INTRODUCTION 3 A line which is composed of portions of straight lines is called a broken line. Other lines, which are neither straight nor broken, are called curves. 5. The part of a straight line contained between two points A and B is called a segment of the line AB, or the distance AB. We can also consider the portion of a line which is not limited in one direction, but limited in the other by a point. This is called a ray. Figure 1 Figure 2 It follows from the remarks above that two arbitrary rays are congruent. The distance AB is said to be equal to the distance A B if the first segment can be moved onto the second in such a way that A falls onto A and B onto B . After such a move, it follows from the two properties which define the notion of a straight line that the two segments will coincide entirely. Consequently, the definition of equal segments agrees with the definition of congruent figures given earlier. Two equal segments AB, A B can be made to coincide in two different ways namely, the point A falling on A and B on B , or A falling on B and B on A . In other words, one can turn the segment AB around in such a way that each of the points A, B takes the place of the other. When two segments AB, BC are on the same line so that one of them is the extension of the other (Fig. 1), the segment AC is called the sum of the first two segments. The sum of two, and therefore of many, segments is independent of the order of the summands.9 In order to compare segments, we move them onto the same line, so that they start from the same point and are oriented in the same direction: for example AB and AC (Figures 1 and 2). If the points are in the order A, B, C (Fig. 1), the segment AC is the sum of AB and another segment BC in this case AC is greater than AB if on the contrary, the order is A, C, B, the segment AC is smaller than AB. In either case, the third segment BC which, when added to one of the first produces the other, is called the difference of the two segments. Finally, the points B and C may coincide, in which case we know that the two segments under consideration are equal. On each line segment AB, there exists a point M , the midpoint of AB, equally distant from A and B every point on the line between A and M is clearly closer to A than to B, and the opposite is true for points between M and B. More generally, a line segment can be divided into any number of equal parts.10 6. The Plane. An infinite surface such that a line joining any two points on it is contained entirely on the surface is called a plane. We will assume that there is a plane passing through any three points in space. A straight line drawn in a plane separates the surface into two regions called half- planes, situated on different sides of the line. One cannot travel along a continuous 9 For two segments this follows immediately from the preceding paragraph. 10 We mean by this that there exist points on AB which divide this segment into equal parts. The question as to whether these points can actually be determined with the help of available instruments will be considered later (Book III).
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