Angle Properties of Parallel Lines

      Recently I have been focusing on skills taught in previous years that were not proved, rather assumed to be true based on the reputation of other mathematical principles for being true.  This led me to produce a few proofs for fundamental principles from which later proofs are based.  My favourite of these is that of equal alternate and corresponding angles and supplementary co-interior angles at the points of intersection of a transversal and a pair of parallel lines.  This is especially so because the initial proof, on which the others are derived, seems much simpler than any I can find on the internet (silly Euclid).  However it is based on a simple assumption that I'm not sure is completely valid.  Here goes it:

Fig. 6 Parallel Lines Cut by a Transversal

         LR || L'R'

         B and C lie on LR and L'R' respectively

         AC ⊥ LR and BD ⊥ LR

         AC = BD  (with parallel lines the distance is maintained)

         BC is common to △ABC and △BCD

         ∴ △ABC ≡ △BCD  (RHS)

         ∴ ∠ABC = ∠BCD  (corresponding ∠s in congruent △s)

         ∴ where a transversal intersects a pair of parallel lines, the alternate ∠s are equal

          From this we can then prove that corresponding angles are equal (using vertically opposite ∠s), and supplementary co-interior ∠s (using supplementary ∠s on a straight line).