HG 18
A SIDE-TRIP (SORT OF!)
E: Euclidean Parallel Postulate (Euclid's version)
If two lines are cut by a transversal in such a way that the sum of two interior angles on one side of the transversal is less than 180°, then the lines will meet on that side of the transversal.
P: Euclidean Parallel Postulate (Playfair's version)
There exists only one line parallel to a given line through a given point not on the line.
Prove: P if and only if E. [Notations: P iff E or P <-> E]
Note: By proving P iff E, we can say the statements are logically equivalent or simply equivalent. This means that if the first statement is taken as an axiom, then the second statement can be deduced as a theorem; and conversely, if the second is taken as an axiom, then the first can be deduced as a theorem. It is thus logically immaterial which of the two statements we assume as an axiom, and which we deduce as a theorem.