NCERT Class 9 Mathematics Textbook for Blind Students made Screen Readable by Professor T K Bansal.

Euclid’s fifth postulate is very significant in the history of mathematics. Recall it again from Section 5.2. We see that by implication, 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°. There are several equivalent versions of this postulate. One of them is ‘Playfair’s Axiom’ (given by a Scottish mathematician John Playfair in 1729), as stated below:

“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.”

From Fig. 5.11, We can see that of all the lines passing through the point P, only line m is parallel to line l.

Fig. 5.11

This result can also be stated in the following form:

Two distinct intersecting lines cannot be parallel to the same line.

[Euclid did not require his fifth postulate to prove his first 28 theorems. Many mathematicians, including him, were convinced that the fifth postulate is actually a theorem that can be proved using just the first four postulates and other axioms. However, all attempts to prove the fifth postulate as a theorem have failed. But these efforts have led to a great achievement - the creation of several other geometries. These geometries are quite different from Euclidean geometry. They are called non-Euclidean geometries. Their creation is considered a landmark in the history of thought because till then everyone had believed that Euclid’s was the only geometry and the world itself was Euclidean. Now the geometry of the universe we live in has been shown to be a non-Euclidean geometry. In fact, it is called spherical geometry. In spherical geometry, lines are not straight. They are parts of great circles (i.e., circles obtained by the intersection of a sphere and planes passing through the centre of the sphere).

In Fig. 5.12, the lines A N and B N (which are parts of great circles of a sphere) are perpendicular to the same line AB. But they are meeting each other, though the sum of the angles on the same side of line AB is not less than two right angles (in fact, it is 90° + 90° = 180°). Also, note that the sum of the angles of the triangle NAB is greater than 180°, as angle A + angle B = 180°. Thus, Euclidean geometry is valid only for the figures in a plane. On the curved surfaces, it fails.

Fig. 5.12

Now, let us consider an example.

Example 3

Consider the following statement :
There exists a pair of straight lines that are everywhere equidistant from one another.
Is this statement a direct consequence of Euclid’s fifth postulate? Explain.

Solution :

Take any line l and a point P not on l. Then, by Playfair’s axiom, which is equivalent to the fifth postulate, we know that there is a unique line m through P which is parallel to l.

Now, the distance of a point from a line is the length of the perpendicular from the point to the line. This distance will be the same for any point on m from l and any point on l from m. So, these two lines are everywhere equidistant from one another.

Remark :
The geometry that We will be studying in the next few chapters is Euclidean Geometry. However, the axioms and theorems used by us may be different from those of Euclid’s.

EXERCISE 5.2

Q1. How would We rewrite Euclid’s fifth postulate so that it would be easier to understand?

A1. Any formulation the student gives should be discussed in the class for its validity.

Q2. Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.

A2. If a straight line l falls on two straight lines m and n such that sum of the interior angles on one side of l is two right angles, then by Euclid’s fifth postulate the line will not meet on this side of l. Next, we know that the sum of the interior angles on the other side of line l will also be two right angles. Therefore, they will not meet on the other side also. So, the lines m and n never meet and are, therefore, parallel.