Kutztown UniversityKutztown, PennsylvaniaMAT 240: Synthetic GeometryI. Three semester hours; three clock hours; a required course for Secondary Education mathematicsmajors.II. Catalog Description:MAT 240: Synthetic Geometry 3 s.h.This course is designed for students who have, in addition to an interest in geometry, some previousexperience in this subject are, either on the high school or college level. Topics include Euclideangeometry using Hilbert’s axioms; neutral geometry; the historical development of non-Euclideangeometries; and hyperbolic geometry. Prerequisite: MAT 224.III. Course Objectives:The student will:A. Recognize some of the flaws in Euclidean geometry.B. Examine those foundations of Euclidean geometry using Hilbert’s axioms.C. Examine historical development of non-Euclidean geometry.D. Study the basic fundamentals of neutral geometry.E. Study the basic fundamentals of hyperbolic geometry.F. Increase his/her capability for carrying out deductive proofs.G. Understand the need for both intuitive and deductive thought processes in the doing ofmathematics.H. Understand and appreciate the nature of geometry.IV. Course Outline:A. The make-up of a Mathematical System1. Undefined and defined terms2. Mathematical statements3. Rule of Logic4. ModelsB. Euclid’s Geometry1. Origins of geometry2. Euclid’s Postulates3. The logical deficiencies of Euclid’s geometry4. The parallel postulate controversyMAT 240: Synthetic Geometry2C. Hilbert’s Axiom’s1. Axioms of incidence2. Axioms of betweenness3. Axioms of congruence4. Axioms of continuity5. Axioms of parallelismD. Neutral Geometry1. Axioms of neutral geometry2. Saccheri - Lengendre Theorem3. Angle sum of a triangle4. Equivalence of a parallel postulatesE. Non-Euclidean Geometries - Hyperbolic Geometry1. Historical events leading up to its discovery2. Angle sum of a triangle and quadrilateral3. Similar triangles4. Classification of parallel lines5. Area and Defect6. The angle of parallelismV. Instructional ResourcesGreenberg, Marvin J. Euclidean and Non-Euclidean Geometries. San Francisco, CA: W. H.Freeman & Company, 1980.Bennett, M. K. Affine and Projective Geometry. New York, NY: John Wiley & Sons, Inc. 1995.Berele, A. and Goldman, J. Geometry: Theorems and Constructions. Upper Saddle River, NJ:Prentice Hall. 2001.Coxeter, H. S. M. & Greitzer, S. L. Geometry Revisited. New York, NY: Random House, 1967.Eves, H. A Survey of Geometry. 2 vols. Boston, MA: Allyn and Bacon, 1965.Hansberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington,D.C.: Mathematical Association of America, 1995.Henderson, D. W. Differential Geometry: A Geometric Introduction. Upper Saddle River, NJ;Prentice Hall. 1998.Henderson, D. W. Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces. SecondEd. Upper Saddle River, NJ: Prentice Hall. 2001.Isaacs. I.M. Geometry for College Students. Brooks/Cole Pub. Co. 2001.Kay, David C. College Geometry: A Discover Approach, Second Ed. Boston, MA: Addison WesleyLongman, Inc. 2001.Meschkowski, H. Non-Euclidean Geometry. New York, NY: Academic Press, 1964.MAT 240: Synthetic Geometry3Moise, E. E. Elementary Geometry form an Advanced Standpoint. Reading, MA: Addison Wesley,1963.O’Daffer, P. G. and Clemens, S. R. Geometry: An Investigative Approach. Reading, MA: AddisonWesley, 1992.Pedoe, D. Circles: A Mathematical View. Washington, D.C.: Mathematical Association of America,1995.Prenowitz, W. and Jordan, M. Basic Concepts of Geometry. New York, NY: Ginn-BlaisdellPublishing Company, 1965.Reid, C. Hilbert. New York, NY: Springer Verlag, 1970.Sibley, T. Q. The Geometric Viewpoint: A Survey of Geometries, Reading, MA: Addison Wesley.1998.Smart, J. R. Modern Geometries, Fourth Ed. Pacific Grove, CA: Brooks/Cole Pub. Co. 1994.Tuller, A. Modern Introduction to Geometries. New York, NY: Van Nostrand Reinhold, 1967.Wallace, E. C. and West, S. E. Road to Geometry, Second Ed. Upper Saddle River, NJ: PrenticeHall, 1998.VI. Methods of Presentation and Evaluation:While conventional methods of instruction - lecture, discussion, problem-solving- are used in thiscourse, special emphasis is placed upon having the student do mathematics. Students are involved inthe development of mathematical proofs.Student performance is based upon class participation, assigned problems and proofs, and