aren't universally accepted among mathematicians. 

• Consequently, these terms are considered undefined in modern mathematics. Axioms or postulates, recognized as evident universal truths, stand as assumptions that do not require proof. 

• Theorems, on the other hand, are statements established through deductive reasoning, using definitions, axioms, previously validated statements, and rigorous proofs.

• Euclid proposed several Axioms, including principles such as:

• When two things are equal to a third thing, they are equal to each other.

• If equal quantities are added to equal quantities, the sums are equal.

• Subtracting equal quantities from equal quantities results in equal remainders.

• Objects that coincide with one another are considered equal.

• The whole is greater than any of its parts.

• Objects that are double of the same entity are considered equal.

• Similarly, objects that represent halves of the same entity are also deemed equal.

• Euclid’s Postulates were: 

• Postulate 1: It is possible to draw a straight line from any one point to any other point.

• Postulate 2: A terminated line can be extended indefinitely.

• Postulate 3: A circle can be drawn with any center 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 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.

• Two equivalent formulations of Euclid’s fifth postulate are:

(i) 'Given any line l and any point P not on l, there exists only one line m through P that doesn't intersect with l, termed parallel to l.'

(ii) Two different lines that intersect cannot both be parallel to a third line.

• Efforts to prove Euclid’s fifth postulate solely from the initial four postulates were unsuccessful. However, these attempts resulted in the revelation of alternative geometries known as non-Euclidean geometries.