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

(i) False
Reason : If we mark a point O on the surface of a paper. Using pencil and scale, we can draw infinite number of straight lines passing
through O.

(ii) False
Reason : In the following figure, there are many straight lines passing through P. There are many lines, passing through Q. But there is one and only one line which is passing through P as well as Q.

(iii) True
Reason: The postulate 2 says that “A terminated line can be produced indefinitely.”

(iv) True
Reason : Superimposing the region of one circle on the other, we find them coinciding. So, their centres and boundaries coincide.
Thus, their radii will coincide or equal.

(v) True
Reason : According to Euclid’s axiom, things which are equal to the same thing are equal to one another.

Yes, we need to have an idea about the terms like point, line, ray, angle, plane, circle and quadrilateral, etc. before defining the required terms.
Definitions of the required t erms are given below :

(i) Parallel Lines :
Two lines l and m in a plane are said to be parallel, if they have no common point and we write them as l ॥ m.

(ii) Perpendicular Lines :
Two lines p and q lying in the same plane are said to be perpendicular if they form a right angle and we write them as p ⊥ q.

(iii) Line Segment :
A line segment is a part of line and having a definite length. It has two end-points. In the figure, a line segment is shown having end points A and B. It is written as 

or

(iv) Radius of a circle :
The distance from the centre to a point on the circle is called the radius of the circle. In the figure, P is centre and Q is a point on the circle, then PQ is the radius.

(v) Square :
A quadrilateral in which all the four angles are right angles and all the four sides are equal is called a square. Given figure, PQRS is a square.

Yes, these postulates contain undefined terms such as ‘Point and Line’. Also, these postulates are consistent because they deal with two different situations as
(i) says that given two points A and B, there is a point C lying on the line in between them. Whereas
(ii) says that, given points A and B, you can take point C not lying on the line through A and B.
No, these postulates do not follow from Euclid’s postulates, however they follow from the axiom, “Given two distinct points, there is a unique line that passes through them.”

We have,

AC = BC [Given]
∴ AC + AC = BC + AC
[If equals added to equals then wholes are equal]
or 2AC = AB [∵ AC + BC = AB]
or AC = 1/2AB

Let the given line AB is having two mid points ‘C’ and ‘D’.

AC = 1/2AB……(i)
and AD = 1/2AB……(ii)
Subtracting (i) from (ii), we have
AD – AC = 1/2AB-1/2AB
or AD – AC = 0 or CD = 0
∴ C and D coincide.
Thus, every line segment has one and only one mid-point.

Given: AC = BD
⇒ AB + BC = BC + CD
Subtracting BC from both sides, we get
AB + BC – BC = BC + CD – BC
[When equals are subtracted from equals, remainders are equal]
⇒ AB = CD

 As statement is true in all the situations. Hence, it is considered a ‘universal truth.’

We need to rewrite Euclid’s fifth postulate so that it is easier to understand.

We know that Euclid’s fifth postulate states that “No intersection of lines will take place when the sum of the measures of the interior angles on the same side of the falling line is exactly 180°.”

We know that Play fair’s axiom states that “For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l”.

The above mentioned Play fair’s axiom is easier to understand in comparison to the Euclid’s fifth postulate.

Let us consider a line l that passes through a point p and another line m. Let these lines be at a same plane.

Let us consider the perpendicular CD on l and FE on m.

From the above figure, we can conclude that CD = EF.

Therefore, we can conclude that the perpendicular distance between lines m and l will be constant throughout, and the lines m and l will never meet each other or in other words, we can say that the lines m and l are equidistant from each other.

We need to verify whether Euclid’s fifth postulate imply the existence of parallel lines or not.

The answer to the above statement is Yes.

Let us consider two lines m and l.

In the figure given below, we can conclude that the lines m and l will intersect further.

From the figure, we can conclude that

and

We know that Euclid’s fifth postulate states that “No intersection of lines will take place when the sum of the measures of the interior angles on the same side of the falling line is exactly 180°.”

Let us consider lines l and m.

From the above figure, we can conclude that lines l and m will never intersect from either side.

Therefore, we can conclude that the lines l and m are parallel.