## College Geometry - MTH 280

https://courses.vccs.edu/courses/MTH280-CollegeGeometry

Effective: 2018-01-01

### Course Description

Presents topics in Euclidean and non-Euclidean geometries chosen to prepare individuals for teaching geometry at the high school level or for other areas of study applying geometric principles. Studies Euclid's geometry and its limitations, axiomatic systems, techniques of proof, and Hilbert's geometry, including the parallel postulates for Euclidean, hyperbolic, and elliptic geometries.

Lecture 3 hours. Total 3 hours per week.

3 credits

The course outline below was developed as part of a statewide standardization process.

### General Course Purpose

Presents topics in Euclidean and non-Euclidean geometries chosen to prepare individuals for teaching geometry at the high school level or to other areas of study applying geometric principles.

### Course Prerequisites/Corequisites

Prerequisite: Completion of MTH 263 with a grade of C or better or equivalent.

### Course Objectives

- Historic Overview of Euclidean Geometry
- Describe the historic role of Euclid's Elements in the development of modern geometry
- State the axioms ("postulates" and "common notions") that are the foundation of Euclid's geometry
- Upon reviewing familiar results of "high school" Euclidean geometry, write basic proofs using the Elements as the source (focus on definitions and theorems of Book One)
- Introduction to Logic and Proof
- Identify the hypothesis and conclusion of a conditional statement
- Write the inverse, converse, and contrapositive of a conditional
- Identify argument structures/rules of inference (e.g. modus ponens)
- Recognize styles of direct and indirect proof, proof by contradiction, proof by cases
- Axiomatic Systems
- Define an axiomatic system and identify its parts
- Determine if a set of axioms is consistent, independent, complete
- Prove theorems using rules of inference within an abstract axiomatic system, given a list of the available definitions, axioms and theorems
- Construct models that satisfy the given axioms of a system and determine if the system is categorical
- State and apply the axioms that define finite projective and affine geometries (e.g. Fano Plane)
- Neutral Geometry
- Progress through the development of a neutral geometry based on Hilbert's (or similar) axioms, starting with incidence, metric, and betweenness axioms, incorporating the SAS Postulate of congruence, and to the proof of the Saccheri-Legendre Theorem
- Write proofs of results within neutral geometry, correctly identifying and citing axioms, definitions, and theorems established within the system as justifications
- Apply the above outcomes in the context of results about congruent triangles and quadrilaterals (including Saccheri quadrilaterals), and circles (including the tangent and secant theorems)
- Illustrate geometric concepts and results by construction through the use of appropriate software
- Parallel Postulates
- Articulate the history and development of various forms of a parallel postulate (Euclid, Riemann, Lobechevski)
- State the parallel postulates that define Euclidean, hyperbolic, and spherical geometries
- State equivalent forms of the Euclidean parallel postulate
- Euclidean Geometry
- Write proofs of results within Euclidean geometry, correctly identifying and citing axioms, definitions, and theorems established within the system as justifications
- Apply the above outcomes in the context of results about congruence and similarity for triangles and quadrilaterals
- Apply the Side-Splitting Theorem to compute segment lengths in triangles
- State, prove, and apply results about circles in Euclidean geometry (e.g. arc and angle measures, Secant-Tangent Theorem)
- Apply terminology related to polygons to classify and describe; compute angle measures and area of polygons
- Hyperbolic Geometry
- State and prove key theorems of hyperbolic geometry, recognizing where the results differ from Euclidean (e.g. angle sum of triangle less than 180, AAA similarity implies congruence)
- Construct segments and angles and measure distances and angle sums within the Poincare disk model of hyperbolic geometry (note this objective assumes the continued existence of the hyperbolic geometry construction application Non-Euclid or comparable)
- Compute the defect and area of a polygon in hyperbolic geometry
- Spherical Geometry
- Define spherical geometry in terms of its parallel postulate
- Contrast spherical geometry with Euclidean and hyperbolic geometries, and identify key theorems of neutral geometry which do not hold in spherical geometry
- Topics in Geometry/Modern Geometry
- Trigonometry (instructors may wish to review trigonometry, but knowledge of definitions and techniques is assumed as prerequisite knowledge):
- Demonstrate a deep understanding of the foundations of trigonometry through derivation of key results, including Law of Sines and Law of Cosines
- Solid figures:
- Define, identify and classify convex polyhedra (Platonic solids, Archimedean solids)
- Compute volume and surface area for selected examples of the above ((instructors may also wish to review volume and surface area formulas for sphere, cone, prism, etc.):
- Transformations:
- Describe transformations as functions that take points in the plane as inputs, and give other points as outputs
- Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

### Major Topics to be Included

- Historic Overview of Euclidean Geometry
- Introduction to Logic and Proof
- Axiomatic Systems
- Neutral Geometry
- Parallel Postulates
- Euclidean Geometry
- Hyperbolic Geometry
- Spherical Geometry
- Topics in Geometry/Modern Geometry