# NCERT Solutions for Class 9 Mathematics Chapter 5 – Introduction to Euclid Geometry

Mathematics is a subject that deals with numbers, shapes, objects, diagrams, graphs and various functions and patterns. Basically, it helps students to develop the reasoning and thinking abilities required in this fast-paced world. Hence, it is quite a demanding subject in almost all fields of sciences and commerce.

Euclid was a Mathematics  teacher from Egypt who formulated some of the important concepts of Geometry. Hence in his honour, the Chapter 5 of Class 9 Mathematics is named as Introduction to Euclid’s Geometry. The terminologies and the points covered in this chapter include introduction, definitions, axioms and postulates.

In our NCERT Solutions for Class 9 Mathematics Chapter 5, you will find some of the important definitions listed by Euclid. It also has the various postulates given by Euclid and his understanding of the role of Geometry in Mathematics. His overview on dealing with various geometrical concepts is also included in it.

You can get a detailed analysis of the chapter along with what concepts to study more in-depth on our Extramarks website. You will learn how to study smartly, how to manage time efficiently, and how to draw quick conclusions if you regularly visit Extramarks website and take full advantage of it.

## Key Topics Covered In NCERT Solutions for Class 9 Mathematics Chapter 5

The term ‘Geometry’ originates from the Greek word. ‘Geo’ means ‘earth’ whereas ‘metry’ means ‘to measure’. In short, Geometry is the study of the measurement of the earth and various components associated with it. As the name of the chapter suggests, the chapter will be more about Euclid’s understanding of the measurement of earth.

In order to understand  this chapter and remember all the definitions and postulates easily, we have provided well-structured academic notes in our NCERT Solutions for Class 9 Mathematics Chapter 5 on the Extramarks’ website. Apart from Class 9 Mathematics solutions, you can also find a lot of material related to Class 9 and other classes for Mathematics as well as other subjects. You can access and take full benefits by registering on our Extramarks’ website.

In our NCERT Solutions for Class 9 Mathematics Chapter 5, we have covered the following topics related to the chapter Introduction to Euclid’s Geometry:

 Exercise Topic 5.1 Introduction 5.2 Euclid’s Definitions, Axioms and Postulates 5.3 Equivalent versions of Euclid’s Fifth Postulate

The definitions and postulates covered in this chapter will go a long way in understanding Geometry as an individual subject and its application in various science and engineering fields. Hence, it is considered a crucial topic.

5.1 Introduction

This section starts with the basic introduction of Geometry, its meaning, where is the word derived from, and its application in day to day life.

You can find the area of regular and irregular shapes, measure the length of the river etc. if you have an idea of the basic applications of Geometry. The different styles of examples based on ancient India help you to grasp these concepts quickly.

Moreover, you will also read about Euclid, a teacher of Mathematics, in this section and try to develop Euclid’s thinking as well as understanding.

5.2 Euclid’s Definitions, Axioms and Postulates

A solid figure has shape, size and well-defined boundaries which can be shifted from one place to another, called surfaces. Each surface has dimensions. Euclid summarised these dimensions into properly structured statements known as definitions.

Further, he gave certain general notions called axioms and stated a few assumptions known as postulates.

The five postulates given by Euclid as given in our Class 9 Mathematics Chapter 5 Solutions are:

Postulate No. 1: A straight line may be drawn from any one point to any other point

Postulate No. 2: A terminated line can be produced indefinitely.

Postulate No. 3: A circle can be drawn with any centre and radius.

Postulate No. 4: All right angles are equal to one another.

Postulate No. 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.

5.3 Equivalent versions of Euclid’s Fifth Postulate

Euclid’s Fifth Postulate is considered one of the most important postulates in the study of Geometry. Due to various uses in the applications of Geometry, it holds a huge significance.

You will find different results and their interpretations associated with it in our NCERT Solutions for Class 9 Mathematics Chapter 5 in detail and develop a solid conceptual understanding.

### NCERT Solutions for Class 9 Mathematics Chapter 5 Exercise &  Solutions

To get through examinations, be they related to your school or any other competitive examinations, you should regularly solve all the questions and exercises given in the NCERT textbooks to test your knowledge on the topic. Hence, in our NCERT Solutions for Class 9 Mathematics Chapter 5, we have compiled all the questions and solutions in one place to provide you with ease. You can get it on the Extramarks website and test  your preparation.

You can find for exercise specific questions and solutions for NCERT Solutions for Class 9 Mathematics Chapter 5 by referring to the following links:

•  Chapter 5: Exercise 5.1 Question and answers
•  Chapter 5: Exercise 5.2 Question and answers

Along with NCERT Solutions for Class 9 Mathematics Chapter 5, students can explore NCERT Solutions on our Extramarks website for all primary and secondary classes.

• NCERT Solutions Class 1
• NCERT Solutions Class 2
• NCERT Solutions Class 3
• NCERT Solutions Class 4
• NCERT Solutions Class 5
• NCERT Solutions Class 6
• NCERT Solutions Class 7
• NCERT Solutions Class 8
• NCERT Solutions Class 9
• NCERT solutions Class 10
• NCERT solutions Class 11
• NCERT solutions Class 12

### NCERT Exemplar for Class 9 Mathematics

NCERT Exemplar Class 9 Mathematics book has a compilation of all the advanced level questions designed as per NCERT textbooks based on the latest CBSE syllabus. Questions are made in a way which requires students to use their comprehension skills.

The trend of questions follows the pattern of the competitive examinations. As a result, students rely a lot on NCERT Exemplar for their studies and use it as a reference book in preparation for the competitive examinations. From developing an interest in the subject to gaining insights into learning various concepts of Mathematics, the book offers multiple benefits. Hence, it is fruitful for the students of all the classes irrespective of their curriculum.

Students develop more practical and logical  thinking after referring to NCERT Solutions and NCERT Exemplar.  It has the key aspect of anything you study helps in better understanding of the things. As a result, students can solve more advanced and higher-level conceptual questions easily.  NCERT Exemplar can boost your performance to a great extent and therefore we highly recommend you to incorporate it in your  study material.

#### Key Features of NCERT Solutions for Class 9 Mathematics Chapter 5

The best performers are those who practice a lot and rectify their mistakes. Hence, NCERT Solutions for Class 9 Mathematics Chapter 5 offers a complete guide to everything that is required in your preparation. The key features are provided:

• You can find all the exercises related to question coverage as well as extra questions in our NCERT study material.
• You will learn to apply the use of all the core concepts and will connect them to all the interrelated topics with the help of Extramarks NCERT solutions.
• After completing the NCERT Solutions for Class 9 Mathematics Chapter 5, you will develop confidence in solving Geometry based questions and hence will be able to solve them with ease.

Q.1 Which of the following statements are true and which are false? Give reasons for your answers.
(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 the sides.
(iv) If two circles are equal, then their radii are equal.
(v) In the below figure, if AB = PQ and PQ = XY, then AB = XY.

Ans.

(i) False, many lines can pass through a single point. In the following figure, it can be seen that there are infinite numbers of lines passing through a single point P.

(ii) False, only one line can pass through two distinct points. 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.

(iii) True, a terminated line can be produced indefinitely on both the sides.

Let AB be a terminated line. It can be seen that it can be produced indefinitely on both the sides.

(iv) True, If you superimpose the region bounded by one circle on the other, then they coincide. So, their centres and boundaries coincide. Therefore, their radii will coincide.

(v) True. It is given that AB and XY are two terminated lines 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 and XY will be equal to each other.

Q.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?

(i) parallel lines (ii) perpendicular lines
(iii) line segment (iv) radius of a circle
(v) square

Ans.

(i) 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.

(ii) Perpendicular lines: If two lines intersect at 90°, then these lines are called perpendicular lines. We should define line and angle before defining perpendicular lines.

(iii) Line segment: A part of a line, whose starting and ending points are known, is called line segment. We should define point and line before defining line segment. AB is a line segment in the figure.

(iv) Radius of a circle: The distance from centre to any point lying on the circumference is called radius of the circle. We should define centre, circle and distance before defining radius of a circle.

(v) Square: A rectangle whose adjacent sides are equal is called a square. In other words, a quadrilateral, whose all sides are equal and each angle is 90°, is called square. We should define quadrilateral, side and agnle before defining the square.

Q.3 Consider two ‘postulates’ given below:

(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.

(ii) There exist 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 which should be listed.
These postulates are consistent, because they deal with two different situations — (i) says that given two points A and B, there is a point C lying on the line in between them; (ii) says that given A and B, you can take C not lying on the line through A and B.
Both these postulates are not related to Euclid’s postulates, so, these postulates do not follow Euclid’s postulates.

Q.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.

$\begin{array}{l}\mathrm{Since},\text{AC}=\text{BC }\left[\mathrm{Given}\right]\\ \text{So, \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{AC}+\mathrm{AC}=\mathrm{BC}+\mathrm{AC}\text{ }\left[\mathrm{equals}\text{are added to equals.}\right]\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}2\mathrm{AC}=\mathrm{AB}\text{ }\left[\because \mathrm{AC}+\mathrm{BC}=\mathrm{AB}\right]\\ \mathrm{AC}=\frac{1}{2}\mathrm{AB}\end{array}$

Q.5 If a point C lies is a mid-point of line segment AB, then prove that every line segment has one and only one mid-point.

Ans.

Let D be the midpoint of AB other than C, then

It is given that C is midpoint of AB, then

AC = CB

AC + AC = AC + DB [equals are added to equals.]

2AC = AB
AC = (1/2)AB … (ii)
From equation (i) and equation (ii), we have

(Things which are double of the same things are equal to one another.)

This is possible only when point C and D are representing a single point. Hence, our assumption is wrong that D is a midpoint of AB other than C.

Therefore, every line has one and only one midpoint.

Q.6 In the below figure, if AC = BD, then prove that AB = CD.

Ans.

Given: AC = BD
To prove: AB = CD
Proof: AC = BD [Given]

AB + BC = BC + CD
AB + BC – BC = BC + CD – BC

(According to Euclid’s axiom, when equals are subtracted from equals, the resultants are also equal.)

AB = CD. Hence proved.

Q.7 Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)

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.

Case I

Let A represent a whole quantity and only x, y, z are parts of it.

A = x + y + z

Clearly, A will be greater than all its parts x, y, and z.

Therefore, it is rightly said that the whole is greater than the part.

Case II

Let us consider the country India. Then, let us consider a state Goa which belongs to India.
If we compare the land area of state Goa with that of India, we will find that land area of India is greater than that of Goa.
That is why we can say that the whole is greater than the part.

This is true for anything in any part of the world and is thus a universal truth.

Q.8 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 each 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.

Q.9 Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.

Ans.

According to Euclid’s 5th postulate, when n line falls on l and m and if

$\begin{array}{l}\angle \text{\hspace{0.17em}}1+\angle \text{\hspace{0.17em}}2<180°\text{then}\angle \text{\hspace{0.17em}}3+\angle \text{\hspace{0.17em}}4>180°.\\ \mathrm{Then}\text{}\mathrm{lines}\text{}l\text{and m will intersect as the sum of}\angle \text{\hspace{0.17em}}1\text{}\mathrm{and}\text{}\angle \text{\hspace{0.17em}}2\text{is}\\ \text{less then 180°}\text{.}\\ \text{If}\angle \text{\hspace{0.17em}}1+\angle \text{\hspace{0.17em}}2=180°\text{then}\angle \text{\hspace{0.17em}}3+\angle \text{\hspace{0.17em}}4=180°\\ \mathrm{Then}\text{}\mathrm{lines}\text{}l\text{and m will not intersect as the sum of}\angle \text{\hspace{0.17em}}1\text{}\mathrm{and}\text{}\angle \text{\hspace{0.17em}}2\text{is}\\ \text{equal to 180°}\text{. Therefore, we can say that lines are parallel}\text{.}\end{array}$