System of Measurement and Arithmetic
Euclidean geometry has two fundamental types of measurements: angle and distance. The angle scale is absolute, and Euclid uses the right angle as his basic unit, so that, e.g., a 45-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 length as the unit, and other distances are expressed in relation to it.
A line in Euclidean geometry is a model of the real number line. A line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Addition 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 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 (as to multiply three numbers in a Euclidean interpretation, a 3 dimensional rectangular prism would have to have been created, and if one would have wanted to a greater amount of numbers, they would have to move into higher dimensions, which Euclid did not accept the existence of), there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, e.g., 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, and similarly for angles. The stronger term "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 similar. Corresponding angles in a pair of similar shapes are congruent and corresponding sides are in proportion to each other.
Read more about this topic: Euclidean Plane Geometry
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