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492 CHAPTER 9 9.1 Geometry Points, Lines, Planes, and Angles The Geometry of Euclid Let no one unversed in geometry enter here. —Motto over the door of Plato’s Academy Euclid’s Elements as translated by Billingsley appeared in 1570 and was the first English language translation of the text—the most influential geometry text ever written. Unfortunately, no copy of Elements exists that dates back to the time of Euclid (circa 300 B.C.), and most current translations are based upon a revision of the work prepared by Theon of Alexandria. Although Elements was only one of several works of Euclid, it is, by far, the most important. It ranks second only to the Bible as the most published book in history. To the ancient Greeks, mathematics meant geometry above all—a rigid kind of geometry from a modern-day point of view. The Greeks studied the properties of figures identical in shape and size (congruent figures) as well as figures identical in shape but not necessarily in size (similar figures). They absorbed ideas about area and volume from the Egyptians and Babylonians and established general formulas. The Greeks were the first to insist that statements in geometry be given rigorous proof. The Greek view of geometry (and other mathematical ideas) was summarized in Elements, written by Euclid about 300 B.C. The influence of this book has been extraordinary; it has been studied virtually unchanged to this day as a geometry textbook and as the model of deductive logic. The most basic ideas of geometry are point, line, and plane. In fact, it is not really possible to define them with other words. Euclid defined a point as “that which has no part,” but this definition is so vague as to be meaningless. Do you think you could decide what a point is from this definition? But from your experience in saying “this point in time” or in sharpening a pencil, you have an idea of what he was getting at. Even though we don’t try to define point, we do agree that, intuitively, a point has no magnitude and no size. Euclid defined a line as “that which has breadthless length.” Again, this definition is vague. Based on our experience, however, we know what Euclid meant. The drawings that we use for lines have properties of no thickness and no width, and they extend indefinitely in two directions. What do you visualize when you read Euclid’s definition of a plane: “a surface which lies evenly with the straight lines on itself”? Do you think of a flat surface, such as a tabletop or a page in a book? That is what Euclid intended. The geometry of Euclid is a model of deductive reasoning. In this chapter, we will present geometry from an inductive viewpoint, using objects and situations found in the world around us as models for study. Points, Lines, and Planes There are certain universally accepted conventions and symbols used to represent points, lines, planes, and angles. A capital letter usually represents a point. A line may be named by two capital letters representing points that lie on the line, or by a single (usually lowercase) letter, such as . Subscripts are sometimes used to distinguish one line from another when a lowercase letter is used. For example, 1 and 2 would represent two distinct lines. A plane may be named by three capital letters representing points that lie in the plane, or by a letter of the Greek alphabet, such as (alpha), (beta), or (gamma). Figure 1 depicts a plane that may be represented either as or as plane ADE. Contained in the plane is the line DE (or, equivalently, line ED), which is also labeled in the figure. Selecting any point on a line divides the line into three parts: the point itself, and two half-lines, one on each side of the point. For example, in Figure 2, point A divides the line into three parts, A itself and two half-lines. Point A belongs to neither An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 9.1 Points, Lines, Planes, and Angles 493 half-line. As the figure suggests, each half-line extends indefinitely in the direction opposite to the other half-line. A Ray AB B Half-line A Line segment AB A α FIGURE 1 Given any three points that are not in a straight line, a plane can be passed through the points. That is why camera tripods have three legs—no matter how irregular the surface, the tips of the three legs determine a plane. On the other hand, a camera support with four legs would wobble unless all four legs were carefully extended just the right amount. B A E D A Ray BA B Half-line FIGURE 2 FIGURE 3 Including an initial point with a half-line gives a ray. A ray is named with two letters, one for the initial point of the ray, and one for another point contained in the half-line. For example, in Figure 3 ray AB has initial point A and extends in the direction of B. On the other hand, ray BA has B as its initial point and extends in the direction of A. A line segment includes both endpoints and is named by its endpoints. Figure 3 shows line segment AB, which may also be designated as line segment BA. The following chart shows these figures along with the symbols used to represent them. Name Figure Symbol i Line AB or line BA Half-line AB Half-line BA Ray AB Ray BA Segment AB or segment BA A B A B A B A B A B A B i AB or BA l AB l BA •l AB •l BA •–• •–• AB or BA For a line, the symbol above the two letters shows two arrowheads, indicating that the line extends indefinitely in both directions. For half-lines and rays, only one arrowhead is used since these extend in only one direction. An open circle is used for a half-line to show that the endpoint is not included, while a solid circle is used for a ray to indicate the inclusion of the endpoint. Since a segment includes both endpoints and does not extend in either direction, solid circles are used to indicate endpoints of line segments. The geometric definitions of “parallel” and “intersecting” apply to two or more lines or planes. (See Figure 4 on the next page.) Parallel lines lie in the same plane and never meet, no matter how far they are extended. However, intersecting lines do meet. If two distinct lines intersect, they intersect in one and only one point. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 494 CHAPTER 9 Geometry We use the symbol to denote parallelism. If 1 and 2 are parallel lines, as in Figure 4, then this may be indicated as 1!2. Parallel planes also never meet, no matter how far they are extended. Two distinct intersecting planes form a straight line, the one and only line they have in common. Skew lines do not lie in the same plane, and they never meet, no matter how far they are extended. 1 2 Intersecting lines Parallel lines Intersecting planes Parallel planes Skew lines FIGURE 4 Vertex A Side X B Side C D E F J K L FIGURE 5 Angles An angle is the union of two rays that have a common endpoint, as shown in Figure 5. It is important to remember that the angle is formed by points on the rays themselves, and no other points. In Figure 5, point X is not a point on the angle. (It is said to be in the interior of the angle.) Notice that “angle” is the first basic term in this section that is actually defined, using the undefined terms ray and endpoint. The rays forming an angle are called its sides. The common endpoint of the rays is the vertex of the angle. There are two standard ways of naming angles. If no confusion will result, an angle can be named with the letter marking its vertex. Using this method, the angles in Figure 5 can be named, respectively, angle B, angle E, and angle K. Angles also can be named with three letters: the first letter names a point on one side of the angle; the middle letter names the vertex; the third names a point on the other side of the angle. In this system, the angles in the figure can be named angle ABC, angle DEF, and angle JKL. The symbol for representing an angle is . Rather than writing “angle ABC,” we may write “ABC.” An angle can be associated with an amount of rotation. For example, in Figure 6(a), •l •l •l we let BA first coincide with BC—as though they were the same ray. We then rotate BA (the endpoint remains fixed) in a counterclockwise direction to form ABC. A A A B C B C B C (a) 150° 360° 90° X C 10° 45° N (b) FIGURE 6 An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. M 9.1 Why 360? The use of the number 360 goes back to the Babylonian culture. There are several theories regarding why 360 was chosen for the number of degrees in a complete rotation around a circle. One says that 360 was chosen because it is close to the number of days in a year, and is conveniently divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, and other numbers. Angles are key to the study of geodesy, the measurement of distances on the earth’s surface. Points, Lines, Planes, and Angles 495 Angles are measured by the amount of rotation, using a system that dates back to the Babylonians some two centuries before Christ. Babylonian astronomers chose the number 360 to represent the amount of rotation of a ray back onto itself. Using 360 as the amount of rotation of a ray back onto itself, one degree, written 1°, is defined to be 1360 of a complete rotation. Figure 6(b) shows angles of various degree measures. Angles are classified and named with reference to their degree measures. An angle whose measure is between 0° and 90° is called an acute angle. Angles M and N in Figure 6(b) are acute. An angle that measures 90° is called a right angle. Angle C in the figure is a right angle. The squared symbol at the vertex denotes a right angle. Angles that measure more than 90° but less than 180° are said to be obtuse angles (angle X, for example). An angle that measures 180° is a straight angle. Its sides form a straight line. Our work in this section will be devoted primarily to angles whose measures are less than or equal to 180°. Angles whose measures are greater than 180° are studied in more detail in trigonometry courses. A tool called a protractor can be used to measure angles. Figure 7 shows a protractor measuring an angle. To use a protractor, position the hole (or dot) of the protractor on the vertex of the angle. The 0-degree measure on the protractor should be placed on one side of the angle, while the other side should extend to the degree measure of the angle. The figure indicates an angle whose measure is 135°. 0 0 12 110 100 90 80 70 0 15 30 0 14 40 20 10 180 170 160 60 50 13 Protractor FIGURE 7 Q R When two lines intersect to form right angles they are called perpendicular lines. Our sense of vertical and horizontal depends on perpendicularity. In Figure 8, the sides of NMP have been extended to form another angle, RMQ. The pair NMP and RMQ are called vertical angles. Another pair of vertical angles have been formed at the same time. They are NMQ and PMR. An important property of vertical angles follows. M N Property of Vertical Angles P Vertical angles have equal measures. FIGURE 8 For example, NMP and RMQ in Figure 8 have equal measures. What other pair of angles in the figure have equal measures? An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 496 CHAPTER 9 Geometry EXAMPLE (4x + 19)° (6x – 5)° 1 Refer to the appropriate figure to solve each problem. (a) Find the measure of each marked angle in Figure 9. Since the marked angles are vertical angles, they have the same measure. Set 4x 19 equal to 6x 5 and solve. 4x 19 6x 5 FIGURE 9 4x 4x 19 4x 6x 5 Add 4x. 19 2x 5 19 5 2x 5 5 Add 5. 24 2x 12 x (3x – 30)° (4x)° FIGURE 10 Divide by 2. Since x 12, one angle has measure 412 19 67 degrees. The other has the same measure, since 612 5 67 as well. Each angle measures 67°. (b) Find the measure of each marked angle in Figure 10. The measures of the marked angles must add to 180° since together they form a straight angle. The equation to solve is 3x 30 4x 180 . 7x 30 180 7x 30 30 180 30 Combine like terms. Add 30. 7x 210 x 30 Divide by 7. To find the measures of the angles, replace x with 30 in the two expressions. 3x 30 330 30 90 30 60 4x 430 120 The two angle measures are 60° and 120°. If the sum of the measures of two acute angles is 90°, the angles are said to be complementary, and each is called the complement of the other. For example, angles measuring 40° and 50° are complementary angles, because 40° 50° 90°. If two angles have a sum of 180°, they are supplementary. The supplement of an angle whose measure is 40° is an angle whose measure is 140°, because 40° 140° 180°. If x represents the degree measure of an angle, 90 x represents the measure of its complement, and 180 x represents the measure of its supplement. A (2x + 20)° (12x)° B C FIGURE 11 EXAMPLE 2 Find the measures of the angles in Figure 11, given that ABC is a right angle. The sum of the measures of the two acute angles is 90° (that is, they are complementary), since they form a right angle. We add their measures to obtain a sum of 90 and solve the resulting equation. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 9.1 Points, Lines, Planes, and Angles 497 2x 20 12x 90 14x 20 90 14x 70 Subtract 20. x5 Divide by 14. The value of x is 5. Therefore, the measures of the angles are 2x 20 25 20 30 degrees 12x 125 60 degrees . and EXAMPLE 3 The supplement of an angle measures 10° more than three times its complement. Find the measure of the angle. Let Then and x the degree measure of the angle. 180 x the degree measure of its supplement, 90 x the degree measure of its complement. 10 more than 180 x 10 three times its complement. measures Moiré Patterns A set of parallel lines with equidistant spacing intersects an identical set, but at a small angle. The result is a moiré pattern, named after the fabric moiré (“watered”) silk. You often see similar effects looking through window screens with bulges. Moiré patterns are related to periodic functions, which describe regular recurring phenomena (wave patterns such as heartbeats or business cycles). Moirés thus apply to the study of electromagnetic, sound, and water waves, to crystal structure, and to other wave phenomena. Supplement Now use the words of the problem to write the equation. 390 x 180 x 10 270 3x Distributive property 180 x 280 3x 2x 100 Add 3x ; subtract 180. x 50 Divide by 2. The angle measures 50°. Since its supplement (130°) is 10° more than three times its complement (40°), that is, 130 10 340 is true, the answer checks. Parallel lines are lines that lie in the same plane and do not intersect. Figure 12 shows parallel lines m and n. When a line q intersects two parallel lines, q is called a transversal. In Figure 12, the transversal intersecting the parallel lines forms eight angles, indicated by numbers. Angles 1 through 8 in the figure possess some special properties regarding their degree measures, as shown in the following chart. q 1 3 5 7 2 4 6 8 m n FIGURE 12 An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 498 CHAPTER 9 Geometry Name Figure Rule Alternate interior angles Angle measures are equal. q m 4 5 n (also 3 and 6) Alternate exterior angles Angle measures are equal. q 1 m n (also 2 and 7) 8 Interior angles on same side of transversal Angle measures add to 180°. q m 4 6 n (also 3 and 5) Corresponding angles Angle measures are equal. q 2 6 m n (also 1 and 5, 3 and 7, 4 and 8) The converses of the above also are true. That is, if alternate interior angles are equal, then the lines are parallel, with similar results valid for alternate exterior angles, interior angles on the same side of a transversal, and corresponding angles. (3x + 2)° m n (5x – 40)° FIGURE 13 EXAMPLE 4 Find the measure of each marked angle in Figure 13, given that lines m and n are parallel. The marked angles are alternate exterior angles, which are equal. This gives 3x 2 5x 40 42 2x Subtract 3x and add 40. 21 x . Divide by 2. One angle has a measure of 3x 2 3 21 2 65 degrees, and the other has a measure of 5x 40 5 21 40 65 degrees. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.