NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry
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Page 1
NCERT
SOLUTIONS
CLASS - 9th
aglase .co
SOLUTIONS
CLASS - 9th
aglase .co
Page 2
Book : Mathematics Ncert Solutions | Chapter-5 Maths
Class : 9th
Subject : Maths
Chapter : 5
Chapter Name : Introduction To Euclid's Geometry
Exercise 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 in nite number of lines which pass through two distinct points.
(iii) A terminated line can be produced inde nitely 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. Since through a single point, in nite number of lines can pass. In the following gure, it can be seen that there are in nite
numbers of lines passing through a single point P.
(ii) False. Since through two distinct points, only one line can pass. In the following gure, 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 inde nitely on both the sides. Let AB be a terminated line. It can be seen that it can be produced
inde nitely on
both the sides.
(iv)True. If two circles are equal, then their centre and circumference will coincide and hence, the radii will also be equal.
(v) True. It is given that AB and XY are two terminated lines and both are equal to a third line PQ. Euclid's rst 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.
Page : 85 , Block Name : Exercise 5.1
Page 1 of 5 Aglasem Schools
Class : 9th
Subject : Maths
Chapter : 5
Chapter Name : Introduction To Euclid's Geometry
Exercise 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 in nite number of lines which pass through two distinct points.
(iii) A terminated line can be produced inde nitely 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. Since through a single point, in nite number of lines can pass. In the following gure, it can be seen that there are in nite
numbers of lines passing through a single point P.
(ii) False. Since through two distinct points, only one line can pass. In the following gure, 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 inde nitely on both the sides. Let AB be a terminated line. It can be seen that it can be produced
inde nitely on
both the sides.
(iv)True. If two circles are equal, then their centre and circumference will coincide and hence, the radii will also be equal.
(v) True. It is given that AB and XY are two terminated lines and both are equal to a third line PQ. Euclid's rst 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.
Page : 85 , Block Name : Exercise 5.1
Page 1 of 5 Aglasem Schools
Page 3
Book : Mathematics Ncert Solutions | Chapter-5 Maths
Q2. Give a de nition for each of the following terms. Are there other terms that need to be de ned rst? What are they, and how might you
de ne them?
(i) parallel lines
(ii) perpendicular lines
(iii) line segment
(iv) radius of a circle
(v) square
Answer. (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 de ne 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 each other at 90°, then these are called perpendicular lines. We are required to de ne line and the angle before de ning
perpendicular lines.
(iii) Line segment
A straight line drawn from any point to any other point is called as line segment. To de ne a line segment, we must know about point and line
segment.
(iv) Radius of a circle
It is the distance between the centres of a circle to any point lying on the circle. To de ne the radius of a circle, we must know about point and
circle.
(v) Square
A square is a quadrilateral having all sides Of equal length and all angles Of same measure, I.e., 90° • To de ne square, we must know about
quadrilateral, side, and angle.
Page 2 of 5 Aglasem Schools
Q2. Give a de nition for each of the following terms. Are there other terms that need to be de ned rst? What are they, and how might you
de ne them?
(i) parallel lines
(ii) perpendicular lines
(iii) line segment
(iv) radius of a circle
(v) square
Answer. (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 de ne 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 each other at 90°, then these are called perpendicular lines. We are required to de ne line and the angle before de ning
perpendicular lines.
(iii) Line segment
A straight line drawn from any point to any other point is called as line segment. To de ne a line segment, we must know about point and line
segment.
(iv) Radius of a circle
It is the distance between the centres of a circle to any point lying on the circle. To de ne the radius of a circle, we must know about point and
circle.
(v) Square
A square is a quadrilateral having all sides Of equal length and all angles Of same measure, I.e., 90° • To de ne square, we must know about
quadrilateral, side, and angle.
Page 2 of 5 Aglasem Schools
Page 4
Book : Mathematics Ncert Solutions | Chapter-5 Maths
Page : 85 , Block Name : Exercise 5.1
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 unde ned terms? Are these postulates
consistent? Do they follow from Euclid’s postulates? Explain.
Answer. There are various unde ned 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".
Page : 85 , Block Name : Exercise 5.1
Q4 If a point C lies between two points A and B such that AC = BC, then prove that AC = . Explain by drawing the gure.
1
AB
2
Answer. It is given that,
AC = BC
Here, (BC + AC) coincides with AB. It is known that things which coincide with one another are equal to one another.
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
2AC = AB
1
∴ AC = AB
2
Page : 86 , Block Name : Exercise 5.1
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.
AC = CB
AC+AC= BC+AC (Equals are added on both sides) …….(1)
Here, (BC + AC) coincides with AB. It is known that things which coincide with one another are equal to one another.
Therefore BC + AC = AB ……(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
2AC = AB….. (3)
Similarly, by taking D as the mid-point of Ad, it can be proved that
2AD = AB…. (4)
From equation (3) and (4), we obtain
2AC = 2AD (Things which are equal to the same thing are equal to One another.)
AC = AD (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 and there can be only one mid-point of a given line Segment.
Page 3 of 5 Aglasem Schools
Page : 85 , Block Name : Exercise 5.1
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 unde ned terms? Are these postulates
consistent? Do they follow from Euclid’s postulates? Explain.
Answer. There are various unde ned 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".
Page : 85 , Block Name : Exercise 5.1
Q4 If a point C lies between two points A and B such that AC = BC, then prove that AC = . Explain by drawing the gure.
1
AB
2
Answer. It is given that,
AC = BC
Here, (BC + AC) coincides with AB. It is known that things which coincide with one another are equal to one another.
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
2AC = AB
1
∴ AC = AB
2
Page : 86 , Block Name : Exercise 5.1
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.
AC = CB
AC+AC= BC+AC (Equals are added on both sides) …….(1)
Here, (BC + AC) coincides with AB. It is known that things which coincide with one another are equal to one another.
Therefore BC + AC = AB ……(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
2AC = AB….. (3)
Similarly, by taking D as the mid-point of Ad, it can be proved that
2AD = AB…. (4)
From equation (3) and (4), we obtain
2AC = 2AD (Things which are equal to the same thing are equal to One another.)
AC = AD (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 and there can be only one mid-point of a given line Segment.
Page 3 of 5 Aglasem Schools
Page 5
Book : Mathematics Ncert Solutions | Chapter-5 Maths
Page : 86 , Block Name : Exercise 5.1
Q6 In Fig, if AC = BD, then prove that AB = CD.
Answer. From the gure, it can be observed that
AC = AB + BC
BD = BC + CD
It is given that AC = BD
AB + BC = BC + CD(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
Page : 86 , Block Name : Exercise 5.1
Q7 Why is Axiom 5, in the list of Euclid's axioms, considered a 'universal truth'? (Note that the question is not about the fth postulate.)
Answer. 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 eld, and
not just in the eld of mathematics. Let us take two cases — one in the eld of mathematics, and one other than that.
Case I
Let t represent a whole quantity and only a, b, c are parts of it.
t=a+b+c
Clearly, t will be greater than all its parts a, b, and C. Therefore, it is rightly said that the whole is greater than the part.
Case II
Let us consider the continent Asia. Then, let us consider a country India which belongs to Asia. India is a part Of Asia and it can also be
observed that Asia is
greater than India. 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.
Page : 86 , Block Name : Exercise 5.1
Exercise 5.2
Q1 How would you rewrite Euclid’s fth postulate so that it would be easier to understand?
Answer 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 I and a point P not on l. Then, by Playfair's axiom (equivalent to the fth 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
I and CD be the distance Of any point on I from m. It can be observed that AB = CO. In this way, the distance will be the same for any point on
m from / and any point on I from m. Therefore, these two lines are everywhere equidistant from one another.
Page : 88 , Block Name : Exercise 5.2
Q2 Does Euclid’s fth postulate imply the existence of parallel lines? Explain.
Page 4 of 5 Aglasem Schools
Page : 86 , Block Name : Exercise 5.1
Q6 In Fig, if AC = BD, then prove that AB = CD.
Answer. From the gure, it can be observed that
AC = AB + BC
BD = BC + CD
It is given that AC = BD
AB + BC = BC + CD(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
Page : 86 , Block Name : Exercise 5.1
Q7 Why is Axiom 5, in the list of Euclid's axioms, considered a 'universal truth'? (Note that the question is not about the fth postulate.)
Answer. 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 eld, and
not just in the eld of mathematics. Let us take two cases — one in the eld of mathematics, and one other than that.
Case I
Let t represent a whole quantity and only a, b, c are parts of it.
t=a+b+c
Clearly, t will be greater than all its parts a, b, and C. Therefore, it is rightly said that the whole is greater than the part.
Case II
Let us consider the continent Asia. Then, let us consider a country India which belongs to Asia. India is a part Of Asia and it can also be
observed that Asia is
greater than India. 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.
Page : 86 , Block Name : Exercise 5.1
Exercise 5.2
Q1 How would you rewrite Euclid’s fth postulate so that it would be easier to understand?
Answer 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 I and a point P not on l. Then, by Playfair's axiom (equivalent to the fth 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
I and CD be the distance Of any point on I from m. It can be observed that AB = CO. In this way, the distance will be the same for any point on
m from / and any point on I from m. Therefore, these two lines are everywhere equidistant from one another.
Page : 88 , Block Name : Exercise 5.2
Q2 Does Euclid’s fth postulate imply the existence of parallel lines? Explain.
Page 4 of 5 Aglasem Schools
Page 6
Book : Mathematics Ncert Solutions | Chapter-5 Maths
Answer. Yes
According to Euclid's 5th postulate, when n line falls on I and m and if ∠1 + ∠2 < 180 , then ∠3 + ∠4 > 180 ,
∘ ∘
producing line I and m further will meet in the side of ∠1 and ∠ 2 which is less than 180° .
The lines I and m neither meet at the side of ∠1 and ∠ 2 nor at the side of ∠3 and ∠ 4. This means that the lines I and m will never intersect
each other. Therefore, it can be said that the lines are parallel.
Page : 88 , Block Name : Exercise 5.2
Page 5 of 5 Aglasem Schools
Answer. Yes
According to Euclid's 5th postulate, when n line falls on I and m and if ∠1 + ∠2 < 180 , then ∠3 + ∠4 > 180 ,
∘ ∘
producing line I and m further will meet in the side of ∠1 and ∠ 2 which is less than 180° .
The lines I and m neither meet at the side of ∠1 and ∠ 2 nor at the side of ∠3 and ∠ 4. This means that the lines I and m will never intersect
each other. Therefore, it can be said that the lines are parallel.
Page : 88 , Block Name : Exercise 5.2
Page 5 of 5 Aglasem Schools
Document Details
| Board / Org | NCERT |
| Exam | Class 9 |
| Type | Solution |
| Pages | 6 |
| Updated | 30 Apr 2026 |