The answers to Chapter 5 are available in the NCERT Class 9 Math Solution. Euclid’s geometry chapter has some introduction to it as a part of the history of Indian geometry. Introduction to Euclid’s Geometry provides you with a way of defining the common geometrical shapes and terms. With a total of two exercises, you will be delving deeper into the relationship between axiom, postulates, and theorems.
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NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry: EX 5.1
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
(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 the following figure, if AB = PQ and PQ = XY, then AB = XY.
Refer to the figure below. As seen in the figure, an infinite number of lines can pass through a single point ‘P’. Thus, the given statement in (i) is false
Refer to the figure here. As seen in the figure, there is only one single line passing through two distinct points P and Q. Thus, the given statement in (ii) is false.
Refer to the figure. Consider AB as a terminated line. As seen, it can be produced indefinitely on both the sides. Thus, the given statement in (iii) is true.
For two equal circles, their center and circumference will coincide and hence, their radii will also be equal. Thus, the given statement in (iv) is true.
Here, consider AB and XY as the two terminated lines (Line Segments) and these lines are equal to a third line PQ.
Now, from the Euclid’s first axiom, things which are equal to the same thing are equal to one another.
AB PQ = and PQ XY = , then,
AB XY = . Thus, the given statement in (v) is true.
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
(ii) Perpendicular lines
(iii) Line segment
(iv) Radius of a circle
Parallel Lines: If the perpendicular distance between the two lines (let us say lines l and m) is always constant, then the lines are called parallel lines.
In other words, parallel lines are the lines which never intersect each other.
Perpendicular lines: If two lines (say l and m as shown in figure) intersect each other at 90o , then these lines are called perpendicular lines.
Line segment: A straight line drawn from any point (say P) to any other point (say Q) is called as line segment.
Radius of a circle: The distance between the centers of a circle to any point lying on the circle is termed as its radius.
Square: A square is a type of quadrilateral that have its all sides as equal and all angles of same measure, i.e., 90o .
To define square, we must know about quadrilateral, angle and side.
Consider the two ‘postulates’ given below:
(i) Given any two distinct points A and B, there exists a third point C,
which is between A and B.
(ii) There exists at least three points that are not on the same line.
Do these postulates contain any undefined terms? Are these postulates
Do they follow from Euclid’s postulates? Explain.
There are undefined terms. They are consistent. Here, it is given only that point C lies between points A and B but it is not mentioned in the problem whether the third point C is lying on the line segment AB or not. Thus, here two different cases are possible –
The third point C lies on the line segment made by joining the points A and B.
The third point C does not lie on the line segment made by joining the points A and B.
Yes, they are consistent as these are two different situations;
(i) The third point C lies on the line segment made by joining points A and B.
(ii) The third point C does not lie on the line segment made by joining points A and B.
No, the above postulates do not follow Euclid’s postulates, actually these are axiom.
If a point C lies between two points A and B such that AC = BC, then prove
that AC = ?/? (AB).Explain by drawing the figure.
From the Figure,
Point C lies between two points A and B.
AC = BC
NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry: EX 5.2
How would you rewrite Euclid’s fifth postulate so that it would be easier to
Two lines (say l and m) are said to be parallel if they do not have any point of intersection and are equidistant from one other.
As shown in figure, consider a point P on the line l and not on m.
Then, by Playfair’s axiom (equivalent to the fifth 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.
Consider AB as the distance of any point on lines m from l and CD be the distance of any point on the lines l from m as shown in figure.
It can be observed that AB = CD. In this way, the distance will be the same for any point on the lines m from l and any point on the lines l from m.
Therefore, these two lines are everywhere equidistant from one another.
Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.
According to Euclid’s 5th postulate, when n line falls on the lines l and m, and if
+ = 1 2 180
+ = 3 4 180
producing line l and m further will meet in the side of
∠1 and ∠2 which is less than 180°.
+ = 1 2 180 , then
The lines l and m neither meet at the side of ∠1 and ∠2 nor at the side of ∠3 and∠4. This means that the lines l and m will never intersect each other. Therefore, it can be said that the lines are parallel.