Introduction to Chapter 3: Euclidean Geometry
We recall some familiar features of plane geometry. First of all, two
triangles are congruent if there is a rigid motion of the plane that
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. Indeed, any
geometric property of a given triangle is automatically shared by
every congruent triangle.
Conversely, there are a number of simple ways to decide whether
two given triangles are congruent--for example, if for each the same
three numbers occur as lengths of sides.
In this chapter we investigate the rigid motions (isometries) of
Euclidean space, and see how these remarks about triangles can
be extended to other geometric objects.
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