Elements I.29 (thing)

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.