NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid’s 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. CBSE 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 angle in space. In this chapter, there are 2 exercises with 9 questions. Solutions of NCERT for class 9 maths chapter 5 Introduction to Euclid’s Geometry consist of a detailed explanations to all the 9 questions. This chapter talks about the basic observation of geometry which is generally ignored by the students. CBSE 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.

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

Q2 (i) 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

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.

Q2 (ii) 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? (ii) 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 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.

Q2 (iii) 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? iii) line segment

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

Q2 (iv) 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? iv) radius of circle 

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.

Q4 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.

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                                        (\because AC = BC )
Therefore,
AC = \frac{1}{2}AB
Hence proved.

Q5 In Question 4, point C is called a mid-point of line segment AB. Prove that every line  segment has one and only one mid-point.

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 

Q7 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.)

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 \angle 1+\angle 2 = 180\degree 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

Chapter No.

Chapter Name

Chapter 1

NCERT solutions for class 9 maths chapter 1 Number Systems

Chapter 2

CBSE NCERT solutions for class 9 maths chapter 2 Polynomials

Chapter 3

Solutions of NCERT class 9 maths chapter 3 Coordinate Geometry

Chapter 4

NCERT solutions for class 9 maths chapter 4 Linear Equations In Two Variables

Chapter 5

CBSE NCERT solutions for class 9 maths chapter 5 Introduction to Euclid's Geometry

Chapter 6

Solutions of NCERT class 9 maths chapter 6 Lines And Angles

Chapter 7

NCERT solutions for class 9 maths chapter 7 Triangles

Chapter 8

CBSE NCERT solutions for class 9 maths chapter 8 Quadrilaterals

Chapter 9

Solutions of NCERT class 9 maths chapter 9 Areas of Parallelograms and Triangles

Chapter 10

NCERT solutions for class 9 maths chapter 10 Circles

Chapter 11

CBSE NCERT solutions for class 9 maths chapter 11 Constructions

Chapter 12

Solutions of NCERT class 9 maths chapter 12 Heron’s Formula

Chapter 13

NCERT solutions for class 9 maths chapter 13 Surface Area and Volumes

Chapter 14

CBSE NCERT solutions for class 9 maths chapter 14 Statistics

Chapter 15

Solutions of NCERT class 9 maths chapter 15 Probability

NCERT solutions for class 9 subject wise

CBSE NCERT Solutions for Class 9 Maths

Solutions of NCERT Solutions for Class 9 Science

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.

 

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