Yes, Euclid’s fifth postulate imply the existence of parallel lines.
If the sum of the interior angles will be equal to sum of the two right angles then two lines will not meet each other on either sides and therefore they will be parallel to each other.

m and n will be parallel if
∠1 + ∠3 = 180° 
Or  ∠3 + ∠4 = 180°