Elements I.27 (thing)

Euclid's Elements: Book I
< Proposition 26 | Proposition 27 | Proposition 28 >

Given:                          

                            /
                       d   /  a
  --- A ------------------E------------- B --- .
                     c   /  b                   \
                        /                        G
                       /                        /
                  h   /   e                    /
  --- C -------------F------------------ D ---'
                g   /   f
                   /

Claim:
  If c = e, then AB and CD are parallel.

Proof by contradiction:
  Assume that AB and CD meet.
  Wlog, let lines AB and CD meet at G on the B, D end.
  c = e contradicts proposition 16 of triangles.
  Therefore AB and CD never touch.
  AB and CD are parallel by definition 23.
  This completes the proof.

This theorem does not rely on the parallel postulate, hence it holds for non-euclidian geometries also. Proposition 28 will be the last theorem that does not depend on the parallel postulate.