NCERT Solutions for Class 9 Maths Chapter 5 Introduction To Euclids Geometry

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



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

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 each other at 90°, then these are called perpendicular lines. We are required to define line and the angle before defining perpendicular lines. 



(iii) Line segment 
A straight line drawn from any point to any other point is called as line segment. To define 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 define 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 define square, we must know about quadrilateral, side, and 
angle. 
 

Ans : There are various undefined 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". 

Ans : 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 
\(\therefore \mathrm{AC}=\frac{1}{2} \mathrm{AB}\) 

Ans : 
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 midpoint 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.