Parallel Postulate... Why?

      Using the method I developed in "Angle Properties of Parallel Lines," we can prove that parallel lines have co-interior ∠s that are supplementary without the need for Euclid's Fifth Postulate in Book 1 of 'Elements,' or the "Parallel Postulate" as it is termed.  Given this we can then prove the ∠ sum △, using the fact that ∠s on a straight line are supplementary, and that alternate ∠s in parallel lines are equal, as in the diagram:

Fig. 11 Triangle formed within Parallel Lines

If this is acceptable, the "Parallel Postulate" can easily be proved using my 'reductio ad absurdum' proof as follows.

Fig. 12 Intersecting lines with transversal

          LR intersects L'R' at I

          Assume the co-interior ∠s between the transversal TT' and the lines LR and L'R' are supplementary

          ∠TIT' + ∠ITT' + ∠IT'T = 180 (∠ sum △TIT')

                                      ∠TIT' = 180 - 180 (∠ITT' and ∠IT'T are supp.)

                                                 = 0

          ∴ LR || L'R'

          But LR and L'R' intersect

          ∴ sum co-interior ∠s < 180 (to give ∠ sum of △ = 180) on side of intersection

          As they are supplementary in parallel lines, any two lines who's co-interior ∠s are not supplementary form a △ on the side where their sum < 180, and ∴ intersect on that side.