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

Answer :

Yes, according to Euclid’s fifth postulate when line x falls on straight line y and z such that sum of the interior angles on one side of line x is two right angles.

\(\therefore \) \(\angle{1}\) + \(\angle{2}\) = \(180^\circ\).

Then, line y and line z on producing further will meet in the side of \(\angle{1}\) and \(\angle{2}\) which is less than \(180^\circ\).


We find that the lines which are not according to Euclid’s fifth postulate.
\(\therefore \) \(\angle{1}\) + \(\angle{2}\) = \(180^\circ\), do not intersect.

So, the lines y and z never meet and are, therefore, parallel.

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