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.