CHARACTERISTICS OF A GOOD DEFINITION

Transcription

1 1.3 arly efinitions and Postulates is a straight angle. Using your protractor, you can show that m 1 m Find m 1 if m *48. In the drawing, m 1 x and m 2 y. If m RSV 67 and x y 17, find x and y. (HINT: m 1 m 2 m RSV.) S 1 2 R V T 1 2 xercises 39, Find m 1 if m 1 2x and m 2 x. (HINT: See xercise 39.) In xercises 41 to 44, m 1 m 2 m. 41. Find m if m 1 32 and m Find m 1 if m 68 and m 1 m Find x if m 1 x, m 2 2x 3, and m Find an expression for m if m 1 x and m 2 y. 45. compass was used to mark off three congruent segments,,, and. Thus, has been trisected at points and. If 32.7, how long is? 1 2 xercises For xercises 49 and 50, use the following information. Relative to its point of departure or some other point of reference, the angle that is used to locate the position of a ship or airplane is called its bearing. The bearing may also be used to describe the direction in which the airplane or ship is moving. y using an angle between 0 and 90, a bearing is measured from the North-South line toward the ast or West. In the diagram, airplane (which is 250 miles from hicago s O Hare airport s control tower) has a bearing of S 53 W. 49. Find the bearing of airplane relative to the control tower. 50. Find the bearing of airplane relative to the control tower. W 250 mi 53 N 300 mi 22 control tower mi 46. Use your compass and straightedge to bisect F. F *47. In the figure, m 1 x and m 2 y. If x y 24, find x and y. xercises 49, 50 S (HINT: m 1 m ) arly efinitions and Postulates KY ONPTS Mathematical System xiom or Postulate Theorem Ruler Postulate istance Segment-ddition Postulate ongruent Segments Midpoint of a Line Segment Ray Opposite Rays Intersection of Two Geometric Figures Parallel Lines Plane oplanar Points Space MTHMTIL SYSTM Like algebra, the branch of mathematics called geometry is a mathematical system. The formal study of a mathematical system begins with undefined terms. uilding on this foundation, we can then define additional terms. Once the terminology is sufficiently developed, Unless otherwise noted, all content on this page is engage Learning. opyright 2014 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

2 20 HPTR 1 LIN N NGL RLTIONSHIPS certain properties (characteristics) of the system become apparent. These properties are known as axioms or postulates of the system; more generally, such statements are called assumptions in that they are assumed to be true. Once we have developed a vocabulary and accepted certain postulates, many principles follow logically as we apply deductive methods. These statements can be proved and are called theorems. The following box summarizes the components of a mathematical system (sometimes called a logical system or deductive system). 1. Undefined terms 2. efined terms 3. xioms or postulates 4. Theorems FOUR PRTS OF MTHMTIL SYSTM f vocabulary f principles iscover lthough we cannot actually define line and plane, we can compare them in the following analogy. Please complete: _? is to straight as a _? is to flat. NSWRS line; plane HRTRISTIS OF GOO FINITION Terms such as point, line, and plane are classified as undefined because they do not fit into any set or category that has been previously determined. Terms that are defined, however, should be described precisely. ut what is a good definition? good definition is like a mathematical equation written using words. good definition must possess four characteristics, which we illustrate with a term that we will redefine at a later time. FINITION n isosceles triangle is a triangle that has two congruent sides. In the definition, notice that: (1) The term being defined isosceles triangle is named. (2) The term being defined is placed into a larger category (a type of triangle). (3) The distinguishing quality (that two sides of the triangle are congruent) is included. (4) The reversibility of the definition is illustrated by these statements: If a triangle is isosceles, then it has two congruent sides. If a triangle has two congruent sides, then it is an isosceles triangle. HRTRISTIS OF GOO FINITION 1. It names the term being defined. 2. It places the term into a set or category. 3. It distinguishes the defined term from other terms without providing unnecessary facts. 4. It is reversible. Figure 1.30 Figure 1.31 The reversibility of a definition is achieved by using the phrase if and only if. For instance, we could define congruent angles by saying Two angles are congruent if and only if these angles have equal measures. The if and only if statement has the following dual meaning: If two angles are congruent, then they have equal measures. If two angles have equal measures, then they are congruent. When represented by a Venn iagram, the definition above would relate set = {congruent angles} to set = {angles with equal measures} as shown in Figure The sets and are identical and are known as equivalent sets. Once undefined terms have been described, they become the building blocks for other terminology. In this textbook, primary terms are defined within boxes, whereas related terms are often boldfaced and defined within statements. onsider the following definition (see Figure 1.31). Unless otherwise noted, all content on this page is engage Learning. opyright 2014 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

3 1.3 arly efinitions and Postulates 21 FINITION XS. 1 4 line segment is the part of a line that consists of two points, known as endpoints, and all points between them. Geometry in the Real World On the road map, driving distances between towns are shown. In traveling from town to town, which path traverses the least distance? Solution to, to, to : F XMPL 1 State the four characteristics of a good definition of the term line segment. 1. The term being defined, line segment, is clearly present in the definition. 2. line segment is defined as part of a line (a category). 3. The definition distinguishes the line segment as a specific part of a line. 4. The definition is reversible. i) line segment is the part of a line between and including two points. ii) The part of a line between and including two points is a line segment. INITIL POSTULTS Recall that a postulate is a statement that is assumed to be true. POSTULT 1 Through two distinct points, there is exactly one line. Figure 1.32 Postulate 1 is sometimes stated in the form Two points determine a line. See Figure 1.32, in which points and determine exactly one line, namely. Of course, Postulate 1 also implies that there is a unique line segment determined by two distinct points used as endpoints. Recall Figure 1.31, in which points and determine. NOT: In geometry, the reference numbers used with postulates (as in Postulate 1) need not be memorized. XMPL 2 In Figure 1.33, how many distinct lines can be drawn through a) point? b) both points and at the same time? c) all points,, and at the same time? SOLUTION a) n infinite (countless) number b) xactly one c) No line contains all three points. Figure 1.33 Recall from Section 1.2 that the symbol for line segment, named by its endpoints, is. Omission of the bar from, as in, means that we are considering the length of the segment. These symbols are summarized in Table 1.3. Unless otherwise noted, all content on this page is engage Learning. opyright 2014 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

4 22 HPTR 1 LIN N NGL RLTIONSHIPS TL 1.3 Symbol Words for Symbol Geometric Figure Line Line segment Length of segment number Geometry in the Real World ruler is used to measure the length of a line segment such as. This length may be represented by or (the order of and is not important). However, must be a positive number. POSTULT 2 Ruler Postulate The measure of any line segment is a unique positive number. In construction, a string joins two stakes. The line determined is described in Postulate 1 on the previous page. We wish to call attention to the term unique and to the general notion of uniqueness. The Ruler Postulate implies the following: 1. There exists a number measure for each line segment. 2. Only one measure is permissible. haracteristics 1 and 2 are both necessary for uniqueness! Other phrases that may replace the term unique include One and only one xactly one One and no more than one more accurate claim than the commonly heard statement The shortest distance between two points is a straight line is found in the following definition. FINITION The distance between two points and is the length of the line segment that joins the two points. X Figure 1.34 s we saw in Section 1.2, there is a relationship between the lengths of the line segments determined in Figure This relationship is stated in the third postulate. The title and meaning of the postulate are equally important! The title Segment-ddition Postulate will be cited frequently in later sections. POSTULT 3 Segment-ddition Postulate If X is a point of and -X-, then X X. Technology xploration Use software if available. 1. raw line segment XY. 2. hoose point P on XY. 3. Measure XP, PY, and XY. 4. Show that XP PY XY. XMPL 3 In Figure 1.34, find if a) X 7.32 and X b) X 2x 3 and X 3x 7. Unless otherwise noted, all content on this page is engage Learning. opyright 2014 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

5 1.3 arly efinitions and Postulates 23 SOLUTION a) , so b) (2x 3) (3x 7), so 5x 4. FINITION ongruent ( ) line segments are two line segments that have the same length. F In general, geometric figures that can be made to coincide (fit perfectly one on top of the other) are said to be congruent. The symbol is a combination of the symbol ~, which means that the figures have the same shape, and =, which means that the corresponding parts of the figures have the same measure. In Figure 1.35,, but F (meaning that and F are not congruent). oes it appear that F? Figure 1.35 XMPL 4 In the U.S. system of measures, 1 foot 12 inches. If 2.5 feet and 2 feet 6 inches, are and congruent? SOLUTION Yes, because 2.5 feet 2 feet 0.5 feet or 2 feet 0.5(12 inches), or 2 feet 6 inches. FINITION The midpoint of a line segment is the point that separates the line segment into two congruent parts. M In Figure 1.36, if, M, and are collinear and M M, then M is the midpoint of. quivalently, M is the midpoint of if M M. lso, if M M, then is described as a bisector of. If M is the midpoint of in Figure 1.36, we can draw any of these conclusions: M M M 1 2 () M 1 2 () 2 (M) 2 (M) Figure 1.36 XMPL 5 GIVN: M is the midpoint of F (not shown). M 3x 9 and MF x 17 FIN: x, M, and MF iscover ssume that M is the midpoint of in Figure an you also conclude that M is the midpoint of? NSWR No SOLUTION ecause M is the midpoint of F, M MF. Then 3x 9 x 17 2x x 8 x 4 y substitution, M 3(4) and MF Thus, x 4 while M MF 21. In geometry, the word union is used to describe the joining or combining of two figures or sets of points. Unless otherwise noted, all content on this page is engage Learning. opyright 2014 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

6 24 HPTR 1 LIN N NGL RLTIONSHIPS FINITION Ray, denoted by!, is the union of and all points X on such that is between and X. In Figure 1.37,,!, and! are shown in that order; note that! and! are not the same ray. Line ( has no endpoints) Ray ( has endpoint ) Ray ( has endpoint ) radley ve. Figure 1.38 Neil St. Figure 1.37 Opposite rays are two rays with a common endpoint; also, the union of opposite rays is a straight line. In Figure 1.39(a),! and! are opposite rays. The intersection of two geometric figures is the set of points that the two figures have in common. In everyday life, the intersection of radley venue and Neil Street is the part of the roadway that the two roads have in common (Figure 1.38). POSTULT 4 If two lines intersect, they intersect at a point. (a) m When two lines share two (or more) points, the lines coincide; in this situation, we say there is only one line. In Figure 1.39(a), and are the same as. In Figure 1.39(b), lines and m intersect at point P. Figure 1.39 P (b) FINITION Parallel lines are lines that lie in the same plane but do not intersect. In Figure 1.40, suppose that and n are parallel; in symbols, n and n. However, and m are not parallel because they intersect at point ; so m and m. XS XMPL 6 In Figure 1.40, n. What is the intersection of a) lines n and m? b) lines and n? n SOLUTION a) Point b) Parallel lines do not intersect. Figure 1.40 m nother undefined term in geometry is plane. plane is two-dimensional; that is, it has infinite length and infinite width but no thickness. xcept for its limited size, a flat surface such as the top of a table could be used as an example of a plane. n uppercase let- Unless otherwise noted, all content on this page is engage Learning. opyright 2014 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

7 1.3 arly efinitions and Postulates 25 ter can be used to name a plane. ecause a plane (like a line) is infinite, we can show only a portion of the plane or planes, as in Figure R S T V Figure 1.42 Figure 1.41 Planes R and S Planes T and V plane is two-dimensional, consists of an infinite number of points, and contains an infinite number of lines. Two distinct points may determine (or fix ) a line; likewise, exactly three noncollinear points determine a plane. Just as collinear points lie on the same line, coplanar points lie in the same plane. In Figure 1.42, points,,, and are coplanar, whereas,,, and are noncoplanar. In this book, points shown in figures are generally assumed to be coplanar unless otherwise stated. For instance, points,,,, and are coplanar in Figure 1.43(a), as are points F, G, H, J, and K in Figure 1.43(b). K G J F H (a) (b) Geometry in the Real World Figure 1.43 The tripod illustrates Postulate 5 in that the three points at the base enable the unit to sit level. ja65/shutterstock.com POSTULT 5 Through three noncollinear points, there is exactly one plane. On the basis of Postulate 5, we can see why a three-legged table sits evenly but a fourlegged table would wobble if the legs were of unequal length. Space is the set of all possible points. It is three-dimensional, having qualities of length, width, and depth. When two planes intersect in space, their intersection is a line. n opened greeting card suggests this relationship, as does Figure 1.44(a). This notion gives rise to our next postulate. R S R S M N (a) (b) (c) Figure 1.44 Unless otherwise noted, all content on this page is engage Learning. opyright 2014 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

8 26 HPTR 1 LIN N NGL RLTIONSHIPS iscover uring a baseball game, the catcher and the third baseman follow the path of a foul pop fly toward the grandstand. oes it appear that there is a play on the baseball? POSTULT 6 If two distinct planes intersect, then their intersection is a line. The intersection of two planes is infinite because it is a line. [See Figure 1.44(a) on page 25.] If two planes do not intersect, then they are parallel. The parallel vertical planes R and S in Figure 1.44(b) may remind you of the opposite walls of your classroom. The parallel horizontal planes M and N in Figure 1.44(c) suggest the relationship between the ceiling and the floor. Imagine a plane and two points of that plane, say points and. Now think of the line containing the two points and the relationship of to the plane. Perhaps your conclusion can be summed up as follows. VISITORS POSTULT 7 Given two distinct points in a plane, the line containing these points also lies in the plane. NSWR No; the baseball will land in the stands. XS ecause the uniqueness of the midpoint of a line segment can be justified, we call the following statement a theorem. The proof of the theorem is found in Section 2.2. THORM The midpoint of a line segment is unique. M Figure 1.45 XS If M is the midpoint of in Figure 1.45, then no other point can separate into two congruent parts. The proof of this theorem is based on the Ruler Postulate. M is the point 1 that is located 2 () units from (and from ). The numbering system used to identify Theorem need not be memorized. However, this theorem number may be used in a later reference. The numbering system works as follows: HPTR STION ORR where where found in found found section summary of the theorems presented in this textbook appears at the end of the book. xercises 1.3 In xercises 1 and 2, complete the statement. xercises 1, 2 1.?_ 2. If, then is the?_ of. In xercises 3 and 4, use the fact that 1 foot 12 inches. 3. onvert 6.25 feet to a measure in inches. 4. onvert 52 inches to a measure in feet and inches. In xercises 5 and 6, use the fact that 1 meter 3.28 feet (measure is approximate) onvert meter to feet. 6. onvert 16.4 feet to meters. 7. In the figure, the 15-mile road from to is under construction. detour from to of 5 miles and then from to of 13 miles must be taken. How much farther is the detour from to than the road from to? xercises 7, 8 Unless otherwise noted, all content on this page is engage Learning. opyright 2014 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

9 1.3 arly efinitions and Postulates cross-country runner jogs at a rate of 15 feet per second. If she runs 300 feet from to, 450 feet from to, and then 600 feet from back to, how long will it take her to return to point? See figure for xercise 7. In xercises 9 to 28, use the drawings as needed to answer the following questions. 9. Name three points that appear to be a) collinear. b) noncollinear. 10. How many lines can be drawn through a) point? b) points and? c) points,, and? d) points,, and? 11. Give the meanings of,,, and!. 12. xplain the difference, if any, between a) and. c) and. b) and. d)! and!. 13. Name two lines that appear to be a) parallel. b) nonparallel. 14. lassify as true or false: a) d) b) e) c) M xercises Given: M is the midpoint of M 2x 1 and M 3x 2 Find: x and M 16. Given: M is the midpoint of M 2(x 1) and M 3(x 2) Find: x and 17. Given: M 2x 1, M 3x 2, and 6x 4 Find: x and 18. an a segment bisect a line? a segment? an a line bisect a segment? a line? 19. In the figure, name a) two opposite rays. b) two rays that are not opposite. p t m xercises 9, 10 O 20. Suppose that (a) point lies in plane X and (b) point lies in plane X. What can you conclude regarding? 21. Make a sketch of a) two intersecting lines that are perpendicular. b) two intersecting lines that are not perpendicular. c) two parallel lines. 22. Make a sketch of a) two intersecting planes. b) two parallel planes. c) two parallel planes intersected by a third plane that is not parallel to the first or the second plane. 23. Suppose that (a) planes M and N intersect, (b) point lies in both planes M and N, and (c) point lies in both planes M and N. What can you conclude regarding? 24. Suppose that (a) points,, and are collinear and (b). Which point can you conclude cannot lie between the other two? 25. Suppose that points, R, and V are collinear. If R 7 and RV 5, then which point cannot possibly lie between the other two? 26. Points,,, and are coplanar;,, and are collinear; point is not in plane M. How many planes contain a) points,, and? b) points,, and? c) points,,, and? d) points,,, and? 27. Using the number line provided, name the point that a) is the midpoint of. b) is the endpoint of a segment of length 4, if the other endpoint is point G. c) has a distance from equal to 3(). F G H xercises 27, onsider the figure for xercise 27. Given that is the midpoint of and is the midpoint of, what can you conclude about the lengths of a) and? c) and? b) and? In xercises 29 to 32, use only a compass and a straightedge to complete each construction. 29. Given: and ( ) onstruct: MN on line so that MN xercises 29, 30 M 30. Given: and ( ) onstruct: F on line so that F Unless otherwise noted, all content on this page is engage Learning. opyright 2014 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

10 28 HPTR 1 LIN N NGL RLTIONSHIPS 31. Given: as shown in the figure onstruct: PQ on line n so that PQ 3() xercises 31, Given: as shown in the figure onstruct: TV on line n so that TV 1 2 () 33. an you use the construction for the midpoint of a segment to divide a line segment into a) three congruent parts? c) six congruent parts? b) four congruent parts? d) eight congruent parts? 34. Generalize your findings in xercise onsider points,,, and, no three of which are collinear. Using two points at a time (such as and ), how many lines are determined by these points? 36. onsider noncoplanar points,,, and. Using three points at a time (such as,, and ), how many planes are determined by these points? 37. Line is parallel to plane P (that is, it will not intersect P even if extended). Line m intersects line. What can you conclude about m and P? P m n 38. and F are said to be skew lines because they neither intersect nor are parallel. How many planes are determined by a) parallel lines and? b) intersecting lines and? c) skew lines and F? d) lines,, and? e) points,, and F? f) points,, and H? g) points,, F, and H? G H xercises F 39. In the box shown for xercise 38, use intuition to answer each question. a) re and parallel? b) re and F skew line segments? c) re and F perpendicular? 40. In the box shown for xercise 38, use intuition to answer each question. a) re G and skew line segments? b) re G and congruent line segments? c) re GF and parallel? *41. Let a and b. Point M is the midpoint of. If N 2 3 (), find the length of NM in terms of a and b. N M 1.4 ngles and Their Relationships KY ONPTS ngle: Sides of ngle, Vertex of ngle Protractor Postulate cute, Right, Obtuse, Straight, and Reflex ngles ngle-ddition Postulate djacent ngles ongruent ngles isector of an ngle omplementary ngles Supplementary ngles Vertical ngles This section introduces you to the language of angles. Recall from Sections 1.1 and 1.3 that the word union means that two sets or figures are joined. FINITION n angle is the union of two rays that share a common endpoint. 1 Figure 1.46 The preceding definition is illustrated in Figure 1.46, in which! and! have the common endpoint. ll content on this page is engage Learning. opyright 2014 engage Learning. ll Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. ue to electronic rights, some third party content may be suppressed from the eook and/or ehapter(s). ditorial review has deemed that any suppressed content does not materially affect the overall learning experience. engage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.