Elements I.27

Euclid's Elements: Book I: Proposition 27

Proposition 27: If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.

For let the straight line EF falling on the two straight lines AB, CD make the alternate angles AEF, EFD equal to one another;

I say that AB is parallel to CD.

For, if not, AB, CD when produced will meet either in the direction of B, D or towards A, C.

Let them be produced and meet, in the direction of B, D, at G.

Then, in the triangle GEF, the exterior angle AEF is equal to the interior and opposite angle EFG: which is impossible. I. 16

Therefore AB, CD when produced will not meet in the direction of B, D.

Similarly it can be proved that neither will they meet towards A, C.

But straight lines which do not meet in either direction are parallel; Def. 23

Therefore AB is parallel to CD.

Therefore etc.

Q.E.D.

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.