#### Description

Hilbert's axioms for Euclidean Geometry. Neutral Geometry: The consistency of the hyperbolic parallel postulate and the inconsistency of the elliptic parallel postulate with neutral geometry. Models of hyperbolic geometry. Existence and properties of isometries.

#### Prerequisites

A C- or better in MATH 240 or MATH 461 or MATH341

#### Topics

Logical deficiencies in Euclidean Geometry

Flawed proofs
Correction of some flawed proofs by clearly stating certain postulates and giving rigorous deductions
Overview of the structure of Euclidean Geometry
Importance of Euclid's Parallel Postulate as opposed to the other postulates
Equivalence of certain postulates (Play fair, etc) to Euclid's Parallel Postulate
Absolute or Neutral geometry. Work of Legendre and Saccheri
Negation of Euclid's Parallel Postulate; non-Euclidean geometry.

Discussion of models for non-Euclidean geometry

Axiom systems
Incidence axiom and ruler postulate
Betweenness
Segments, Rays and Convex sets
Angles and Triangles
Pasch's Postulate and Plane Separation Postulate
Perpendiculars and inequalities
SAS postulate
Parallel postulates
Models for non-Euclidean geometry
Proof of the consistency of non-Euclidean geometry by means of models