The graduate course
of Advanced Topics in Geometry differs from the undergraduate course called College Geometry not so much in the
content but the philosophy of each course. The undergraduate course is all about writing formal proofs. It is a
rigorous course focusing on absolute geometry where each step of the argument is justified by previously proven
axioms and theorems. The graduate course also deals with formal proofs; however the focus of this course is to
make justification of theorems as visual as possible. The formal proof is the end product of a process of exercises
where the student discovers fundamental properties of geometry. In the undergraduate course, a model of the theorem
is not required; in some cases a model is undesirable since students tend to rely on the model as justification
of the theorem. In the graduate course, we deal with how to select a model to use to illustrate the theorem to
be proven. The graduate course deals with properties of good models and how we can encourage students to discover
the many principles and phenomena of geometry for themselves. In the graduate course, students, who usually teach
geometry in high school, are taught the transition of proof-writing, how to start with proofs that are largely
intuitive and go through the process of developing a more formal proof.
In general, the graduate course assumes the student has mastered the proof-writing procedures covered in the undergraduate
course in geometry. The purpose of the graduate course is how to go from observation or discovery of a principle
to a formal justification.
Course content
Procedures for discovery in geometry
Foundation of geometry (points, lines, segments, angles)
Introduction to writing proofs and modeling
Area axioms and theorems
Incidence axioms and theorems
Metric axioms and theorems
Plane separation axiom
Angle, ruler, and protractor axioms
The Crossbar theorem
Axioms and theorems associated with triangles, quadrilaterals, and circles
Euclidean parallel axiom
Selected advanced topics as time permits
Other Information
Any student with a documented disability needing academic adjustments
is requested to speak directly to Disability Support Services and the instructor, as early in the semester (preferably
within the first class week) asx possible. All discussions will remain confidential.
This publication is available in alternative formats upon request. Please
contact Mary Helen Walker, Disability Support Services, Career Services Center, 521-6270
