456 CHAPTER 8 Geometry Plane The term plane is one of Euclid’s undefined terms. For our purposes, we can think of a plane as a two-dimensional surface that extends infinitely in all directions, like an infinitely large white board. Euclidean geometry is called plane geometry because it is the study of two-dimensional figures in a plane. Two lines in the same plane that do not intersect are called parallel lines. Fig. 8.4(a) illustrates two parallel lines in a plane ( gAB is parallel to gCD). Properties of planes include the following: 1. Any three points that are not on the same line (noncollinear points) determine a unique plane (Fig. 8.4b). 2. A line in a plane divides the plane into three parts, the line itself and two half planes (Fig. 8.4c). 3. Any line and a point not on the line determine a unique plane. 4. The intersection of two distinct planes is a line (Fig. 8.4d). Two planes that do not intersect are said to be parallel planes. For example, in Fig. 8.5 plane ABE is parallel to plane GHF. Two lines that do not lie in the same plane and do not intersect are called skew lines. Fig. 8.5 illustrates many skew lines (for example, gAB and gCD). c) h AB CD > Line segment AB and ray CD have no points in common, so their intersection is the empty set, ∅. A C D B CD → AB ⎯ Solution: = ∅ h AB CD > d) < AD CA The union of line segment AD and half line CA is ray DA, hDA (or, equivalently, hDB or hDC). → → AD CA Solution: DA A B C D = h AD CA DA < 7 Now try Exercise 19 Learning Catalytics Keyword: Angel-SOM-8.1 (See Preface for additional details.) (b) (c) (d) (a) B B D A A C C Half plane Half plane Figure 8.4 A B C D E F G H Figure 8.5 Example 2 Working with Lines and Planes Fig. 8.6 suggests several lines and planes. The lines may be described by naming two points, and the planes may be described by naming three points. Determine the following. a) > ABC BCF plane plane b) plane g ABC CF > c) plane g ABC DF > d) two parallel lines e) two parallel planes f) two skew lines
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