Medium
Does Euclid's fifth postulate imply the existence of parallel lines? Explain.
(N/A) Yes,Euclid's fifth postulate implies the existence of parallel lines.
If a straight line $l$ falls on two lines $m$ and $n$ such that the sum of the interior angles on one side of $l$ is equal to two right angles $(180^{\circ})$,then by Euclid's fifth postulate,the lines $m$ and $n$ will not meet on this side of $l$.
Since the sum of the interior angles on the other side of the line $l$ will also be equal to two right angles $(180^{\circ})$,the lines will not meet on the other side either.
$\therefore$ The lines $m$ and $n$ never meet,which means they are parallel to each other.
If a straight line $l$ falls on two lines $m$ and $n$ such that the sum of the interior angles on one side of $l$ is equal to two right angles $(180^{\circ})$,then by Euclid's fifth postulate,the lines $m$ and $n$ will not meet on this side of $l$.
Since the sum of the interior angles on the other side of the line $l$ will also be equal to two right angles $(180^{\circ})$,the lines will not meet on the other side either.
$\therefore$ The lines $m$ and $n$ never meet,which means they are parallel to each other.
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