Euclid’s Five Postulates

Postulate 1 : A straight line may be drawn from any one point to any other point.

Postulate 2 : A terminated line can be produced indefinitely.

Postulate 3 : A circle can be drawn with any centre and any radius.

Postulate 4 : All right angles are equal to one another.

Postulate 5 : 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.

Two equivalent versions of Euclid’s fifth postulate are:

(i) ‘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’.

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

All the attempts to prove Euclid’s fifth postulate using the first 4 postulates failed. But they led to the discovery of several other geometries, called non-Euclidean geometries.