CHAPTER III - Euclidean Geometry

Abstract

This chapter focuses on Euclidean geometry. Two triangles are congruent if there is a rigid motion of the plane which carries one triangle exactly onto the other. Corresponding angles of congruent triangles are equal, corresponding sides have the same length, the areas enclosed are equal, and so on. Any geometric property of a given triangle is automatically shared by every congruent triangle. Conversely, there are a number of simple ways in which one can decide if two given triangles are congruent—for example, if for each triangle the same three numbers occur as lengths of sides. An isometry, or rigid motion, of Euclidean space is a special type of mapping that preserves the Euclidean distance between points. Various theorems are proven in the chapter. Intuitively, it is the orientation that distinguishes between a right-handed glove and a left-handed glove in ordinary space. To handle this concept mathematically, gloves have been replaced by frames, and all the frames on Euclidean 3-space E3 are separated into two classes.