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

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.

Yes, if ‘l’ and ‘m’ are two straight lines that are intersected by another line ‘c’, and the sum of co-interior angles is equal to 180°, then l || m.

According to Euclid’s 5^th postulate,

∠1 + ∠2 < 180 then ∠3 + ∠4 > 180

[The interior angles on the same side of two straight lines which are intersected by another line taken together are less than two right angles]

Producing the line ‘l’ and m’ further will meet in the side of which is less than 180°.

If ∠1 + ∠2 = 180 then ∠3 + ∠4 = 180

The lines ‘l’ and ‘m’ do not meet on the side where the angle is lesser than 180°

Thus, they will never intersect with each other. Hence the two lines are said to be parallel to each other i.e. l || m