This theorem does not depend on the parallel postulate. It is true in both Euclidean and Non-Euclidean Geometry. In fact, it was the last result proved by Euclid in his Elements before stating the parallel postulate, which is essentially the converse of this theorem. We use contradiction. Suppose l and m do meet at some point C. C could either be on the right side of t or the left.
Lets say they meet on the right as shown in the figure below.
A similar argument works if they meet on the left. TAC has measure a by the vertical angle theorem. So l and m cannot meet as assumed. Therefore they are parallel. QED D. Showing Statements are Equivalent Let P and Q be statements.