Parallelism

In Euclidean Geometry, parallelism is a geometrical reasoning based on Euclid’s fifth postulate. It is

1. One of the invariant introduced by Flex Klein (the five invariants are: distance, angle, parallelism, straight-ness, continuity)
2. Origin of the birth of non-Euclidean Geometry
3. Demarcation of Euclidean and Non- Euclidean Geometry
4. Extends Euclidean to Projective Geometry

The idea of parallelism are mention in following works

1. Euclid: Fifth postulate
2. Playfair axiom
3. Hilbert: postulate 14
4. Brikhoff: postulate 4
5. SMSG: postulate 16

The details are given below.

Euclid’s Fifth postulate

If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Play-fair

Let l is a line and P be a point not on l, then there is at most one line on P parallel to l.

Hilbert-IV-14

Let l be any line and P be a point not on it. Then there is at most one line in the plane, determined by l and P, that passes through P and does not intersect l.

Brikhoff’s 4

Given two triangles ABC and A'B'C' and some constant k > 0, 

if

d(A', B' ) = kd(A, B), 

d(A', C' ) = kd(A, C and 

<B'A'C'  = <BAC, 

then

d(B', C' ) = kd(B, C), 

<C'B'A'  = <CBA, nd 

<A'C'B'  = <ACB

SMSG Group VI, 16

Through a given external point there is at most one-line parallel to a given line.