Euclid's fifth postulate:
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.
This means that, when the sum of the measures of the interior angles on the same side of the transversal line is , then no intersection will take place.
Equivalent postulates of Euclid's fifth postulate:
Several equivalent versions of this postulates have been developed. One of them is ‘Playfair’s Axiom’ (given by a Scottish mathematician John Playfair in \(1729\)). He stated as:
For every line '\(q\)' and for every point '\(O\)' not lying on '\(q\)', there exists a unique line '\(p\)' passing through '\(O\)' and parallel to '\(q\)'.
Another version of Euclid's postulate as stated below:
Two distinct intersecting lines cannot be parallel to the same line.
National Council of Educational Research and Training (2006). Mathematics. Chapter-5 Introduction to Euclid's Geometry(pp. 78-88). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.