Introduction

Summary

Geometry is the study of spatial relationships, such as the familiar assertion from elementary plane Euclidean geometry that, if two triangles have sides of the same lengths, then they are “congruent.” What does congruent mean here? One possibility, which is rather abstract and very much in the spirit of the axiomatic approach usually attributed to Euclid, is to say:

Call two straight line segments “congruent” if they have the same length. Call two triangles “congruent” if each side of one can be paired with a side of equal length on the other.

A more concrete way to say this is that one can take the first line segment and move it “rigidly” from wherever it is in the plane to wherever the second line segment is, in such a way that it overlies the second exactly; similarly, one can take the first triangle and move it rigidly so that it overlies the second exactly.

One of the key insights of modern geometry is that the rigid motions are precisely those maps from the plane onto itself that preserve lengths of line segments. The point is that it is just the notion of “length” that counts: all the angles, the area and other stuff follow once you preserve lengths.

The simplest rigid motion of the plane is reflection in a line: that is, pick a line and, for every point off the line, draw the perpendicular to the line through the point and find the point on the other side that is the same distance from the line; points on the line itself are fixed.