Euclid Geometry

Euclid, a teacher of mathematics in Alexandria in Egypt gave us a remarkable idea regarding the basics of geometry, through his book called ‘Elements’. His axioms and postulates are studied till now for a better understanding of the subject. In this article we will be concentrating on the equivalent version of his 5^{th} postulate given by John Playfair, a Scottish mathematician in 1729.

Before understanding the equivalent version of Euclid’s fifth postulate, let us focus on fifth postulate first.

##### Fifth postulate of Euclid geometry:

If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two **straight lines**, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

In the given diagram, the sum of angle 1 and angle 2 is less than 180°, so lines n and m will meet on the side of angle 1 and angle 2.

Now let us focus on the equivalent version of Euclid’s fifth postulate given by John Playfair. As per him:

‘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’.

In the above figure, consider line l and a point P not lying on l. Now, we know that infinite number of lines can pass through a given point. Thus, infinite lines can pass through point P. But, are all the lines parallel to line l or are some of them parallel to line l or are none of them parallel to line l? According to Playfair there is only line passing through P which will be parallel to line m.

Draw any line passing through point P, say line s. If you draw a transversal cutting lines s and l and passing through point P, you can see that on one of the sides, the sum of co-interior angles will be less than 180° thus both the lines will meet in that direction. There is only one possibility, i.e. line m, in which the sum of the co-interior angles if exactly 180° on both the sides. Hence in this case both the lines will never meet.

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