#### 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