454 CHAPTER 8 Geometry Point and Line Three basic terms in geometry are point , line , and plane . These three terms are not given a formal definition, but we recognize points, lines, and planes when we see them. Let’s consider some properties of a line. Assume that a line means a straight line unless otherwise stated. 1. A line is a set of points. Each point is on the line and the line passes through each point. When we wish to refer to a specific point, we will label it with a single capital letter. For example, in Fig. 8.1(a) three points are labeled A, B, and C, respectively. 2. Any two distinct points determine a unique line. Fig. 8.1(a) illustrates a line. The arrows at both ends of the line indicate that the line continues in each direction. The line in Fig. 8.1(a) may be symbolized with any two points on the line by placing a line with a double-sided arrow above the letters that correspond to the points, such as ggggg AB BA AC CA BC , , , , , or gCB. 3. Any point on a line separates the line into three parts: the point itself and two half lines (neither of which includes the point). For example, in Fig. 8.1(a) point B separates the line into the point B and two half lines. Half line BA, symbolized BA, is illustrated in Fig. 8.1(b). The open circle above the B indicates that point B is not included in the half line. Fig. 8.1(c) illustrates half line BC, symbolized BC. Compass Straightedge Did You Know? Compass and Straightedge Constructions Geometric constructions were central to ancient Greek mathematics. Although these constructions are often referred to as Euclidean constructions , they were used centuries before Euclid wrote his classic work, Elements . The tools allowed in geometric constructions are a pencil, an unmarked straightedge, and a drawing compass. The straightedge is used to draw line segments, and the compass is used to draw circles and arcs. The Internet has many sites devoted to classic geometric constructions. Exercise 101 shows you how to construct a triangle with sides of equal length (an equilateral triangle). B A B A C B C (a) (b) (c) Figure 8.1 Look at the half line AB in Fig. 8.2(b). If the end point , A, is included with the set of points on the half line, the result is called a ray . Ray AB, symbolized AB, h is illustrated in Fig. 8.2(c). Ray BA, symbolized hBA, is illustrated in Fig. 8.2(d). A line segment is that part of a line between two points, including the end points. Line segment AB, symbolized AB, is illustrated in Fig. 8.2(e). An open line segment is the set of points on a line between two points, excluding the end points. Open line segment AB, symbolized AB, is illustrated in Fig. 8.2(f). Fig. 8.2(g) illustrates two half open line segments, symbolized AB and AB. Description Diagram Symbol (a) Line AB (b) Half line AB (c) Ray AB (d) Ray BA (e) Line segment AB (f) Open line segment AB (g) Half open line segments AB AB AB AB AB BA AB AB AB B A B A B A B A B A B A B A B A Figure 8.2
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