Informal Geometry and Measurement

Transcription

1 HP LIN N NGL LIONHIP In xercises 56 and 57, P is a true statement, while Q and are false statements. lassify each of the following statements as true or false. 56. a) (P and Q) or b) (P or Q) and 57. a) (P and Q) or ~ b) (P or Q) and ~.2 Informal Geometry and easurement KY ONP Point Line Plane ollinear Points Line egment etweenness of Points idpoint ongruence Protractor Parallel isect Intersect Perpendicular ompass onstructions ircle rc adius Figure.8 X (c) (d) Figure.9 m In geometry, the terms point, line, and plane are described but not defined. Other concepts that are accepted intuitively, but never defined, include the straightness of a line, the flatness of a plane, the notion that a point on a line lies between two other points on the line, and the notion that a point lies in the interior or exterior of an angle. ome of the terms found in this section are formally defined in later sections of hapter. he following are descriptions of some of the undefined terms. point, which is represented by a dot, has location but not size; that is, a point has no dimensions. n uppercase italic letter is used to name a point. Figure.8 shows points,, and. he second undefined geometric term is line. line is an infinite set of points. Given any two points on a line, there is always a point that lies between them on that line. Lines have a quality of straightness that is not defined but assumed. Given several points on a line, these points form a straight path. Whereas a point has no dimensions, a line is onedimensional; that is, the distance between any two points on a given line can be measured.! Line, represented symbolically by, extends infinitely far in opposite directions, as suggested by the arrows on the line. line may also be represented by a single lowercase letter. Figures.9 and show the lines and m. When a lowercase letter is used to name a line, the line symbol is omitted; that is, and m can name the same line.!! Note the position of point X on in Figure.9(c). When three points such as, X, and are on the same line, they are said to be collinear. In the order shown, which is symbolized -X- or -X-, point X is said to be between and. When a drawing is not provided, the notation -- means that these points are collinear, with between and. When a drawing is provided, we assume that all points in the drawing that appear to be collinear are collinear, unless otherwise stated. Figure.9(d) shows that,, and are collinear; in Figure.8, points,, and are noncollinear. t this time, we informally introduce some terms that will be formally defined later. You have probably encountered the terms angle, triangle, and rectangle many times. n example of each is shown in Figure.. W X ngle riangle F F Z ectangle WXYZ (c) Y Figure. Unless otherwise noted, all content on this page is engage Learning. opyright 204 engage Learning. ll ights eserved. ay 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 .2 Informal Geometry and easurement Figure. Using symbols, we refer to Figures.,, and (c) as, F, and WXYZ, respectively. ome caution must be used in naming figures; although the angle in Figure. can be called, it is incorrect to describe the angle as because that order implies a path from point to point to point... a different angle! In, the point at which the sides meet is called the vertex of the angle. ecause there is no confusion regarding the angle described, is also known as (using only the vertex) or as. he points,, and F at which the sides of F (also called F, F, etc.) meet are called the vertices (plural of vertex) of the triangle. imilarly, W, X, Y, and Z are the vertices of the rectangle; the vertices are named in an order that traces the rectangle. line segment is part of a line. It consists of two distinct points on the line and all points between them. (ee Figure..) Using symbols, we indicate the line segment by ; note that is a set of points but is not a number. We use (omitting the segment symbol) to indicate the length of this line segment; thus, is a number. he sides of a triangle or rectangle are line segments. XPL an the rectangle in Figure.(c) be named a) XYZW? b) WYXZ? OLUION a) Yes, because the points taken in this order trace the figure. b) No; for example, WY is not a side of the rectangle. iscover In converting from U.. units to the metric system, a known conversion is the fact that inch 2.54 cm. What is the cm equivalent of 3.7 inches? NW 9.4 cm UING LIN GN he instrument used to measure a line segment is a scaled straightedge such as a ruler, a yardstick, or a meter stick. Line segment ( in symbols) in Figure.2 measures 5 centimeters. ecause we express the length of by (with no bar), we write = 5 cm. o find the length of a line segment using a ruler:. Place the ruler so that 0 corresponds to one endpoint of the line segment. 2. ead the length of the line segment by reading the number at the remaining endpoint of the line segment. ecause manufactured measuring devices such as the ruler, yardstick, and meter stick may lack perfection or be misread, there is a margin of error each time one is used. In Figure.2, for instance, may actually measure 5.02 cm (and that could be rounded from cm, etc.). easurements are approximate, not perfect NI 6 Figure.2 In xample 2, a ruler (not drawn to scale) is shown in Figure.3. In the drawing, the distance between consecutive marks on the ruler corresponds to inch. he measure of a line segment is known as linear measure. Unless otherwise noted, all content on this page is engage Learning. opyright 204 engage Learning. ll ights eserved. ay 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 2 HP LIN N NGL LIONHIP XPL 2 In rectangle of Figure.3, the line segments and shown are the diagonals of the rectangle. How do the lengths of the diagonals compare? Figure.3 OLUION s shown on the ruler,. s intuition suggests, the lengths of the diagonals are the same, so it follows that. NO: In linear measure, means inches, and means feet. Figure.4 In Figure.4, point lies between and on. If =, then is the midpoint of. When =, the geometric figures and are said to be congruent; in effect, geometric figures are congruent when one can be placed over the other (a perfect match). Numerical lengths may be equal, but the actual line segments (geometric figures) are congruent. he symbol for congruence is ; thus, if is the midpoint of. xample 3 emphasizes the relationship between,, and when lies between and. XPL 3 X. 8 In Figure.5, the lengths of and are 4 and 8. What is, the length of? Figure.5 OLUION s intuition suggests, the length of equals. hus, iscover he word geometry means the measure (from metry) of the earth (from geo). Words that contain meter also suggest the measure of some quantity. What is measured by each of the following objects? odometer, pedometer, thermometer, altimeter, clinometer, anemometer NW Vehicle mileage, distance walked, temperature, altitude, angle of inclination, wind speed UING NGL lthough we formally define an angle in ection.4, we consider it intuitively at this time. n angle s measure depends not on the lengths of its sides but on the amount of opening between its sides. In Figure.6, the arrows on the angles sides suggest that the sides extend indefinitely. N Q Figure.6 Unless otherwise noted, all content on this page is engage Learning. opyright 204 engage Learning. ll ights eserved. ay 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 Informal Geometry and easurement 3 30 Figure he instrument shown in Figure.7 (and used in the measurement of angles) is a protractor. For example, we would express the measure of by writing m 50 ; this statement is read, he measure of is 50 degrees. easuring the angles in Figure.6 with a protractor, we find that m 55 and m 90. If the degree symbol is missing, the measure is understood to be in degrees; thus, m 90. In practice, the protractor shown will measure an angle that is greater than 0 but less than or equal to 80. o find the degree measure of an angle using a protractor:. Place the notch of the protractor at the point where the sides of the angle meet (the vertex of the angle). ee point in Figure Place the edge of the protractor along a side of the angle so that the scale reads 0. ee point in Figure.8 where we use 0 on the outer scale. 3. Using the same (outer) scale, read the angle size by reading the degree measure that corresponds to the second side of the angle. Warning any protractors have dual scales, as shown in Figure.8. XPL 4 For Figure.8, find the measure of Figure.8 OLUION Using the protractor, we find that the measure of angle is 3. (In symbols, m 3 or m 3.) ome protractors show a full 360 ; such a protractor is used to measure an angle whose measure is between 0 and 360. n angle whose measure is between 80 and 360 is known as a reflex angle. Just as measurement with a ruler is not perfect, neither is measurement with a protractor. he lines on a sheet of paper in a notebook are parallel. Informally, parallel lines lie on the same page and will not cross over each other even if they are extended indefinitely. We say that lines and m in Figure.9 are parallel; note here the use of a lowercase letter to name a line. We say that line segments are parallel if they are parts of parallel lines;!! if is parallel to N, then is parallel to N in Figure.9. Unless otherwise noted, all content on this page is engage Learning. opyright 204 engage Learning. ll ights eserved. ay 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 4 HP LIN N NGL LIONHIP m N Figure.9 For {, 2, 3} and {6, 8, }, there are no common elements; for this reason, we say that the intersection of and is the empty set (symbol is ). Just as, the parallel lines in Figure.9 are characterized by m. XPL 5 In Figure.20, the sides of angles and F are parallel ( to and to F). Use a protractor to decide whether these angles have equal measures. Figure.20 F OLUION he angles have equal measures. oth measure Figure.2 wo angles with equal measures are said to be congruent. In Figure.20, we see that F. In Figure.2,. In Figure.2, angle has been separated into smaller angles and ; if the two smaller angles are congruent (have equal measures), then angle has been bisected. In general, the word bisect means to separate a line segment (or an angle) into two parts of equal measure; similarly, the word trisect means that the line segment (or angle) is separated into three parts of equal measure. ny angle having a 80 measure is called a straight angle, an angle whose sides are in opposite directions. ee straight angle in Figure.22. When a straight angle is bisected, as shown in Figure.22, the two angles formed are right angles (each measures 90 ). When two lines have a point in common, as in Figure.23, they are said to intersect. When two lines intersect and form congruent adjacent angles, they are said to be perpendicular. In Figure.23, lines r and t are perpendicular if 2 or 2 3, and so on V 90 Figure.22 Figure.23 r t Unless otherwise noted, all content on this page is engage Learning. opyright 204 engage Learning. ll ights eserved. ay 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 .2 Informal Geometry and easurement 5 XPL 6 In Figure.23, suppose that a) 3 4 b) 4 c) 3. re lines r and t perpendicular? X. 9 3 OLUION a) Yes b) Yes c) No Figure.24 enter O O adius O rc Figure.25 ONUION nother tool used in geometry is the compass. his instrument, shown in Figure.24, is used to draw circles and parts of circles known as arcs. he ancient Greeks insisted that only two tools (a compass and a straightedge) be used for geometric constructions, which were idealized drawings assuming perfection in the use of these tools. he compass was used to create perfect circles and for marking off segments of equal length. he straightedge could be used to draw a straight line through two designated points. circle is the set of all points in a plane that are at a given distance from a particular point (known as the center of the circle). he part of a circle between any two of its points is known as an arc. ny line segment joining the center to a point on the circle is a radius (plural: radii) of the circle. ee Figure.25. onstruction, which follows, is quite basic and depends only on using arcs of the same radius length to construct line segments of the same length. he arcs are created by using a compass. onstruction 2 is more difficult to perform and explain, so we will delay its explanation to a later chapter (see ection 3.4). ONUION o construct a segment congruent to a given segment. GIVN: in Figure.26. ONU: on line m so that (or ) ONUION: With your compass open to the length of, place the stationary point of the compass at and mark off a length equal to at point, as shown in Figure.26. hen. Figure.26 m (c) Figure.27 X. 4 7 he following construction is shown step by step in Figure.27. Intuition suggests that point in Figure.27(c) is the midpoint of. ONUION 2 o construct the midpoint of a given line segment. GIVN: in Figure.27 ONU: on so that ONUION: Figure.27: Open your compass to a length greater than one-half of. Figure.27: Using as the center of the arc, mark off an arc that extends both above and below segment. With as the center and keeping the same length of radius, mark off an arc that extends above and below so that two points ( and ) are determined where the arcs cross. Figure.27(c): Now draw. he point where crosses is the midpoint. Unless otherwise noted, all content on this page is engage Learning. opyright 204 engage Learning. ll ights eserved. ay 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 6 HP LIN N NGL LIONHIP XPL 7 In Figure.28, is the midpoint of. a) Find if 5. b) Find if 4.3. c) Find if 2x. Figure.28 OLUION a) is one-half of, so 7 2. b) is twice, so 2(4.3) or 8.6. c) is twice, so 2(2x ) or 4x 2. he technique from algebra used in xample 8 and also needed for xercises 47 and 48 of this section depends on the following properties of addition and subtraction. If a b and c d, then a c b d. Words: quals added to equals provide equal sums. Illustration: ince and 0.2 2, it follows that ; that is, If a b and c d, then a c b d. Words: quals subtracted from equals provide equal differences. Illustration: ince and 0.2 2, it follows that ; that is, XPL 8 x Figure.29 y In Figure.29, point lies on between and. If and is 2 units longer than, find the length x of and the length y of. OLUION ecause, we have x y. ecause 2, we have x y 2. dding the left and right sides of these equations, we have x y x y 2 2x 2 so x 6. X. 8, 9 If x 6, then x y becomes 6 y and y 4. hus, 6 and 4. Unless otherwise noted, all content on this page is engage Learning. opyright 204 engage Learning. ll ights eserved. ay 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 .2 Informal Geometry and easurement 7 xercises.2. If line segment and line segment are drawn to scale, what does intuition tell you about the lengths of these segments? 2. If angles and F were measured with a protractor, what does intuition tell you about the degree measures of these angles? F 2. triangle is named. an it also be named? an it be named? 3. onsider rectangle NPQ. an it also be named rectangle PQN? an it be named rectangle NQP? 4. uppose and F have the same measure. Which statements are expressed correctly? a) m m F b) F c) m m F d) F 5. uppose and have the same length. Which statements are expressed correctly? a) b) c) d) 6. When two lines cross (intersect), they have exactly one point in common. In the drawing, what is the point of intersection? How do the measures of and 2 compare? 3. How many endpoints does a line segment have? How many midpoints does a line segment have? 2 Q 4. o the points,, and appear to be collinear? P N xercises How many lines can be drawn that contain both points and? How many lines can be drawn that contain points,, and? 6. onsider noncollinear points,, and. If each line must contain two of the points, what is the total number of lines that are determined by these points? 7. Name all the angles in the figure. 8. Which of the following measures can an angle have? 23, 90, 200,.5, 5 9. ust two different points be collinear? ust three or more points be collinear? an three or more points be collinear?. Which symbol(s) correctly expresses the order in which the points,, and X lie on the given line, -X- or --X? 7. Judging from the ruler shown (not to scale), estimate the measure of each line segment. a) b) xercises 7, 8 8. Judging from the ruler, estimate the measure of each line segment. a) F b) GH 9. Judging from the protractor provided, estimate the measure of each angle to the nearest multiple of 5 (e.g., 20, 25, 30, etc.). a) m b) m G F H X. Which symbols correctly name the angle shown?,, xercises 9, Unless otherwise noted, all content on this page is engage Learning. opyright 204 engage Learning. ll ights eserved. ay 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 8 HP LIN N NGL LIONHIP 20. Using the drawing for xercise 9, estimate the measure of each angle to the nearest multiple of 5 (e.g., 20, 25, 30, etc.). a) m 3 b) m 4 2. onsider the square at the right, V. It has four right angles and four sides of the same length. How are sides and related? How are sides and V related? 22. quare V has diagonals and V (not shown). If the diagonals are drawn, how will their lengths compare? xercises 2, 22 o the diagonals of a square appear to be perpendicular? 23. Use a compass to draw a circle. raw a radius, a line segment that connects the center to a point on the circle. easure the length of the radius. raw other radii and find their lengths. How do the lengths of the radii compare? 24. Use a compass to draw a circle of radius inch. raw a chord, a line segment that joins two points on the circle. raw other chords and measure their lengths. What is the largest possible length of a chord in this circle? 25. he sides of the pair of angles are parallel. re and 2 congruent? V 30. On a piece of paper, use your protractor to draw a triangle that has two angles of the same measure. ut the triangle out of the paper and fold the triangle in half so that the angles of equal measure coincide (one lies over the other). What seems to be true of two of the sides of that triangle? 3. trapezoid is a four-sided figure that contains one pair of parallel sides. Which sides of the trapezoid NPQ appear to be parallel? Q 32. In the rectangle shown, what is true of the lengths of each pair of opposite sides? N P 2 V 26. he sides of the pair of angles are parallel. re 3 and 4 congruent? he sides of the pair of angles are perpendicular. re 5 and 6 congruent? he sides of the pair of angles are perpendicular. re 7 and 8 congruent? 33. line segment is bisected if its two parts have the same length. Which line segment, or, is bisected at point X? 34. n angle is bisected if its two parts have the same measure. Use three letters to name the angle that is bisected. 3 cm X 5 cm 3 cm 5 3 cm In xercises 35 to 38, with -- on, it follows that. 7 8 xercises On a piece of paper, use your compass to construct a triangle that has two sides of the same length. ut the triangle out of the paper and fold the triangle in half so that the congruent sides coincide (one lies over the other). What seems to be true of two angles of that triangle? 35. Find if 9 and Find if 25 and. 37. Find x if x, x 3, and Find an expression for (the length of ) if x and y. Unless otherwise noted, all content on this page is engage Learning. opyright 204 engage Learning. ll ights eserved. ay 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 .3 arly efinitions and Postulates is a straight angle. Using your protractor, you can show that m m Find m if m *48. In the drawing, m x and m 2 y. If m V 67 and x y 7, find x and y. (HIN: m m 2 m V.) 2 V 2 xercises 39, Find m if m 2x and m 2 x. (HIN: ee xercise 39.) In xercises 4 to 44, m m 2 m. 4. Find m if m 32 and m Find m if m 68 and m m Find x if m x, m 2 2x 3, and m Find an expression for m if m x and m 2 y. 45. compass was used to mark off three congruent segments,,, and. hus, has been trisected at points and. If 32.7, how long is? 2 xercises 4 44 For xercises 49 and 50, use the following information. elative 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. he 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-outh 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 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 x and m 2 y. If x y 24, find x and y. xercises 49, 50 (HIN: m m 2 80.) 2.3 arly efinitions and Postulates KY ONP athematical ystem xiom or Postulate heorem uler Postulate istance egment-ddition Postulate ongruent egments idpoint of a Line egment ay Opposite ays Intersection of wo Geometric Figures Parallel Lines Plane oplanar Points pace HIL Y Like algebra, the branch of mathematics called geometry is a mathematical system. he 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 204 engage Learning. ll ights eserved. ay 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.