#### Individual Presentation or Panel Title

Triangles in Hyperbolic Geometry

#### Abstract

Euclid introduced five postulates as the fundamentals for the study of geometry. Over time his fifth postulate has been singled out as the most controversial. In fact, mathematicians have attempted unsuccessfully to prove this postulate, aka the parallel postulate, based on the first four for more than 2000 years. In the 1820s, Bolyai, Gauss, and Lobachevsky each decided to assume the negation of Euclid’s fifth postulate and this assumption led to a new branch of geometry, non-Euclidean geometry. In this paper, I will discuss the discovery of non-Euclidean geometry to provide a background for understanding several ideas in hyperbolic geometry. We will construct basic tools for the Poincare disc model of hyperbolic geometry. The goals of this paper are to explain the angle sum theorem of hyperbolic triangle and prove several triangle congruence theorems in neutral geometry. We will highlight several hyperbolic results that are quite different from results that we learned in high school Euclidean geometry.

#### Location

Glass Dining Room

#### Start Date

3-5-2014 3:30 PM

#### End Date

3-5-2014 4:20 PM

Triangles in Hyperbolic Geometry

Glass Dining Room

Euclid introduced five postulates as the fundamentals for the study of geometry. Over time his fifth postulate has been singled out as the most controversial. In fact, mathematicians have attempted unsuccessfully to prove this postulate, aka the parallel postulate, based on the first four for more than 2000 years. In the 1820s, Bolyai, Gauss, and Lobachevsky each decided to assume the negation of Euclid’s fifth postulate and this assumption led to a new branch of geometry, non-Euclidean geometry. In this paper, I will discuss the discovery of non-Euclidean geometry to provide a background for understanding several ideas in hyperbolic geometry. We will construct basic tools for the Poincare disc model of hyperbolic geometry. The goals of this paper are to explain the angle sum theorem of hyperbolic triangle and prove several triangle congruence theorems in neutral geometry. We will highlight several hyperbolic results that are quite different from results that we learned in high school Euclidean geometry.