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Unformatted text preview: A FOUNDATIONS OF GEOMETRY STUDY GUIDE DAVID ANDREW SMITH Abstract. An introduction to the foundations of geometry is an undergraduate course at many universities. Topics included in such a course can vary widely but a common approach is to study the basics of logic first, and then with an understanding of what a mathematical statement is and how to use quantifiers, different methods of proofs can be detailed. Helping students learn how to write mathematical proofs by using logical reasoning according to established guidelines can be a difficult challenge. The following topics provide material for students to practice writing proofs. Rigorous geometry proofs are constructed in the areas of incidence, betweenness, congruence, and continuity. The propositions listed below, in order of dependency, are proven using Hilberts axioms contained in his celebrated Foundations of Geometry book, with slight modifications given by Greenberg. Foundations of Geometry presents students with a challenge to explain and to construct geometric proofs with the axiomatic development of a consistent math- ematical system and identify or describe consistent mathematical proofs. Students will be able to write and communicate proofs of geometric statements us- ing direct or indirect arguments; and will be able to discuss the historical development of new geometries. Students will also be able to explain differences be- tween Euclidean and non-Euclidean geometries. The idea is to familiarize students with Euclids and Hilberts approach to axiomatic geometry by in- troducing the student to basic propositional logic, un- defined terms, and postulates. The goal is to have students prove given theorems in a rigorous column format; thereby establishing the foundations of ge- ometry and also providing a proof-writing experience. Consistency of the given axioms is a point that must be emphasized and expounded upon, considering the choice of axioms could be exchanged or refined. The foundations of geometry course is an important part of any math students skill set as most people who work with mathematics learn to write technically and potentially with a great deal of rigor. During an undergraduate course on the foundations of geometry undergraduate students should be learn- ing the following. (1) Students will be able to explain the axiomatic development of consistent mathematical sys- tems and identify or describe consistent math- ematical systems. (2) Students will be able to write and communi- cate proofs of geometric statements using di- rect or indirect arguments. (3) Students will be able to illustrate geometric objects and demonstrate geometric relation- ships using technology....
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