# NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclidâs Geometry

**NCERT solutions for class 9 maths chapter 5 Introduction to Euclid’s Geometry- **Geometry can be written as Geo + Metry. 'Geo' is a Greek word which means 'earth' and 'Metry' means 'to measure'. This chapter will discuss Euclid’s approach to geometry and its real-life applications. NCERT solutions for class 9 maths chapter 5 Introduction to Euclid's Geometry has been designed to provide you assistance while solving the exercise problems related to this chapter. Euclidean Geometry was introduced around 300 BC by a Greek Mathematician Euclid. Euclidean geometry is based on his theorems and axioms to study the relationship between lines, solid figures, and angles in space. In this chapter, there are 2 exercises with 9 questions. NCERT solutions for class 9 maths chapter 5 Introduction to Euclid’s Geometry consist of detailed explanations for all the 9 questions. This chapter talks about the basic observation of geometry which is generally ignored by the students. NCERT solutions for class 9 maths chapter 5 Introduction to Euclid’s Geometry is covering each and every single step to answer the practice exercise questions. In this chapter, you will study definitions like a point, a line, and a plane which are defined by Euclid’s and Euclid’s five postulates. NCERT solutions for other classes and subjects are also available which can be downloaded by clicking on the given link. Here you will get NCERT solutions for class 9 also.

** Important point of NCERT Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry **

Some of Euclid’s axioms are given below-

- If things are equal to the same thing then things are equal to one another.
- If equals are added to the equals, then the wholes are also equal.
- If equals are subtracted from the equals, then the remainders are also equal.
- If things coincide with one another, then things are equal to one another.
- If things which are double of the same things, then things are equal to one another.
- If things which are halves of the same things, then things are equal to one another.
- The whole is greater than the part.

## ** NCERT solutions for class 9 maths chapter 5 Introduction To Euclid's Geometry Excercise: 5.1 **

** Q1 ** 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 Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.

** Answer: **

i) ** FALSE **

Because there is the infinite number of lines that can be passed through a single point. As shown in the diagram below

ii) ** FALSE **

Because only one line can pass through two distinct points. As shown in the diagram below

iii) ** TRUE **

Because a terminated line can be produced indefinitely on both sides. As shown in the diagram below

iv) ** TRUE **

Because if two circles are equal, then their centre and circumference will coincide and hence, the radii will also be equal.

v) ** TRUE **

By Euclid’s first axiom things which are equal to the same thing, are equal to one another

** Answer: **

Yes, there are other terms that are needed to be defined first which are:

Plane: A plane is a flat surface on which geometric figures are drawn.

Point: A point is a dimensionless dot which is drawn on a plane surface.

Line: A line is the collection of n number of points which can extend in both the directions and has only one dimension.

i) Parallel line:-

If the perpendicular distance between two lines is always constant and they never intersect with each other in a plane. Then, two lines are called parallel lines.

** Answer: **

Yes, there are other terms that are needed to be defined first which are:

Plane: A plane is a flat surface on which geometric figures are drawn.

Point: A point is a dimensionless dot which is drawn on a plane surface.

Line: A line is the collection of n number of points which can extend in both the directions and has only one dimension.

ii) perpendicular line:-

If two lines intersect with each other and make a right angle at the point of intersection. Then, two lines are called perpendicular lines.

** Answer: **

Yes, there are other terms that are needed to be defined first which are:

Plane: A plane is a flat surface on which geometric figures are drawn.

Point: A point is a dimensionless dot which is drawn on a plane surface.

Line: A line is collection of n number of points which can extend in both the directions and has only one dimension.

iii) line segment : -

A straight line with two end points that cannot be extended further and has a definite length is called line segment

** Answer: **

iv) Radius of the circle : -

The distance between the centre of the circle and any point on the circumference of the circle is called the radius of a circle.

** Q2 (v) ** 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? v) square

** Answer: **

v) Square:-

A square is a quadrilateral in which all the four sides are equal and each internal angle is a right angle.

To define the square, we must know about quadrilateral.

** Q3 ** 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.

** Answer: **

There are various undefined terms in the given postulates.:

1) There is no information given about the plane whether the points are in the same plane or not.

2) There is the infinite number of points lie in a plane. But here the position of the point C has not specified whether it lies on the line segment joining AB or not.

Yes, these postulates are consistent when we deal with these two situations:

(i) Point C is lying in between and on the line segment joining A and B.

(ii) Point C does not lie on the line segment joining A and B.

No, they don’t follow from Euclid’s postulates. They follow the axioms.

** Answer: **

It is given that ** AC = BC **

Now,

In the figure given above, AB coincides with AC + BC.

Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AC + BC = AB

Now,

2AC = AB

Therefore,

** Hence proved. **

** Answer: **

Let's assume that there are two midpoints C and D

Now,

If C is the midpoint then, AC = BC

And

In the figure given above, AB coincides with AC + BC.

Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AC + BC = AB

From this, we can say that

2AC = AB -(i)

Similarly,

If D is the midpoint then, AD = BD

And

In the figure given above, AB coincides with AD + BD.

Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AD + BD = AB

From this, we can say that

2AD = AB -(ii)

Now,

From equation (i) and (ii) we will get

AD = AC

and this is only possible when C and D are the same points

Hence, our assumption is wrong and there is only one midpoint of line segment AB.

** Q6 ** In Fig. 5.10, if AC = BD, then prove that AB = CD.

** Answer: **

From the figure given in the problem,

We can say that

AC = AB + BC and BD = BC + CD

Now,

It is given that AC = BD

Therefore,

AB + BC = BC + CD

Now, According to Euclid's axiom, when equals are subtracted from equals, the remainders are also equal. Subtracting BC from both sides.

We will get

AB + BC - BC = BC + CD - BC

AB = CD ** Hence proved **

** Answer: **

** Axiom 5 ** states that the whole is greater than the part.

Lets take A = x + y + z

where A , x , y , z all are positive numbers

Now, we can clearly see that A > x , A > y , A > z

Hence, by this we can say that the whole (A) is greater than the parts. (x , y , z)

## ** NCERT solutions for class 9 maths chapter 5 Introduction To Euclid's Geometry Excercise: 5.2 **

** Q1 ** How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?

** Answer: **

Euclid's 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.

Now, in an easy way

Let the line PQ in falls on lines AB and CD such that the sum of the interior angles 1 and 2 is less than 180° on the left side of PQ. Therefore, the lines AB and CD will eventually intersect on the left side of PQ.

** Q2 ** Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.

** Answer: **

According to Euclid's 5 postulates, the line PQ falls on lines AB and CD such that the sum of the interior angles 1 and 2 is less than 180° on the left side of PQ. Therefore, the lines AB and CD will eventually intersect on the left side of PQ

Now,

If then, the line never intersects with each other.

Therefore, we can say that lines AB and CD are parallel to each other

** NCERT solutions for class 9 maths chapter wise **

** NCERT solutions for class 9 subject wise **

** **

**How to use NCERT solutions for class 9 maths chapter 5 Introduction to Euclid's Geometry **

- Go through each and every theorem and property have given at the start of the chapter.
- Have a glance through some examples given using those theorem and properties.
- Now, you can jump to the practice exercises to implement the acquired knowledge.
- During the practice, if you stuck anywhere then you can take the help of NCERT solutions for class 9 maths chapter 5 Introduction to Euclid's Geometry.
- Once you complete the above three points, then you can do some more practice using past papers.

## Frequently Asked Question (FAQs) - NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclidâs Geometry

**Question: **What are the important topics in chapter Chapter 5 Introduction to Euclid's Geometry ?

**Answer: **

Euclid’s definition, and axioms and postulates are two important topics covered in this chapter

**Question: **How many chapters are there in the CBSE class 9 maths ?

**Answer: **

There are 15 chapters starting from number system to probability in the CBSE class 9 maths.

**Question: **Which are the most difficult chapters of NCERT Class 9 Maths syllabus?

**Answer: **

Most of the students consider geometry especially triangles, and surface areas and volumes are the most difficult chapter in the CBSE class 9 maths.

**Question: **Does CBSE provides the solutions of NCERT for class 9 maths ?

**Answer: **

No, CBSE doesn’t provide NCERT solutions for any class or subject.

**Question: **Where can I find the complete solutions of NCERT for class 9 maths ?

**Answer: **

Here you will get the detailed NCERT solutions for class 9 maths by clicking on the link.

**Question: **How does the NCERT solutions are helpful ?

**Answer: **

NCERT solutions are provided in a very detailed manner which is helpful for the students if they stuck while solving NCERT problems. These solutions will give them conceptual clarity.

## Latest Articles

##### OFSS Bihar Admission 2020 Started on ofssbihar.in - Apply Now

OFSS Bihar admission 2020 Online - Bihar School Examination Bo...

##### Bihar Board Compartmental Result 2020 Declared!- Check BSEB 10...

Bihar Board Compartmental Result 2020 has released in on Aug 7...

##### NCERT Books for Class 11 English 2020-21 - Download Pdf here

NCERT Books for Class 11 English 2020-21 - NCERT books of Engl...

##### Mahadbt Scholarship 2020 -Dates, Eligibility, Application Form

Mahadbt Scholarship 2020 - Students can check eligibility, ben...

##### PRMO Registration 2020- Check Application Dates, Fee, Details

PRMO Registration 2020 - HBCSE will release the registration f...

##### Looking Ahead - Planning for Post COVID-19 Learning and Teachi...

Looking Ahead - Planning for Post COVID-19 Learning and Teachi...

##### Kerala Plus One Result 2020 Date and Time- Check DHSE Kerala ...

Kerala Plus One Result 2020 - DHSE, Kerala has declared the pl...

##### JAC Board 2021 - Jharkhand Academic Council Exam Date, Admit ...

Jharkhand Academic Council 2021 - JAC board has started 2020-2...

##### JAC 9th Syllabus 2020-21 - Download Jharkhand Board Class 9th ...

JAC Cass 9th Syllabus 2020-21 - Students can download Jharkhan...

##### Meghalaya Board of School Education 2021 - MBOSE Board Date Sh...

Meghalaya Board of School Education (MBOSE) 2021 - Know more d...