Elements I.28 (thing)

Euclid's Elements: Book I
< Proposition 27 | Proposition 28 | Proposition 29 >

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

Claim:
If a = e, then AB parallel to CD.

Proof:
a = e is given.
a = c by proposition 15.
Hence e = c.
AB parallel to CD by proposition 27.
This completes the proof.

This is the last of the 28 propositions that do not depend on the parallel postulate in Euclid's Elements. Hence it is one of the postulates that holds for non-euclidian geometries also.