The Angle Sum Problem

        In my most recent post, "Angle Properties Of Parallel Lines: A Second Method," I proved properties of parallel lines using the formula for the angle between two lines.  The problem with this is that the way I prove that formula uses the angle sum of a triangle, however the way that is proved USES the properties of parallel lines.  In order to counter this I have tried several things to either: prove the angle formula a different way; or find the angle sum of a triangle by some other method.  The only thing I tried that did work was this:

Fig. 8 Scalene Triangle

          In △ ABD,

          ∠ sum of △ = a + b + 90

          In △ BCD,

          ∠ sum of △ = c + d + 90

          In △ ABC,

          ∠ sum of △ = a + b + c + d

                               = (a + b + 90) + (c + d + 90) - 180

                               = 2(∠ sum of △) - 180

                        180 = 2(∠ sum of △) - ∠ sum of △

          ∠ sum of △ = 180

          The problem with this proof is that it assumes the ∠ sum of △ is constant ...   and I don't know how to prove that - anyone else got an idea?