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.