By Gerard A. Venema

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**Example text**

Suppose again that we want to prove the theorem P implies Q. We assume the hypothesis P. Then we also suppose not Q (RAA hypothesis). After that we proceed to prove that P implies Q. At that point in the proof we have both Q and not Q. That is a contradiction, so we reject the RAA hypothesis and conclude Q. In this case the structure of an indirect argument has been erected around a direct proof, thus obscuring the real proof. Again this is sloppy thinking. It is an abuse of indirect proof and should always be avoided.

This is exactly how it was eventually shown that Euclid’s ﬁfth postulate is independent of his other postulates. Understanding that proof is one of the major goals of this course. Later in the book we will construct two models for geometry, both of which satisfy 22 Chapter 2 Axiomatic Systems and Incidence Geometry all of Euclid’s assumptions other than his ﬁfth postulate. One of the models satisﬁes Euclid’s ﬁfth postulate while the other does not. ) This shows that it is impossible to prove Euclid’s ﬁfth postulate using only Euclid’s other postulates and assumptions.

B) Some triangles have an angle sum of less than 180◦ . (c) Not every triangle has angle sum 180◦ .