Euclid's Elements: Book I

<
Proposition 28
| Proposition 29 |
Proposition 30
>

Given:
E
/
d / a
A ------------------G------------- B
c / b
/
/
h / e
C -------------H------------------ D
g / f
/
F
Claim:
If AB parallel to CD, then c = e, a = e, and b + e = 180
Proof by Contradiction:
Suppose c not equal to e.
Wlog, let e < c.
b + e < b + c
b + e < 180 (proposition 13 on supplementary angles)
Hence AB, CD meet (parallel postulate).
Contradiction with hypothesis that AB, CD are parallel.
Therefore c = e.
c = a (proposition 15)
e = a
b + e = b + a
b + e = 180 (proposition 13)
This completes the proof.

This proposition is the converse of the previous two propositions.
All propositions hereupon in

Book I rely on the

parallel postulate.