# NCERT Solutions for Class 9 Maths Unit 5

## Introduction to Euclid's Geometry Class 9

### Unit 5 Introduction to Euclid's Geometry Exercise 5.1, 5.2 Solutions

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.

For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only where the gravitational field is weak.

### Exercise 5.1 : Solutions of Questions on Page Number : 85

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 the following figure, if AB = PQ and PQ = XY, then AB = XY.

(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.

Q2 :

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 (v) square

(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, 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.

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., To define square, we must know about quadrilateral, side, and angle.

Q3 :

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 consistent?

Do they follow from Euclid's postulates? Explain.

Q4 :

If a point C lies between two points A and B such that AC = BC, then prove that. Explain by drawing the figure.

Q5 :

In the above question, point C is called a mid-point of line segment AB, prove that every line segment has one and only one mid-point.

Q6 :

In the following figure, if AC = BD, then prove that AB = CD.

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.)

### Exercise 5.2 : Solutions of Questions on Page Number : 88

Q1 :

How would you rewrite Euclid's fifth postulate so that it would be easier to understand?

Q2 :

Does Euclid's fifth postulate imply the existence of parallel lines? Explain.