Euclid Geometry: Euclid's Fifth Postulate

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 5th 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.

Euclid geometry

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’.

Euclid Geometry

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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Practise This Question

Bella states that a line segment is a part of a line. True/False