Math 342
Modern Geometry II
Parallel Postulates

On page 93 of the book Gödel, Escher, Bach, Douglas Hofstadter claims, without explanation, that Euclid’s Fifth Postulate (EFP) and the Parallel Postulate (PP) are “equivalent.”  However, he doesn't say what he means by the word "equivalent" in this context.  Show that this definition matters by giving a careful definition of the word "equivalent" under which these statements are not equivalent, and explain why; then give a (different) careful definition of this word under which these statements are equivalent, and prove that they are in fact equivalent under your definition.

In answering this question, you may find it helpful to read Henderson's problem 10.3; the assumption that Hofstadter calls the Parallel Postulate is equivalent (in both senses!) to what Henderson calls the High School Parallel Postulate (HSP), because it is commonly assumed in high school geometry textbooks.