Editing Euclidean geometry (section)

==System of measurement and arithmetic==
Euclidean geometry has two fundamental types of measurements: [[angle]] and [[Euclidean distance|distance]]. The angle scale is absolute, and Euclid uses the [[right angle]] as his basic unit, so that, for example, a 45-[[degree (angle)|degree]] angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction.

Measurements of [[area (geometry)|area]] and [[volume]] are derived from distances. For example, a [[rectangle]] with a width of 3 and a length of 4 has an area that represents the product, 12. Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, for example in the proof of book IX, proposition 20.

Euclid refers to a pair of lines, or a pair of planar or solid figures, as "equal" (ἴσος) if their lengths, areas, or volumes are equal respectively, and similarly for angles. The stronger term "[[congruence (geometry)|congruent]]" refers to the idea that an entire figure is the same size and shape as another figure. Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly. (Flipping it over is allowed.) Thus, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their differing sizes are referred to as [[similarity (geometry)|similar]]. [[corresponding sides and corresponding angles|Corresponding angles]] in a pair of similar shapes are equal and [[corresponding sides]] are in proportion to each other.