Math 550
Fundamental Concepts of Geometry
Professor: Dr. Curtis Bennett
Office: UH 2724
Phone: 338-5112
Email: cbennett@lmu.edu
Office Hours: TR: 1:35- 3:20, W: 10:00-12:00 and by appointment
Textbook: Euclidean and Non-Euclidean Geometries Development and History,
Marvin Jay Greenberg, W. H. Freeman and Company, 3rd edition.
Prerequisites: MATH 248
Tentative Topics:
Introduction
 Euclid’s postulates – The first four postulates, the need for further
definitions, the danger of diagrams.
 Incidence Geometry - Finite Geometries, models, isomorphism of models.
Hilbert Axioms:
 Betweenness – Pasch’s Theorem
 Congruence – Side Angle Side Axiom, ASA criteria, Trichotomy law,
SSS congruence, Euclid’s Fourth Postulate.
 Continuity
Neutral Geometry:
 Basic axioms of neutral geometry – Euclid’s first four axioms.
 Neutral triangles – SSS congruence, alternating interior angle theorem,
and the triangle inequality, Saccheri-Legendre Theorem.
 Basic geometric constructions – angle bisector, perpendicular bisector,
replication of shapes.
Euclidean Plane Geometry:
 The parallel postulate – Equivalents to the parallel postulate including the
alternate interior angle theorem, the angle sum theorem, the mutual
perpendicular line theorem.
 Euclidean triangles and circles – concurrence theorems, exterior angle
theorem, AAA similarity theorem, Pythagorean theorem (and its
converse), geometric mean theorem.
 Trigonometric functions – sine, cosine, law of sines, and law of cosines.
 Euclidean geometric constructions – rational number construction, basic
geometric shapes.
Non-Euclidean Geometries
 Spherical geometry – change of axioms, area of triangles, AAA
congruence theorem.
 Hyperbolic geometry – change of axioms, area of triangles, meaning of
parallel.
Geometric Transformations:
 Rigid motions – isometries in 2-dimensional space (rotations, reflections,
translations).
 Similarities and inversions – dilations, similarity transformations.

Coordinate systems – definitions, proving theorems using coordinate
systems.
Perimeter, Area, and Volume:
 Perimeter and circumference – derive perimeter formulas for circles from
polygons compare to the method using integration.
 Area – compute areas of polygonal regions (signed and unsigned) and use
to find area of circles.
 Volumes and surface areas – find volumes of three-dimensional objects,
Cavalieri’s principle.
Instruction and Technology: In this course, we will be using classroom Socratic
lectures roughly 2 days out of 3 and in-class activity/group work the other day. We will
use the Lénárt sphere for investigations in spherical geometry and students will construct
a model of hyperbolic space for investigations into hyperbolic geometry. Students will
also need to familiarize themselves with Geometer’s Sketchpad, which we will use for
some activities.
Grading:
There will be approximately weekly homework assignments, in class assignments that
may involve some time spent completing them at home, a midterm and a cumulative
final. At the end of the term, students will hand in a portfolio for the class containing all
of their assignments, reflective writings (of no more than a page each) and a comparative
journal. Grades will be determined according to the following weightings:
Homework & in class work
Midterm (Thursday, Oct. 13th)
Portfolio
Participation
Final Exam (Tuesday, Dec. 13th: 2-4)
40%
15%
10%
5%
30%
90% of all points will guarantee students an A or A-, 80% will guarantee students a B+,
B, or B-, 70% a C+, or C. However, at the end of the term, these percentages may be
reduced slightly.
Objectives:
(1) Students will be able to solve routine and complex problems in geometry drawing
form a variety of strategies.
(2) Students will be able to clearly communicate arguments in geometry in everyday and
mathematical languages both orally and in writing. In particular, they will be able to
solve these problems in context and explore their relation with other problems.
(3) Students will be able to use technologies (including Geometer’s Sketchpad and the
Lénárt Sphere) appropriately to investigate and solve problems involving Euclidean
and non-Euclidean geometry and enhance their understanding of these geometries.
(4) Students will be able to articulate mathematical ideas orally and in writing using
appropriate terminology and technologies.
(5) Students will show an understanding of the foundations and proofs of Euclidean
geometry. In particular, students will gain an understanding of
(6) Students will solve problems in Euclidean and non-Euclidean geometry and present
formal and informal proofs of theorems in Euclidean and non-Euclidean geometry
both orally and in writing.
(7) Students will be able to discuss how geometry relates to other topics in mathematics
like linear algebra, complex analysis, and calculus.
(8) Students will see varied teaching strategies in geometry and see how they help
content be conceived and organized for instruction, fostering conceptual
understanding and procedural knowledge.
(9) Students will see multiply ways to solve problems in geometry.
(10) Students will formulate and test conjectures in Euclidean and non-Euclidean
geometry and judge the validity of mathematical arguments.