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– 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