8.1 Points, Lines, Planes, and Angles 457 Angles An angle, denoted , is the union of two rays with a common end point (Fig. 8.7): = h h BA BC ABC CBA (or ) < An angle can be formed by the rotation of a ray about its end point. An angle has an initial side and a terminal side. The initial side indicates the position of the ray prior to rotation; the terminal side indicates the position of the ray after rotation. The point common to both rays is called the vertex of the angle. The letter designating the vertex is always the middle one of the three letters designating an angle. The rays that make up the angle are called its sides. There are several ways to name an angle. The angle in Fig. 8.7 may be denoted ABC CBA B , , or An angle divides a plane into three distinct parts: the angle itself, its interior, and its exterior. In Fig. 8.7, the angle is represented by the blue lines, the interior of the angle is shaded pink, and the exterior is shaded tan. The measure of an angle, symbolized m, is the amount of rotation from its initial side to its terminal side. In Fig. 8.7, the letter x represents the measure of ABC; therefore, we may write = m ABC x. Angles can be measured in degrees, radians, or gradients. In this text, we will discuss only the degree unit of measurement. Consider a circle whose circumference is divided into 360 equal parts. If we draw a line from each mark on the circumference to the center of the circle, we get 360 wedge-shaped pieces. The measure of an angle formed by the straight sides of each wedge-shaped piece is defined to be 1 degree, written 1°. An angle of 45 degrees is written 45°. A protractor is used to measure angles. The angle shown being measured by the protractor in Fig. 8.8 is 50°. D A B C E F Figure 8.6 Solution a) Plane ABC can be described as containing the “front” of Fig. 8.6. Plane BCF can be described as containing the “bottom” of Fig. 8.6. The intersection of these two planes is the line that contains point B and point C. Thus, plane ABC > plane = BCF gBC b) From Fig. 8.6, we see that gCF, the line that contains the points C and F, intersects plane ABC at the single point C. Thus, plane { } = g ABC CF C . > c) From Fig. 8.6, we see that gDF, the line that contains the points D and F, has no points in common with plane ABC. Thus, plane = ∅ g ABC DF , > the empty set. d) Parallel lines are lines that are in the same plane but do not intersect. From Fig. 8.6, we see that gAD and gCF are parallel, gAD and gBE are parallel, gAC and gDF are parallel, and gBC and gEF are parallel. There are several other parallel lines in Fig. 8.6. e) Parallel planes are planes that do not intersect. From Fig. 8.6, we see that plane ABC and plane DEF do not intersect. Thus, plane ABC and plane DEF are parallel planes. f) Skew lines are two lines that do not lie in the same plane and do not intersect. From Fig. 8.6, we see that gAB and gDF are not in the same plane and do not intersect. Thus, gAB and gDF are skew lines. Other examples of skew lines are gAC and gDE, gAD and gBC, and gAB and gCF. There are several other skew lines in Fig. 8.6. 7 Now try Exercise 83 Vertex Terminal side Initial side B C A x Figure 8.7
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