### Transcription

GEOMETRYSUCCESSIN 20 MINUTESA DAY2nd Edition NEWYORK

Copyright 2005 LearningExpress, LLC.All rights reserved under International and Pan-American Copyright Conventions.Published in the United States by LearningExpress, LLC, New York.Library of Congress Cataloging-in-Publication Data:Geometry success : in 20 minutes a day—2nd ed.p. cm.ISBN 1-57685-526-0 (pbk.)1. Geometry—Study and teaching. I. Title.QA461.T46 2005516—dc222005047189Printed in the United States of America9 8 7 6 5 4 3 2 1Second EditionISBN 1-57685-526-0For information on LearningExpress, other LearningExpress products, or bulk sales, please write to us at:LearningExpress55 Broadway8th FloorNew York, NY 10006Or visit us at:www.learnatest.com

ContentsINTRODUCTIONviiPRETEST1LESSON 1The Basic Building Blocks of GeometryExplains the basic building blocks of geometry:points, lines, rays, line segments, and planes13LESSON 2Types of AnglesDescribes right, acute, obtuse, and straight angles23LESSON 3Working with LinesDescribes perpendicular, transversal, parallel, and skew linesand shows how to solve problems involving them29LESSON 4Measuring AnglesDescribes how to measure and draw a variety of anglesusing a protractor and how to add and subtract angle measures37LESSON 5Pairs of AnglesExplains the special relationships that exist betweencomplementary, supplementary, and vertical angles45LESSON 6Types of TrianglesShows how to identify scalene, isosceles, equilateral, acute,equilangular, right, and obtuse triangles51iii

– CONTENTS –LESSON 7Congruent TrianglesDefines congruent triangles and shows how to prove thattriangles are congruent59LESSON 8Triangles and the Pythagorean TheoremShows how the various parts of a right triangle are relatedand how to use the Pythagorean theorem and its converseto solve problems67LESSON 9Properties of PolygonsDescribes plane figures such as decagons, hexagons, pentagons,and octagons and shows how to find their interior and exteriorangle measurements77LESSON 10QuadrilateralsDiscusses the various properties of parallelograms andtrapezoids and explains why they are quadrilaterals85LESSON 11Ratio, Proportion, and SimilarityExplains ratio, proportion, and similarity and shows how todiscern if triangles are similar93LESSON 12Perimeter of PolygonsExplains how to find the perimeter of concave and convex polygons103LESSON 13Area of PolygonsExplains how to find the area of polygons111LESSON 14Surface Area of PrismsShows how to find the surface area of prisms121LESSON 15Volume of Prisms and PyramidsExplains how to find the volume of prisms and pyramids127LESSON 16Working with Circles and Circular FiguresExplains how to find the circumference and area of circlesand the surface area and volume of cylinders and spheres;includes a thorough discussion of pi135LESSON 17Coordinate GeometryIntroduces coordinate geometry, describes the coordinate plane,and shows how to plot points and determine distancebetween two points145LESSON 18The Slope of a LineDefines slope and shows how to determine the slope of a line151iv

– CONTENTS –LESSON 19The Equation of a LineDescribes the equation of a line and shows how to writeequations of lines, including horizontal and vertical lines157LESSON 20Trigonometry BasicsShows how to find the trigonometric ratios sine, cosine,and tangent by using right triangles161POSTTEST169ANSWER KEY179GLOSSARY191APPENDIX AList of postulates and theorems from the book195APPENDIX BList of additional resources readers can consult for moreinformation on topics covered in the book199v

IntroductionThis book will help you achieve success in geometry. Reading about math is often slower than readingfor amusement. The only assignment more difficult than working math problems is reading aboutmath problems, so I have included numerous figures and illustrations to help you understand thematerial. Although the title of this book suggests studying each lesson for 20 minutes a day, you should work atyour own pace through the lessons.This book is the next best thing to having your own private tutor. I have tutored for 20 years and have taughtadults and high school students in classroom settings for several years. During that time, I have learned as muchfrom my students about teaching as they have learned from me about geometry. They have given me an insightinto what kinds of questions students have about geometry, and they have shown me how to answer these questions in a clear and understandable way. As you work through the lessons in this book, you should feel as if someone is guiding you through each one. How to Use This BookGeometry Success in 20 Minutes a Day teaches basic geometry concepts in 20 self-paced lessons. The book alsoincludes a pretest, a posttest, a glossary of mathematical terms, an appendix with postulates and theorems, andan appendix of additional resources for further study. Before you begin Lesson 1, take the pretest, which will assessyour current knowledge of geometry. You’ll find the answers in the answer key at the end of the book. Taking thepretest will help you determine your strengths and weaknesses in geometry. After taking the pretest, move on toLesson 1.vii

– INTRODUCTION –If you feel that you need more help with geometry after you complete this book, see Appendix II foradditional resources to help you continue improvingyour geometry skills.Lessons 1–19 offer detailed explanations of basicgeometry topics, and Lesson 20 introduces basictrigonometry. Each lesson includes example problemswith step-by-step solutions. After you study the examples, you’re given a chance to practice similar problems.The answers to the practice problems are in the answerkey located at the back of the book. At the end of eachlesson is an exercise called Skill Building until NextTime. This exercise applies the lesson’s topic to anactivity you may encounter in your daily life sincegeometry is a tool that is used to solve many real-lifeproblems.After you have completed all 20 lessons, take theposttest, which has the same format as the pretest, butthe questions are different. Compare your scores tosee how much you’ve improved or to identify areas inwhich you need more practice. Make a CommitmentSuccess in geometry requires effort. Make a commitment to improve your geometry skills. Work for understanding. Why you do a math operation is as importantas how you do it. If you truly want to be successful,make a commitment to spend the time you need to doa good job. You can do it! When you achieve success ingeometry, you will have laid a solid foundation forfuture challenges and success.So sharpen your pencil and get ready to begin thepretest!viii

PretestBefore you begin the first lesson, you may want to find out how much you already know and howmuch you need to learn. If that’s the case, take the pretest in this chapter, which includes 50multiple-choice questions covering the topics in this book. While 50 questions can’t cover everygeometry skill taught in this book, your performance on the pretest will give you a good indication of your strengthsand weaknesses.If you score high on the pretest, you have a good foundation and should be able to work your way throughthe book quickly. If you score low on the pretest, don’t despair. This book will take you through the geometry concepts, step by step. If you get a low score, you may need more than 20 minutes a day to work through a lesson.However, this is a self-paced program, so you can spend as much time on a lesson as you need. You decide whenyou fully comprehend the lesson and are ready to go on to the next one.Take as much time as you need to complete the pretest. When you are finished, check your answers in theanswer key at the end of the book. Along with each answer is a number that tells you which lesson of this bookteaches you about the geometry skills needed for that question.1

– LEARNINGEXPRESS ANSWER SHEET bbbbbbbbbbbbbccccccccccccccccdddddddddddddddd

– PRETEST – 5. Which is NOT a correct name for the angle?PretestR1. Which is the correct notation for line AB? a. ABb. A B c. BA d. ABSa.b.c.d.2. Which is a correct name for this line?XYZ a. XZ b. ZXc. X Yd. Y ZT RST TSR S R6. Which line is a transversal?rslm3. Which is not a property of a plane?a. is a flat surfaceb. has no thicknessc. has boundariesd. has two dimensionsna.b.c.d.4. Which is a correct name for the angle?line lline mline nline rR7. Which pairs of lines are parallel?rslSa.b.c.d.mT R T TSR RTSna.b.c.d.5lines r and slines l and mlines r and nlines s and l

– PRETEST –10. Find the measure of BOD.8. What is the best way to describe the pair of lines land m?90 Cl55 BD130 m180 skew0 AO55 90 75 130 11. What is the measure of RUS?R9. Find the measure of AOD.S90 C55 BD130 180 Ea.b.c.d.Oa.b.c.d.0 A55 90 75 130 T25 U50 65 75 90 12. What is the measure of JTK?AJB60 TKa.b.c.d.6180 120 60 30

– PRETEST –13. What is the measure of ATJ?17. Given ΔABC ΔXYZ, Y Z corresponds toa. B C.b. A B.c. A C.d. Z Y.JAB60 18. Which postulate could you use to prove ΔFGH ΔPQR?TKa.b.c.d.30 60 120 180 GHa.b.c.d.14. Classify the triangle by its sides.43PFQRAAASASASASSS19. The hypotenuse of ΔHLM is5Ha.b.c.d.scaleneisoscelesequilateralnone of the aboveLMa. H L.b. L M.c. H M.d. L H.15. Classify ΔABC with the following measurements,AB 3, BC 7, and AC 7.a. scaleneb. isoscelesc. equilaterald. none of the above20. A leg of ΔHLM isH16. Given ΔABC ΔXYZ, A corresponds toa. B.b. C.c. X.d. Z.La. H.b. L.c. H L.d. H M.7M

– PRETEST –21. Which figure is a convex polygon?a.22. Which figure is a concave polygon?a.b.b.c.c.d.23. What is the name for a polygon with six sides?a. pentagonb. hexagonc. octagond. decagond.24. Which of the following is not necessarily aparallelogram?a. quadrilateralb. rectanglec. rhombusd. square25. Which of the following is NOT a quadrilateral?a. trapezoidb. parallelogramc. decagond. square8

– PRETEST –L in simplest form.26. Express the ratio JJM3a.b.c.d.7LJ30. Find the perimeter of a square that measures 16inches on one side.a. 16 in.b. 32 in.c. 64 in.d. not enough informationM3 77 33 1 010 331. Find the area of a rectangle with base 7 inchesand height 11 inches.a. 77 in.2b. 36 in.2c. 18 in.2d. 154 in.2LM 27. Express the ratio JL in simplest form.3a.b.c.d.7LJM3 77 33 1 010 332. Find the area of a parallelogram with base 5 cmand height 20 cm.a. 100 cm 2b. 50 cm 2c. 25 cm 2d. 15 cm 228. Which of the following is a true proportion?a. 3:7 5:8b. 1:2 4:9c. 6:3 10:6d. 2:5 4:1033. Find the area of a square that measures 11 yardson an edge.a. 44 yd.2b. 121 yd.2c. 22 yd.2d. 242 yd.229. Find the perimeter of the polygon.2 cm2 cm3 cm34. Which segment does not equal 3 m?2 cmIa.b.c.d.9 cm18 cm36 cm72 cmJKH1mMLa. L Ob. H Kc. I Jd. L M9N2m3mO

– PRETEST –35. Use the formula SA 2(lw wh lh) to find thesurface area of the prism.IJKH1mMLa.b.c.d.39. Find the circumference of a circle with adiameter of 21 inches. Use 3.14 for π.a. 32.47 in.b. 129.88 in.c. 756.74 in.d. 65.94 in.N2m3m40. Find the area of a circle with a diameter of 20 cm.Use 3.14 for π.a. 628 cm 2b. 62.8 cm 2c. 314 cm 2d. 31.4 cm 2O22 m 211 m 244 m 288 m 236. Find the volume of a prism with length 12inches, width 5 inches, and height 8 inches.a. 60 in.3b. 40 in.3c. 480 in.3d. 96 in.341. Which quadrant would you graph the point(5,–6)?a. Ib. IIc. IIId. IV37. Find the volume of the triangular prism.42. The coordinates for point A arey7 cmA3 cma.b.c.d.4 cmx14 cm 321 cm 342 cm 384 cm 3Ba.b.c.d.38. Find the volume of a prism with base area 50 m 2and height 7 m. Use V bh.a. 17,500 m 3b. 57 m 3c. 175 m 3d. 350 m 310(3,–2).(–2,3).(–3,2).(2,–3).

– PRETEST –47. Which of the following is not a linear equation?a. x 4b. y –4c. 12 x 13 y 743. The coordinates for point B areyAd.x48. Which ordered pair satisfies the equation3x 4y 12?a. (0,0)b. (2,2)c. (1,4)d. (0,3)Ba.b.c.d.2 yx(3,–2).(–2,3).(–3,2).(2,–3).49. The ratio of the opposite leg to an adjacent leg isthe trigonometric ratioa. sine.b. cosine.c. tangent.d. hypotenuse.44. A line that points up to the right has a slopethat isa. positive.b. negative.c. zero.d. undefined.50. The ratio of the adjacent leg to the hypotenuse isthe trigonometric ratioa. sine.b. cosine.c. tangent.d. hypotenuse.45. A vertical line has a slope that isa. positive.b. negative.c. zero.d. undefined.46. Which of the following is a linear equation?a. 1x 0 yb. 3x 2y 2 10c. 3x 2 2y 10d. 3x 2y 1011

L E S S O N1The BasicBuilding Blocksof GeometryLESSON SUMMARYThis lesson explains the basic building blocks of geometry: points, lines,rays, line segments, and planes. It also shows you the basic properties you need to understand and apply these terms.The term geometry is derived from the two Greek words geo and metric. It means “to measure the Earth.”The great irony is that the most basic building block in geometry, the point, has no measurement atall. You must accept that a point exists in order to have lines and planes because lines and planes aremade up of an infinite number of points. If you do not accept the assumption that a point can exist without sizeor dimension, then the rest of this lesson (and book) cannot exist either. Therefore, this is a safe assumption tomake! Let’s begin this lesson by looking at each of the basic building blocks of geometry. PointsA point has no size and no dimension; however, it indicates a definite location. Some examples are: a pencil point,a corner of a room, and the period at the end of this sentence. A series of points are what make up lines, linesegments, rays, and planes. There are countless points on any one line. A point is named with an italicized uppercase letter placed next to it: AIf you want to discuss this point with someone, you would call it “point A.”13

– THE BASIC BUILDING BLOCKS OF GEOMETRY – LinesA line is straight, but it has no thickness. It is an infinite set of points that extends in both directions. Imagine astraight highway with no end and no beginning; it is an example of a line. A line is named by any one italicizedlowercase letter or by naming any two points on the line. If the line is named by using two points on the line, asmall symbol of a line (with two arrows) is written above the two letters. For example, this line could be referred to as line AB or line BA:BAPractice 1. Are there more points on AB than point A and point B?2. What are three examples of a point in everyday life?3. Why would lines, segments, rays, or planes not exist if points do not exist?4. Write six different names for this line.XYZ5. How many points are on a line?6. Why do you think the notation for a line has two arrowheads?14

– THE BASIC BUILDING BLOCKS OF GEOMETRY – RaysA ray is a part of a line with one endpoint. It has an infinite number of points on it. Flashlights and laser beams are good examples of rays. When you refer to a ray, you always name the endpoint first. The ray below is ray AB, not ray BA.BA Line SegmentsA line segment is part of a line with two endpoints. It has an infinite number of points on it. A ruler and a baseboard are examples of line segments. Line segments are also named with two italicized uppercase letters, but thesymbol above the letters has no arrows. This segment could be referred to as AB or BA :BAPracticeR7. Name two different rays with endpoint S.ST8. Why is it important to name the endpoint of a ray first? 9. Why are ray AB and ray BA not the same?L10. Name six different line segments shown.11. Why are arrowheads not included in line segment notation?12. How many points are on a line segment?15MNP

– THE BASIC BUILDING BLOCKS OF GEOMETRY – PlanesA plane is a flat surface that has no thickness or boundaries. Imagine a floor that extends in all directions as faras you can see. Here is what a plane looks like:BCAWhen you talk to someone about this plane, you could call it “plane ABC.” However, the more common practice is to name a plane with a single italicized capital letter (no dot after it) placed in the upper-right corner of thefigure as shown here:XIf you want to discuss this plane with someone, you would call it “plane X.”16

– THE BASIC BUILDING BLOCKS OF GEOMETRY –THE BASIC BUILDING BLOCKS OF GEOMETRYFIGURENAMESYMBOL READ ASPROPERTIES APoint A has no size pencil point has no dimension corner of a room indicates a definitelocation named with an italicizeduppercase letterA BLineAB or BAA BRayABA B— point A l BA EXAMPLESline AB or BA is straight highway without has no thicknessboundariesor line l an infinite set of points hallway without boundsthat extends inopposite directions one dimensionray AB(endpoint isalways first) is part of a line flashlight has only one endpoint laser beam an infinite set of pointsthat extends in onedirection one dimensionLineA B or B A segmentsegment ABor BA is part of a line edge of a ruler has two endpoints base board an infinite set of points one dimensionPlaneplane ABCor plane X is a flat surface floor without boundaries has no thickness surface of a football field an infinite set of points without boundariesthat extends in alldirections two dimensionsNoneCX17

– THE BASIC BUILDING BLOCKS OF GEOMETRY –Practice13. A line is different from a ray because.14. A line is similar to a ray because.15. A ray is different from a segment because.16. A ray is similar to a segment because.17. A plane is different from a line because. Working with the Basic Building Blocks of Geometr yPoints, lines, rays, line segments, and planes are very important building blocks in geometry. Without them, youcannot work many complex geometry problems. These five items are closely related to each other. You will usethem in all the lessons that refer to plane figures—figures that are flat with one or two dimensions. Later in thebook, you will study three-dimensional figures—figures that occur in space. Space is the set of all possible pointsand is three-dimensional. For example, a circle and a square are two-dimensional and can occur in a plane. Therefore, they are called plane figures. A sphere (ball) and a cube are examples of three-dimensional figures that occurin space, not a plane.One way you can see how points and lines are related is to notice whether points lie in a straight line. Collinearpoints are points on the same line. Noncollinear points are points not on the same line. Even though you may nothave heard these two terms before, you probably can correctly label the following two figures based on the soundof the names collinear and noncollinear.Collinear pointsNoncollinear points18

– THE BASIC BUILDING BLOCKS OF GEOMETRY –A way to see how points and planes are related is to notice whether points lie in the same plane. Forinstance, see these two figures:Coplanar pointsNoncoplanar pointsAgain, you may have guessed which name is correct by looking at the figures and seeing that in the figurelabeled Coplanar points, all the points are on the same plane. In the figure labeled Noncoplanar points, the pointsare on different planes.Practice18. Can two points be noncollinear? Why or why not?19. Can three points be noncollinear? Why or why not?20. Can coplanar points be noncollinear? Why or why not?21. Can collinear points be coplanar? Why or why not?19

– THE BASIC BUILDING BLOCKS OF GEOMETRY – Postulates and TheoremsYou need a few more tools before moving on to other lessons in this book. Understanding the terms of geometryis only part of the battle. You must also be able to understand and apply certain facts about geometry and geometric figures. These facts are divided into two categories: postulates and theorems. Postulates (sometimes calledaxioms) are statements accepted without proof. Theorems are statements that can be proved. You will be using bothpostulates and theorems in this book, but you will not be proving the theorems.Geometry is the application of definitions, postulates, and theorems. Euclid is known for compiling all thegeometry of his time into postulates and theorems. His masterwork, The Elements, written about 300 b.c., is thebasis for many geometry books today. We often refer to this as Euclidean geometry.Here are two examples of postulates:Postulate: Two points determine exactly one line.Postulate: Three noncollinear points determine exactly one plane.PracticeState whether the points are collinear.DCMBAEF22. A, B, C23. A, E, F24. B, D, F25. A, E20

– THE BASIC BUILDING BLOCKS OF GEOMETRY –State whether the points are coplanar.DCMBAEF26. A, B, C, E27. D, B, C, E28. B, C, E, F29. A, B, EState whether the following statements are true or false. 30. XY and YX are the same line.31. XY and YX are the same ray.Y and Y X are the same segment.32. X 33. Any four points W, X, Y, and Z must lie in exactly one plane.Draw and label a figure to fit each description, if possible. Otherwise, state “not possible.”34. four collinear points35. two noncollinear points36. three noncoplanar points21

– THE BASIC BUILDING BLOCKS OF GEOMETRY –Skill Building until Next TimeWhen you place a four-legged table on a surface and it wobbles, then one of the legs is shorter than theother three. You can use three legs to support something that must be kept steady. Why do you think thisis true?Throughout the day, be on the lookout for some three-legged objects that support this principle.Examples include a camera tripod and an artist’s easel. The bottoms of the three legs must represent threenoncollinear points and determine exactly one plane.22

L E S S O N2Types of AnglesLESSON SUMMARYThis lesson will teach you how to classify and name several types ofangles. You will also learn about opposite rays.People often use the term angle in everyday conversations. For example, they talk about camera angles,angles for pool shots and golf shots, and angles for furniture placement. In geometry, an angle isformed by two rays with a common endpoint. The symbol used to indicate an angle is . The tworays are the sides of the angle. The common endpoint is the vertex of the angle. In the following figure, the sidesare RD and RY, and the vertex is R.DR YNaming AnglesPeople call you different names at different times. Sometimes, you are referred to by your full name, and othertimes, only by your first name or maybe even a nickname. These different names don’t change who you are—justthe way in which others refer to you. You can be named differently according to the setting you’re in. For example,23

– TYPES OF ANGLES –you may be called your given name at work, but your friends might call you by a nickname. There can be confusion sometimes when these names are different.Similar to you, an angle can be named in different ways. The different ways an angle can be named may beconfusing if you do not understand the logic behind the different methods of naming.If three letters are used to name an angle, then the middle letter always names the vertex. If the angle doesnot share the vertex point with another angle, then you can name the angle with only the letter of the vertex. If anumber is written inside the angle that is not the angle measurement, then you can name the angle by that number. You can name the following angle any one of these names: WET, TEW, E, or 1.W1ETPractice1. Name the vertex and sides of the angle.XZY2. Name the vertex and sides of the angle.TJA3. Name the angle in four different ways.KT1B24

– TYPES OF ANGLES –Right AnglesAngles that make a square corner are called right angles. In drawings, the following symbol is used to indicate aright angle:Straight AnglesOpposite rays are two rays with the same endpoint and that form a line. They form a straight angle. In the following figure, HD and HS are opposite rays.DHSPracticeUse the following figure to answer practice problems 4–7.PNOM4. Name two right angles.5. Name a straight angle.6. Name a pair of opposite rays.7. Why is O an incorrect name for any of the angles shown?25

– TYPES OF ANGLES –Use the following figure to answer practice problems 8 and 9.LNM 8. Are LM and MN opposite rays? Why or why not?9. If two rays have the same endpoints, do they have to be opposite rays? Why or why not? Classifying AnglesAngles are often classified by their measures. The degree is the most commonly used unit for measuring angles.One full turn, or a circle, equals 360 .Acute AnglesAn acute angle has a measure between 0 and 90 . Here are two examples of acute angles:89 45 Right AnglesA right angle has a 90 measure. The corner of a piece of paper will fit exactly into a right angle. Here are two examples of right angles:Obtuse AnglesAn obtuse angle has a measure between 90 and 180 . Here are two examples of obtuse angles:91 170 26

– TYPES OF ANGLES –Straight AnglesA straight angle has a 180 measure. This is an example of a straight angle ( ABC is a straight angle):180 ABCPracticeUse the following figure to answer practice problems 10–13.JKNOLM10. Name three acute angles.11. Name three obtuse angles.12. Name two straight angles.13. If MON measures 27 , then JOK measures degrees.Complete each statement.14. An angle with measure 90 is called a(n) angle.15. An angle with measure 180 is called a(n) angle.16. An angle with a measure between 0 and 90 is called a(n) angle.17. An angle with a measure between 90 and 180 is called a(n) angle.27

– TYPES OF ANGLES –Questions 18–21 list the measurement of an angle. Classify each angle as acute, right, obtuse, or straight.18. 10 20. 98 19. 180 21. 90 You may want to use a corner of a piece of scratch paper for questions 22–25. Classify each angle as acute,right, obtuse, or straight.22.23.24.25.Skill Building until Next TimeIn conversational English, the word acute can mean sharp and the word obtuse can mean dull or not sharp.How can you relate these words with the drawings of acute and obtuse angles?28

L E S S O N3Working withLinesLESSON SUMMARYThis lesson introduces you to perpendicular, transversal, parallel, andskew lines. The angles formed by a pair of parallel lines and a transversal are also explained.Both intersecting and nonintersecting lines surround you. Most of the time you do not pay muchattention to them. In this lesson, you will focus on two different types of intersecting lines: transversals and perpendicular lines. You will also study nonintersecting lines: parallel and skew lines.You will learn properties about lines that have many applications to this lesson and throughout this book. You’llsoon start to look at the lines around you with a different point of view. Intersecting LinesOn a piece of scratch paper, draw two straight lines that cross. Can you make these straight lines cross at morethan one point? No, you can’t, because intersecting lines cross at only one point (unless they are the same line).The point where they cross is called a point of intersection. They actually share this point because it is on both lines.Two special types of intersecting lines are called transversals and perpendicular lines.29

– WORKING WITH LINES –TransversalsA transversal is a line that intersects two or more other lines, each at a different point. In the following figure, line tis a transversal, line s is not.stmnThe prefix trans means to cross. In the previous figure, you can see that line t cuts across the two lines m andn. Line m is a transversal for lines s and t. Also, line n is a transversal across lines s and t. Line s crosses lines m and nat the same point (their point of intersection); therefore, line s is not a transversal. A transversal can cut acrossparallel as well as intersecting lines, as shown here:tmnPracticeUse the following figure to answer questions 1–4.dytr1. Is line d a transversal? Why or why not?2. Is line y a transversal? Why or why not?3. Is line t a transversal? Why or why not?4. Is line r a transversal? Why or why not?30

– WORKING WITH LINES –Perpendicular LinesPerpendicular lines are another type of intersecting lines. Everyday examples of perpendicular lines include the horizontal and vertical lines of a plaid fabric and the lines formed by panes in a window. Perpendicular lines meet toform right angles. Right angles always measure 90 . In the following figure, lines x and y are perpendicular:yxThe symbol “ ” means perpendicular. You could write x y to show these lines are perpendicular. Also, thesymbol that makes a square in the corner where lines x and y meet indicates a right angle. In geometry, youshouldn’t assume anything without being told. Never assume a pair of lines are perpendicular without one of thesesymbols. A transversal can be perpendicular to a pair of lines, but it does not have to be. In the following figure,line t is perpendicular to both line l and line m.tlmPracticeState whether the following statements are true or false.5. Perpendicular lines always form right angles.6. The symbol “ ” means parallel.7. Transversals must always be parallel.31

– WORKING WITH LINES –Nonintersecting LinesIf lines do not intersect, then they are either parallel or skew.jlmkLines l and m are parallel.Lines l and m are coplanar lines.Lines l and m do not intersect.Lines j and k are skew.Lines j and k are noncoplanar lines.Lines j and k do not intersect.The symbol “ ” means parallel. So you can abbreviate the sentence, “Lines l and m are parallel,” by writing“l m.” Do not assume a pair of lines are parallel unless it is indicated. Arrowheads on the lines in a figure indicate that the lines are parallel. Sometimes, double arrowheads are ne

LESSON 1 The Basic Building Blocks of Geometry 13 Explains the basic building blocks of geometry: points, lines, rays, line segments, and planes LESSON 2 Types of Angles 23 Describes right, acute, obtuse, and straight angles LESSON 3 Working with Lines 29 Descr