FM Geo C&A SE-872987 - Glencoe

GeometryConcepts and ApplicationsContributing AuthorDinah ZikeConsultantDouglas Fisher, PhDDirector of Professional DevelopmentSan Diego State UniversitySan Diego, CA

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Printed in theUnited States of America. Except as permitted under the United States CopyrightAct, no part of this book may be reproduced in any form, electronic or mechanical,including photocopy, recording, or any information storage or retrieval system,without prior written permission of the publisher.Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027ISBN: 0-07-872987-4Geometry: Concepts and Applications (Student Edition)Noteables: Interactive Study Notebook with Foldables3 4 5 6 7 8 9 10 024 09 08 07 0

ContentsFoldables. . . . . . . . . . . . . . . . . . . . . . . . . . 1Vocabulary Builder . . . . . . . . . . . . . . . . . . 21-1 Patterns and Inductive Reasoning . . 41-2 Points, Lines, and Planes . . . . . . . . . 71-3 Postulates . . . . . . . . . . . . . . . . . . . . 91-4 Conditional Statements . . . . . . . . . 111-5 Tools of the Trade . . . . . . . . . . . . . 131-6 A Plan for Problem Solving . . . . . . 16Study Guide . . . . . . . . . . . . . . . . . . . . . . 19Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 87Vocabulary Builder . . . . . . . . . . . . . . . . . 885-1 Classifying Triangles. . . . . . . . . . . . 905-2 Angles of a Triangle . . . . . . . . . . . 935-3 Geometry in Motion . . . . . . . . . . . 955-4 Congruent Triangles . . . . . . . . . . . 985-5 SSS and SAS . . . . . . . . . . . . . . . . . 1005-6 ASA and AAS . . . . . . . . . . . . . . . . 102Study Guide . . . . . . . . . . . . . . . . . . . . . 104Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 23Vocabulary Builder . . . . . . . . . . . . . . . . . 242-1 Real Numbers and Number Lines . 262-2 Segments and Properties ofReal Numbers . . . . . . . . . . . . . . . . 282-3 Congruent Segments . . . . . . . . . . . 312-4 The Coordinate Plane . . . . . . . . . . 342-5 Midpoints . . . . . . . . . . . . . . . . . . . 37Study Guide . . . . . . . . . . . . . . . . . . . . . . 40Foldables. . . . . . . . . . . . . . . . . . . . . . . . 107Vocabulary Builder . . . . . . . . . . . . . . . . 1086-1 Medians . . . . . . . . . . . . . . . . . . . . 1106-2 Altitudes and Perp. Bisectors . . . . 1126-3 Angle Bisectors of Triangles. . . . . 1156-4 Isosceles Triangles . . . . . . . . . . . . 1176-5 Right Triangles. . . . . . . . . . . . . . . 1196-6 The Pythagorean Theorem . . . . . 1216-7 Distance on a Coordinate Plane . . 123Study Guide . . . . . . . . . . . . . . . . . . . . . 125Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 43Vocabulary Builder . . . . . . . . . . . . . . . . . 443-1 Angles . . . . . . . . . . . . . . . . . . . . . . 463-2 Angle Measure . . . . . . . . . . . . . . . 483-3 The Angle Addition Postulate . . . . 513-4 Adjacent Angles & Linear Pairs . . . 543-5 Comp. and Supp. Angles . . . . . . . . . 563-6 Congruent Angles. . . . . . . . . . . . . . 583-7 Perpendicular Lines. . . . . . . . . . . . . 60Study Guide . . . . . . . . . . . . . . . . . . . . . . 62Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 65Vocabulary Builder . . . . . . . . . . . . . . . . . 664-1 Parallel Lines and Planes . . . . . . . . . 684-2 Parallel Lines and Transversals . . . . . 704-3 Transversals and Corres. Angles. . . . 734-4 Proving Lines Parallel . . . . . . . . . . . . 764-5 Slope . . . . . . . . . . . . . . . . . . . . . . . . . 784-6 Equations of Lines . . . . . . . . . . . . . 80Study Guide . . . . . . . . . . . . . . . . . . . . . . 83Foldables. . . . . . . . . . . . . . . . . . . . . . . . 129Vocabulary Builder . . . . . . . . . . . . . . . . 1307-1 Segments, Angles, Inequalities . . . 1317-2 Exterior Angle Theorem . . . . . . . 1347-3 Inequalities Within a Triangle . . . 1377-4 Triangle Inequality Theorem . . . . 139Study Guide . . . . . . . . . . . . . . . . . . . . . 142Foldables. . . . . . . . . . . . . . . . . . . . . . . . 145Vocabulary Builder . . . . . . . . . . . . . . . . 1468-1 Quadrilaterals . . . . . . . . . . . . . . . 1488-2 Parallelograms . . . . . . . . . . . . . . . 1508-3 Tests for Parallelograms. . . . . . . . 1528-4 Rectangles, Rhombi, and Squares . 1548-5 Trapezoids . . . . . . . . . . . . . . . . . . 156Study Guide . . . . . . . . . . . . . . . . . . . . . 159Geometry: Concepts and Applicationsiii

Organizing Your FoldablesMake this Foldable to help you organize andstore your chapter Foldables. Begin with onesheet of 11" 17" paper.FoldFold the paper in half lengthwise. Then unfold.Fold and GlueFold the paper in half widthwise and glue all of the edges.Glue and LabelGlue the left, right, and bottom edges of the Foldableto the inside back cover of your Noteables notebook.FoldablesOrganizer© Glencoe/McGraw-HillReading and Taking Notes As you read and study each chapter, recordnotes in your chapter Foldable. Then store your chapter Foldables insidethis Foldable organizer.Geometry: Concepts and Applicationsv

This note-taking guide is designed to help you succeed in Geometry: Conceptsand Applications. Each chapter includes:C H A P T E R12Surface Area and VolumeUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with a plain piece of 11" 17" paper.FoldFold the paper inthirds lengthwise.OpenOpen and fold a 2"tab along the shortside. Then fold therest in fifths.The Chapter Openercontains instructions andillustrations on how to makea Foldable that will help youto organize your notes.DrawDraw lines alongfolds and label asshown.SurfaceArea Volume Ch. 12PrismsCylindersPyramidsConesSpheres© Glencoe/McGraw-HillA Note-Taking Tipprovides a helpfulhint you can usewhen taking notes.NOTE-TAKING TIP: When taking notes, explaineach new idea or concept in words and give oneor more examples.Geometry: Concepts and Applications 227The Build Your Vocabularytable allows you to writedefinitions and examplesof important vocabularyterms together in oneconvenient place.Chapter 12C H A P T E R12BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 12. Asyou complete the study notes for the chapter, you will see Build Your Vocabularyreminders to complete each term’s definition or description on these pages.Remember to add the textbook page number in the second column for referencewhen you study.axisVocabulary Termcomposite solidconecubecylinder[SIL-in-dur]edgefacelateral area[LAT-er-ul]lateral edgelateral facenetoblique cone[oh-BLEEK]oblique cylinderoblique prismFoundon PageDefinitionDescription orExampleWithin each chapter,Build Your Vocabularyboxes will remind youto fill in this table.© Glencoe/McGraw-Hill© Glencoe/McGraw-Hill228 Geometry: Concepts and ApplicationsviGeometry: Concepts and Applications

C H A P T E R1Reasoning in GeometryUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Chapter 1Begin with a sheet of 8 1 2 " 11" paper. sequences patterns conjectures conclusionsFoldFold lengthwise in fourths.DrawDraw lines along thefolds and label eachcolumn sequences,patterns, conjectures,and conclusions.© Glencoe/McGraw-HillNOTE-TAKING TIP: When you are taking notes, besure to be an active listener by focusing on whatyour teacher is saying.Geometry: Concepts and Applications 1

C H A P T E R1BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 1.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplecollinear[co-LIN-ee-ur]compassconclusionconditional statementconjecture[con-JEK-shoor]constructioncontrapositive[con-tra-PAS-i-tiv]conversecoplanar[co-PLAY-nur]counterexampleendpoint© Glencoe/McGraw-Hillformula2 Geometry: Concepts and Applications

1–1Your TurnDraw the next figure in the pattern.. . .BUILD YOUR VOCABULARY (page 2)A conjecture is areasoning.based on inductiveAn example that shows that a conjecture is notis a counterexample.Minowa studied the data below and made the followingconjecture. Find a counterexample for her conjecture.Multiplying a number by1 produces a productthat is less than 1.Number (1)5(1)15(1)100(1)300(1)Product515100300The product of 2 and 1 is 2 but 21. So, theconjecture is .HOMEWORKASSIGNMENTPage(s):Exercises:Your TurnFind a counterexample for this statement:Division of a positive number by another positive numberproduces a quotient less than the dividend.© Glencoe/McGraw-Hill6 Geometry: Concepts and Applications

1–2Points, Lines, and PlanesBUILD YOUR VOCABULARY (pages 2–3)WHAT YOU’LL LEARN• Identify and drawmodels of points,lines, and planes,and determine theircharacteristics.A point is the basic unit of geometry.A series of points that extends without end indirections is a line.Points that lie on the samecollinear.are said to bePoints that do not lie on the same line are said to benoncollinear.A ray is part of a line that has a definite starting pointand extends without end indirection.A line segment has a definite beginning and .Name two points on the line.Two points are point anddABpoint .© Glencoe/McGraw-HillGive three names for the line.Any two points on the line or the script letter can be used toname it. Three names are .Your TurnRefer to the figure shown.a. Name two points on the line.Lmb. Give three names for the line.KGeometry: Concepts and Applications 7

1–2Name three points that are collinearand three points that are noncollinear.Points M, P, and Q, are .Points N, P, and Q are.NMPQREMEMBER ITThe order of theletters that identify aline can be switched butthe order of the lettersthat identify a raycannot.Name three segments and one ray.Three of the segments are.One ray is ray .D E FGYour TurnRefer to the figure.a. Name three collinear points andthree noncollinear points.GCFBDAEb. Name three segments and one ray.HOMEWORKASSIGNMENTBUILD YOUR VOCABULARY (pages 2–3)A plane is a surface that extends withoutend in all directions.© Glencoe/McGraw-HillPage(s):Exercises:Points that lie on the samePoints that do not lie on the samenoncoplanar.are coplanar.are8 Geometry: Concepts and Applications

1–3PostulatesWHAT YOU’LL LEARN• Identify and use basicpostulates about points,lines, and planes.BUILD YOUR VOCABULARY (page 3)Postulates arein geometry that areaccepted as .Postulate 1–1Postulate 1–2Postulate 1–3Two points determine a unique line.If two distinct lines intersect, then theirintersection is a point.Three noncollinear points determine aunique plane.In the figure, points K, L, and M arenoncollinear.Name all of the different lines that canbe drawn through these points.There is only one line through each pair of points.Therefore, the lines that contain points K, L, and M,taken two at a time, are .Name the intersection of KL and KM.KLM© Glencoe/McGraw-HillThe intersection of KL and KM is .Your TurnRefer to the figure.a. Name three different lines.AGJBKHCILb. Name the intersection of AC and BH.DEFGeometry: Concepts and Applications 9

1–3REMEMBER ITThree noncollinearpoints determine aunique plane.Name all of the planes that are represented in the prism.A BEFCDGHThere are eight points, A, B, C, D, E, F, G, and H.There is onlyplane that contains three noncollinearpoints. The different planes are planes.Your TurnName four different planes in the figure.GHIAJBKCLDEFPostulate 1–4If two distinct planes intersect, then theirintersection is a line.HOMEWORKASSIGNMENTPage(s):Exercises:Name the intersection of planeABC and plane DEF.The intersection is .Your TurnName theintersection of plane ABD andplane DJK.CBAGJBADKHHCGILFE© Glencoe/McGraw-HillDEF10 Geometry: Concepts and Applications

1–4Conditional Statements and Their ConversesWHAT YOU’LL LEARN• Write statements inif-then form and writethe converses of thestatements.BUILD YOUR VOCABULARY (pages 2–3)If-then statements join two statements based on acondition.If-then statements are also known as conditionalstatements.In a conditional statement the part following if is thehypothesis. The part following then is the conclusion.Identify the hypothesis and conclusion in thisstatement.If it is raining, then we will read a book.Hypothesis:Conclusion:Write two other forms of this statement.If two lines are parallel, then they never intersect.Allnever intersect.Lines never if they are .© Glencoe/McGraw-HillREVIEW ITHow can you show thata conjecture is false?(Lesson 1-1)Your Turna. Identify the hypothesis and conclusion in this statement.If you ski, then you like snow.b. Write two other forms of this statement. If a figure is arectangle, then it has four angles.Geometry: Concepts and Applications 11

1–4BUILD YOUR VOCABULARY (page 2)The converse of a conditional statement is formed byexchanging theand the conclusion.Write the converse of this statement.REMEMBER ITThe converse of atrue statement is notnecessarily true.If today is Saturday, then there is no school.If there is , then .Your TurnWrite the converse of this statement.If it is 30° F, then it is cold.Write the statement in if-then form. Then write theconverse of the statement.Every member of the jazz band must attend therehearsal on Saturday.If-then form: If ais a member of the jazz band,then he or she must attend.Converse:If a studentHOMEWORKASSIGNMENTPage(s):Exercises:on Saturday, then he or she is amember.Your TurnWrite the statement in if-then form. Then writethe converse of the statement. People who live in glass housesshould not throw stones.© Glencoe/McGraw-Hill12 Geometry: Concepts and Applications

1–5Tools of the TradeWHAT YOU’LL LEARN• Use geometry tools.BUILD YOUR VOCABULARY (pages 2–3)A straightedge is an object used to draw aA compass is commonly used for drawing arcs and.In geometry, figures drawn using only aline.and aare constructions.The midpoint is the in the of aline segment.Find two lines or segments in a classroom that appearto be parallel. Use a ruler to determine whether theyare parallel.The opposite sides of a textbook represent two segments thatappear to be parallel.• Choose two points on one side of the textbook.• Place the 0 mark of the ruler on each point. Make sure theruler is perpendicular to the side at each chosen point.© Glencoe/McGraw-Hill• Measure the distance to the second side. If the distances are, then the sides are .Your TurnFind another pair of lines or segments in aclassroom that appear to be parallel. Use a ruler or a yardstickto determine if they are parallel.Geometry: Concepts and Applications 13

1–5On the figure shown, mark a point C on line that youjudge will create BC that is the same length as AB. Thenmeasure to determine how accurate your guess was.ABTo draw an exact recreation of the length, place the point of acompass on point B. Place the point of the pencil on point. Then draw a small arc on line without changing thesetting of the compass. This duplicates the measure of .REMEMBER ITAn arc is part of acircle.Use a compass and a straightedge to construct asix-pointed star.Use the compass to draw a circle. Then using the samecompass setting, put the compass point on the circle and drawa small arc on the circle.Move the compass point to the arc and, without changing thecompass setting, draw another arc along the circle. Continueuntil there are six arcs.Draw two triangles by connecting alternating marks, resultingin a six-pointed star.© Glencoe/McGraw-Hill14 Geometry: Concepts and Applications

1–5Your Turna. On the figure given, mark point Z on line m that you judgewill create WZ that is the same length as XY. Thenmeasure to determine the accuracy of your guess.XWYmb. Use a compass and a straightedge to construct a trianglewith sides of equal length.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 15

1–6A Plan for Problem SolvingWHAT YOU’LL LEARN• Use a four-step plan tosolve problems thatinvolve the perimetersand areas of rectanglesand parallelograms.BUILD YOUR VOCABULARY (page 2)A formula is anthat shows how certainquantities are related.KEY CONCEPTSPerimeter of a RectangleThe perimeter P of arectangle is the sum ofthe measures of its sides.It can also be expressedas two times the length plus two times thewidth w.Area of a RectangleThe area A of arectangle is the productof the length and thewidth w.a. Find the perimeter of a rectangle with length12 centimeters and width 3 centimeters.P 2 2wP 2 2P or centimetersb. Find the perimeter of a square with side 10 feet long.P 2 2wP 2(10) 2(10)P or feeta. Find the area of a rectangle with length12 kilometers and width 3 kilometers.A wA A square kilometersb. Find the area of a square with sides 10 yards long.A wA © Glencoe/McGraw-HillA square yards16 Geometry: Concepts and Applications

1–6WRITE ITWhat is the differencebetween perimeter andarea?Your Turna. Find the perimeter of a rectangle with length 11 meters andwidth 4 meters.b. Find the perimeter of a square with sides 7 centimeters long.c. Find the area of a rectangle with length 14 inches andwidth 4 inches.d. Find the area of a square with sides 11 feet long.© Glencoe/McGraw-HillKEY CONCEPTArea of a ParallelogramThe area of aparallelogram is theproduct of the base band the height h.Find the area of a parallelogram with a height of4 meters and a base of 5.5 meters.A bhA A square metersYour TurnFind the area of a parallelogram with a heightof 6.4 inches and a base length of 10 inches.Geometry: Concepts and Applications 17

1–6KEY CONCEPTProblem-Solving Plan1. Explore the problem.2. Plan the solution.3. Solve the problem.A door is 3-feet wide and 6.5-feet tall. Chad wants topaint the front and back of the door. A one-pint can ofpaint will cover about 15 ft 2 . Will two one-pint cans ofpaint be enough?EXPLORE You know the dimensions of the door and thatone-pint can of paint covers about .4. Examine the solution.PLANSOLVEUse the formula for the area of ato find the total area of the two sides of the doorto be covered with paint.Area of both sides of the doorA 2wA 2 REMEMBER ITAbbreviations forunits of area haveexponent 2.Square foot ft 2Square meter m 2EXAMINEOne pint covers 15 ft 2 . Two one-pint cans cover2(15) or ft 2 . So, two one-pint cans willbe enough.Since the area of one side of the door is (3)(6.5) or19.5 ft 2 the answer is reasonable.Chad will needone-pint cans of paint.Your TurnA building contractor needs to build arectangular deck with an area of 484 ft 2 . The side lengthsmust be whole numbers. The perimeter must be less than260 ft. What are the possible dimensions for the deck?HOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill18 Geometry: Concepts and Applications

C H A P T E R1BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 1 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 1, go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder (pages 2–3)to help you solve the puzzle.1-1Patterns and Inductive ReasoningFind the next three terms in the sequence.1. 1, 1, 2, 3, 5, . . . 2. 1, 2, 4, 8, 16, . . .3. Draw the next figure in the pattern.1-2Points, Lines, and PlanesUse the figure to match the example to the correct term.© Glencoe/McGraw-Hill4. collinear points5. segment6. plane7. raya. G, F, Cb. PBc. ADd. PEe. GBEGAFPBECDGeometry: Concepts and Applications 19

Chapter1BRINGING IT ALL TOGETHER1-3PostulatesComplete the sentence.8. A(n) is a statement in geometry that isaccepted as true without proof.Identify three planes in the figure shown.B9. 10.AC11.ED12. Refer to the above figure. Where do planes ACF andFDEF intersect?a. point F b. DF--- c. plane DEF d. point D1-4Conditional Statements and Their ConversesUnderline the correct term that completes each sentence.13. The “if” part of the if-then statement is thehypothesis/conclusion.14. The “then” part of the if-then statement is thehypothesis/conclusion.15. Rewrite the statement in if-then form.Students who complete all assignments score higher on tests.16. Write the converse of the statement.If it is Saturday, then there is no school.© Glencoe/McGraw-Hill20 Geometry: Concepts and Applications

Chapter1BRINGING IT ALL TOGETHER1-5Tools of the TradeMatch the geometry tool to its function.17. compass18. straightedge19. protractor20. patty papera. to plot pointsb. to draw arcs and circlesc. to measure anglesd. to draw lines in constructionse. to find the midpoint in constructions21. Indicate whether the statement is true or false.A conjecture is a special drawing that is created using only astraightedge and compass.1-6A Plan for Problem SolvingComplete each sentence.22. The is the distance around the edges of a figure.23. The formula for the area of a rectangle is .24. is the formula to find the area of aparallelogram.© Glencoe/McGraw-Hill25. Find the area of a rectangle with length 8 feet and width9 feet.26. A framer must frame a piece of art. The frame is 1 1 2 incheswide, and its outer edge measures 24 inches by 36 inches.What is the area of the piece of art displayed in the center ofthe frame?Geometry: Concepts and Applications 21

C H A P T E R1ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 1.• You may want to take the Chapter 1 Practice Test on page 45of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 1 Study Guide and Reviewon pages 42–44 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 1 Practice Test onpage 45 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 1 Foldable.• Then complete the Chapter 1 Study Guide and Review onpages 42–44 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 1 Practice Test onpage 45 of your textbook.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature22 Geometry: Concepts and Applications

C H A P T E R2Segment Measure and Coordinate GraphingUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with a sheet of notebook paper.FoldFold lengthwise to theholes.Chapter 2CutCut along the top line andthen cut 10 tabsLabelLabel each tab with ahighlighted term from thechapter. Store the Foldablein a 3-ring binder.© Glencoe/McGraw-HillNOTE-TAKING TIP: When taking notes, it ishelpful to record the main ideas as you listen toyour teacher, or read through a lesson.Geometry: Concepts and Applications 23

C H A P T E R2BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 2.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExampleabsolute valuebetweennessbisectcongruent segments[con-GROO-unt]coordinate[co-OR-duh-net]coordinate planecoordinatesgreatest possible errormeasure© Glencoe/McGraw-Hillmeasurements24 Geometry: Concepts and Applications

Chapter2BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExamplemidpointordered pairorigin[OR-a-jin]percent of errorprecision[pree-SI-zhun]quadrants[KWAH-druntz]theorem[THEE-uh-rem]unit of measure© Glencoe/McGraw-Hillvectorx-axisx-coordinatey-axisy-coordinateGeometry: Concepts and Applications 25

2–1Real Numbers and Number LinesWHAT YOU’LL LEARN• Find the distancebetween two points ona number line.Postulate 2-1 Number Line PostulateEach real number corresponds to exactly one point on anumber line. Each point on a number line corresponds toexactly one real number.For each situation, write a real number with ten digitsto the right of the decimal point.ORGANIZE ITOn the first tab of yourFoldable, write RationalNumbers and on thesecond tab, writeIrrational Numbers.Under each tab, describethe sets of rational andirrational numbers andgive several examplesof each.a rational number between 6 and 8 with a 2-digitrepeating patternSample answer: 7.3232323232 . . .an irrational number greater than 5Sample answer: 5.4344334443 . . .Your TurnFor each situation, write a real numberwith ten digits to the right of the decimal point.a. a rational number between 4 and 1 with a 3-digitrepeating patternb. an irrational number less than 7Postulate 2-2 Distance PostulateFor any two points on a line and a given unit of measure,there is a unique positive real number called the measureof the distance between the points.Postulate 2-3 Ruler PostulateThe points on a line can be paired with the real numbers sothat the measure of the distance between correspondingpoints is the positive difference of the numbers.© Glencoe/McGraw-Hill26 Geometry: Concepts and Applications

2–1BUILD YOUR VOCABULARY (pages 24–25)The number that corresponds to a point on a numberline is called the coordinate of the point.A point with coordinateis known as the origin.The absolute value of a number is the number of units anumber is fromon the number line.REMEMBER ITXY represents themeasure of the distancebetween points X and Y.Use the number line below to find CE.3A B C D EF G21The coordinate of C is , and the coordinate of E is .CE 1 1 3 1 1 3 0123 1 1 3 or© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Erin traveled on I-85 from Durham, North Carolina, toCharlotte. The Durham entrance to I-85 that she used isat the 173-mile marker, and the Charlotte exit she usedis at the 39-mile marker. How far did Erin travel on I-85?173 39 134 She traveledYour Turnmiles on I-85.a. Refer to Example 3. Find AE.b. Rahmi’s drive starts at the 263-mile markerof I-35 and finishes at the 287-mile marker. How far didRahmi drive on I-35?Geometry: Concepts and Applications 27

2–2Segments and Properties of Real NumbersBUILD YOUR VOCABULARY (page 24)WHAT YOU’LL LEARN• Apply properties ofreal numbers to themeasure of segments.Point R is between points P and Q if and only if R, P and Qareand PR RQ PQ.ORGANIZE ITOn the third tab of yourFoldable write Measureand on the fourth tabwrite Unit of Measure.Under each tab, explainthe differences betweenthe terms and giveexamples of each.Points K, L, and J are collinear. If KL 31, JL 16,and JK 47, determine which point is between theother two.Check to see which two measures add to equal the third.KL JL JKTherefore, is between and .Your TurnPoints A, B, and C are collinear. If AB 54,BC 33, and AC 21, determine which point is between theother two.If FG 12 and FJ 47, find GJ.F G H JFG GJ FJDefinition of betweenness GJ Substitution Property© Glencoe/McGraw-Hill12 GJ 47 Subtraction PropertyGJ Substitution Property28 Geometry: Concepts and Applications

2–2KEY CONCEPTSYour TurnIf BE 17 and AE 25, find AB.Properties of Equalityfor Real NumbersABCDE• Reflexive PropertyFor any number a,a a.• Symmetric PropertyFor any numbers a andb, if a b, then b a.• Transitive PropertyFor any numbers a, b,and c, if a b andb c, then a c.• Addition andSubtraction PropertiesFor any numbers a, b,and c, if a b, thena c b c, anda c b c.• Multiplication andDivision PropertiesFor any numbers a, b,and c, if a b, thena c b c, and ifc 0, then a c b c .• Substitution PropertyFor any numbers a andb, if a b, then a maybe replaced by b in anyequation.BUILD YOUR VOCABULARY (pages 24–25)Measurements are composed of parts; a numbercalled the measure and the unit of measure.The precision of a measurement depends on theunit used to make the measurement.The greatest possible error is the smaller unit usedto make the measurement.The percent of error is the of thegreatest possible error with the measurement itself,multiplied by .Use a ruler to draw a segment 8 centimeters long. Thenfind the length of the segment in inches.© Glencoe/McGraw-HillUse a metric ruler to draw the segment. Mark a point and callit X. Then put the 0 point at point X and draw a line segmentextending to the 8 centimeter mark. Mark the endpoint Y.XY0 1 2 3 4 5 6 7 8 9 10centimeters (cm)The length of XY iscentimeters.Geometry: Concepts and Applications 29

2–2Use a customary ruler to measure XY in inches. Put the 0point at X and measure the distance to Y.XY0 1 2 3 4inches (in.)The length of XY is aboutinches.Your TurnUse a ruler to draw a segment 3 centimeterslong. Then find the length of the segment in inches.HOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill30 Geometry: Concepts and Applications

2–3Congruent SegmentsWHAT YOU’LL LEARN• Identify congruentsegments.Use the figure below to determine whether eachstatement is true or false. Explain your reasoning.D E FGH• Find midpoints ofsegments.8 7 6 5 4 3 2 1 0 12 3 4 5 6 78a. DE GHBecause DE 4 and GH , .So, is a true statement.KEY CONCEPTDefinition of CongruentSegments Two segmentsare congruent if andonly if they havethe same length.On the fifthtab of you Foldable,write CongruentSegments. Under thetab, write the definitionand draw examples ofcongruent segments.b. EF FGBecause EF and FG , EF ≠ FG. So, EF isnot congruent to FG, and the statement is false.Your TurnUse the figure below to determine whethereach statement is true or false. Explain your reasoning.A B C D E F G H I J8 7 6 5 4 3 2 1 0 1a. AE BG2 3 4 5 6© Glencoe/McGraw-Hillb. DG FJBUILD YOUR VOCABULARY (pages 24–25)Theorems are statements that can be justified by usingreasoning.Geometry: Concepts and Applications 31

2–3REVIEW ITWrite the converse ofTheorem 2-2. Is theconverse true?(Lesson 1-4)Theorem 2-1Congruence of segments is reflexive.Theorem 2-2Congruence of segments is symmetric.Theorem 2-3Congruence of segments is transitive.Determine whether the statement is true or false.Explain your reasoning.CD is congruent to CD.Congruence of segments is , so .Therefore, the statement is .Your TurnDetermine whether the statement is true orfalse. Explain your reasoning.MN is congruent to NM.MN is congruent to NM.BUILD YOUR VOCABULARY (pages 24–25)A unique point on every segment that separates thesegment into segments of lengthis known as the midpoint.© Glencoe/McGraw-HillTo bisect something means to separate it into twoparts.32 Geometry: Concepts and Applications

2–3In the figure, K is the midpoint of JL.Find the value of d.KEY CONCEPTDefinition of Midpoint Apoint M is the midpointof a segment ST if andonly if M is betweeen Sand T and SM MT.You need to find the value of d. Since K is the midpoint of JL,JK KL. Write and solve an equation involving d, and solvefor d.JK KLd 52dJ K LDefinition ofOn the sixthtab of your Foldable,write Midpoint. Underthe tab, write thedefinition and draw anexample showing themidpoint of a linesegment. dSubstitutionSubtraction Property of EqualityYour TurnIn the figure, D is the midpoint of XY.Find the value of a.7a 8 5aX D Y© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 33

2–4The Coordinate PlaneBUILD YOUR VOCABULARY (pages 24–25)WHAT YOU’LL LEARN• Name and graphordered pairs on acoordinate plane.Thethe coordinate plane.of the grid used to locate points is known asTheThe x-axis is thenumber line is the y-axis.number line.ORGANIZE ITOn the seventh tab ofyour Foldable, writeCoordinate Plane.Under the tab, drawa coordinate plane,labeling the fourquadrants and thetwo axes.On the eighth tab ofyour Foldable, writeOrdered Pair andCoordinates. Under thetab, give an example ofan ordered pair. Labelthe x-coordinate andthe y-coordinate forthe pair.The two axes separate the coordinate plane intoregions known as quadrants.The two axes at a called theorigin.An ordered pair of real numbers, called the coordinates ofa point, locates aon the coordinate plane.The number of the ordered pair is called thex-coordinate.The y-coordinate is thenumber of the orderedpair.Graph point K at (4, 1).Start at the origin. Moveto the left. Then, moveunitsunit up.KyOx© Glencoe/McGraw-HillLabel this point K.34 Geometry: Concepts and Applications

2–4Postulate 2-4Completeness Property for Points in the PlaneEach point in a coordinate plane corresponds to exactlyone ordered pair of real numbers. Each ordered pair ofreal numbers corresponds to exactly one point in acoordinate plane.Your TurnGraph point L at (1, 4).yOxName the coordinates of points L and M.MyOLxPoint L is units to the right of the origin and unitbelow the origin. Its coordinates are .© Glencoe/McGraw-HillWRITE ITExplain how to graphany ordered pair (x, y).Describe which directionyou move when x or yare either positive ornegative.Point M is to the left of the origin and unitsabove the origin. Its coordinates are .Your TurnName the coordinates of points P and Q.yPOQxGeometry: Concepts and Applications 35

2–4Theorem 2-4If a and b are real numbers, a vertical line contains allpoints (x, y) such that x a, and a horizontal line containsall points (x, y) such that y b.Graph y 2.yOxy 2The graph of y 2 is aline that intersectsthe y-axis at .Your TurnGraph x 1.yOxHOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill36 Geometry: Concepts and Applications

2–5MidpointsWHAT YOU’LL LEARN• Find the coordinates ofthe midpoint of asegment.Theorem 2-5 Midpoint Formula for a Number LineOn a number line, the coordinate of the midpoint of asegment whose endpoints have coordinates a and b is a b.2Theorem 2-6 Midpoint Formula for a Coordinate PlaneOn a coordinate plane, the coordinates of the midpoint ofa segment whose endpoints have coordinates (x 1 , y 1 ) and(x 2 , y 2 ) are x 1 x 2 y,1 y 2 .2 2Find the coordinate of the midpoint of AB.ABORGANIZE ITOn the ninth tab ofyour Foldable, writeMidpoint for a NumberLine. Under the tab,explain how to find themidpoint of a segmenton a number line.Use the Midpoint Formula to find the coordinate of themidpoint of AB. a b 4 12 2or4 3 2 1 0 1The coordinate of the midpoint is .© Glencoe/McGraw-HillYour TurnO6 5 4 3Find the coordinate of the midpoint of OK.K2 1 0 1 2 3 4 5 6 7 8Geometry: Concepts and Applications 37

2–5Find the coordinates of D, the midpoint of CE, givenendpoints C(2, 1) and E(16, 8).Use the Midpoint Formula to find the coordinates of D. x 1 x 2 y, 1 y 2 2 2 , 2 2 2, 2The coordinates of D are .Your TurnFind the coordinates of Y, the midpoint of XZ,given endpoints X (3, 5) and Z (6, 1).Suppose L (2, 5) is the midpoint of KM and thecoordinates of K are (4, 3). Find the coordinatesof M.Let (x 1, y 1) or (4, 3) be the coordinates of K and let (x 2, y 2)be the coordinates of M. So, x 1 and y 1 .Use the Midpoint Formula.© Glencoe/McGraw-Hill x 1 x 2 y, 1 y 2 2 238 Geometry: Concepts and Applications

2–5REMEMBER ITThe x-coordinateof the midpoint isthe average of thex-coordinates ofthe endpoints. They-coordinate of themidpoint is the averageof the y-coordinates ofthe endpoints.x-coordinate of M4 x 224 x 22 Replace x 1 with 4. 2 Multiply each side by .Add to isolate the variable.x 2y-coordinate of M 3 y 2 Replace x21 with 3. 3 y 25 Multiply each side by .2Add to isolate the variable.y 2The coordinates of M are .Your TurnSuppose S 1 2 , 3 2 is the midpoint of RT andthe coordinates of R are (2, 5). Find the coordinates of T.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 39

C H A P T E R2BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 2 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 2, go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 24–25) to help you solvethe puzzle.2-1Real Numbers and Number LinesChoose the term that best completes the statement.1. The set of non-negative integers is also called the set of[natural/whole] numbers.2. The quotient of two integers, where the denominator is notzero, is a(n) [rational/irrational] number.3. Decimals that do not repeat or terminate are called[rational/irrational] numbers.Find.4. 4 1 5. (12) 6. 11 22-2Segments and Properties of Real Numbers7. Points X, Y, and Z are collinear. If XY 10 and XZ 3, find YZ.8. Points A, B, and C are collinear. If AB 6, BC 8, andAC 14, which point is between the other two points?© Glencoe/McGraw-Hill9. Points M, N, and P are collinear. If P lies between M and N,MP 2, and PN 1, find MN.40 Geometry: Concepts and Applications

Chapter2BRINGING IT ALL TOGETHER2-3Congruent SegmentsComplete the statement.10. Two segments are if they are equal in length.11. When a segment is separated into two congruent segments,the segment is .12. Statements known as can be justified usinglogical reasoning.13. Points A, B, and C are collinear. If AC CB, then the point C istheof AB.2-4The Coordinate PlaneRefer to the graph and name theordered pair for each point.14. point P15. point LLPOyAx16. point AGraph and label the following points on the abovecoordinate plane.17. point N (4, 2) 18. point E (3, 1) 19. point S (1, 5)© Glencoe/McGraw-Hill2-5Midpoints20. On a number line, if X 2 and Y 4, what is the coordinateof midpoint Z?21. Find the coordinates of the midpoint of a segment whoseendpoints are (5, 1) and (3, 3).22. Find the coordinates of the other endpoint of a segment whosemidpoint has coordinates (4, 5) and second endpoint at (2, 1).Geometry: Concepts and Applications 41

C H A P T E R2ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 2.• You may want to take the Chapter 2 Practice Test on page 85of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 2 Study Guide and Reviewon pages 82–84 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 2 Practice Test onpage 85 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 2 Foldable.• Then complete the Chapter 2 Study Guide and Review onpages 82–84 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 2 Practice Test onpage 85 of your textbook.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature42 Geometry: Concepts and Applications

C H A P T E R3AnglesUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with a sheet of plain 8 1 " 11" paper.2FoldFold in half lengthwise.FoldFold again in thirds.OpenOpen and cut along thesecond fold to makethree tabs.Chapter 3© Glencoe/McGraw-HillLabelLabel as shown. Makeanother 3-tab fold andlabel as shown.Angles and Pointsinterior exterior on anangleNOTE-TAKING TIP: When you take notes, listen orread for main ideas. Then record those ideas insimplified form for future reference.Geometry: Concepts and Applications 43

C H A P T E R3BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 3.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExampleacute angle[a-KYOOT]adjacent angles[uh-JAY-sent]angleangle bisectorcomplementary angles[kahm-pluh-MEN-tuh-ree]congruent anglesdegreesexteriorinterior© Glencoe/McGraw-Hilllinear pair[LIN-ee-ur]44 Geometry: Concepts and Applications

Chapter3BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleobtuse angle[ob-TOOS]opposite raysperpendicular[PER-pun-DI-kyoo-lur]protractorquadrilateral[KWAD-ruh-LAT-er-ul]right anglesidesstraight angle© Glencoe/McGraw-Hillsupplementary angles[SUP-luh-MEN-tuh-ree]trianglevertex[VER-teks]vertical anglesGeometry: Concepts and Applications 45

3–1AnglesBUILD YOUR VOCABULARY (pages 44–45)WHAT YOU’LL LEARN• Name and identify partsof an angle.Opposite rays are two rays that are part of the sameand have only theirin common.The figure formed byas a straight angle.is referred toAny case where two rays have a commonis known as angle.The commonis called the vertex.The two rays that make up thesides of the angle.are called theREMEMBER ITRead the symbol as angle.Name the angle in four ways. Then identifyits vertex and its sides.The angle can be named in four ways:.HB 1 ZIts vertex is . Its sides are and .REVIEW ITName the sides ofABC. (Lesson 1-2)Your TurnName the angle in fourways. Then identify its vertex and its sides.Name all angles having D as their vertex.XBY5Z© Glencoe/McGraw-HillThere aredistinct angles withCvertex D: .A1 2 D46 Geometry: Concepts and Applications

3–1Your TurnName all angles having Aas their vertex.XA3 4 5YZWBUILD YOUR VOCABULARY (page 44)An angle separates a into parts: theinterior of the angle, the exterior of the angle, and theangle itself.CORGANIZE ITIn your first Foldable,explain and drawexamples of interiorpoints, exterior points,and points on the angle.Include underappropriate tab.Angles and Pointsinterior exterior on anangleTell whether each point is in the interior,exterior, or on the angle.Point A: Point A is on thePoint B: Point B is on thePoint C: Point C is .of the angle.Bof the angle.AYour TurnTell whether each point is in the interior,exterior, or on the angle.a. Point T b. Point NTN© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:c. Point DmD nGeometry: Concepts and Applications 47

3–2Angle MeasureBUILD YOUR VOCABULARY (pages 44–45)WHAT YOU’LL LEARN• Measure, draw, andclassify angles.Angles are measured in units called degrees.A protractor is a tool used to measure angles and sketchangles of a given measure.Postulate 3-1 Angle Measurement PostulateFor every angle, there is a unique positive number between0 and 180 called the degree measure of the angle.Use a protractor to measure KLM.STEP 1 Place the center point of the protractor on vertex L.Align the straightedge90with side . LMKSTEP 2 Use the scale that beginswith 0 at . LM Read whereLK crosses this scale.Angle KLM measures .20100301601701804015050140130601207011080100L10080110706012050130401403020150101600170180MYour TurnUse a protractor to measure XYZ.XFind the measures ofEHG, and FHG.DHE,4014030150201601017001805013060120E70110D HYZ8010090100801107012060F130501404015030G16020170101800© Glencoe/McGraw-HillmDHE HD is at 0° on the left.mEHG HG is at 0° on the right.mFHG HG is at 0° on the right.48 Geometry: Concepts and Applications

3–2Your TurnFind mPQR, mRQS, and mSQT.RSPQTPostulate 3-2 Protractor PostulateOn a plane, given AB and a number r between 0 and 180,there is exactly one ray with endpoint A, extending on eachside of AB such that the degree measure of the angleformed is r.Use a protractor to draw an angle having a measure of 35°.REMEMBER ITRead mPQR 75as the degree measureof angle PQR is 75.STEP 1 Draw BC.STEP 2 Place the center point of theprotractor on B. Align the marklabeledwith the ray.B35ACSTEP 3 Locate and draw point A at the mark labeled .Draw BA .Your TurnUse aprotractor to draw anangle having ameasure of 78°.© Glencoe/McGraw-HillREMEMBER ITThe symbol isused to indicate a rightangle.BUILD YOUR VOCABULARY (pages 44–45)A right angle has a degree measure of 90.The degree measure of an acute angle is greater than 0and less than 90.An obtuse angle has a degree measure greater than 90and less than 180.A three-sided closed figure with three interior angles isa triangle.A four-sided closed figure with four interior angles is aquadrilateral.Geometry: Concepts and Applications 49

3–2ORGANIZE ITIn your second Foldable,explain and drawexamples of rightangles, acute angles,and obtuse angles.Include underappropriate tab.Angles and Pointsinterior exterior on anangleClassify each angle as acute, obtuse, or right.9030Your TurnClassify each angle as acute, obtuse, or right.a. b.41127The measure of A is 100. Solve for x.You know that mA 100and mA 10.3x 10 AWrite and solve an equation. 3x 10Substitution100 3x 10 Subtract fromeach side. 3xHOMEWORKASSIGNMENTPage(s):Exercises:903xDivide each side by . xYour TurnThe measure of N is 135. Solve for x.© Glencoe/McGraw-HillN(7x 5)50 Geometry: Concepts and Applications

3–3The Angle Addition PostulateWHAT YOU’LL LEARN• Find the measure andbisector of an angle.Postulate 3-3 Angle Addition PostulateFor any angle PQR, if A is in the interior of PQR thenmPQA mAQR mPQR.If mKNL 110 and mLNM 25, find mKNM.mKNM mKNL mLNML25110 25 Substitution K NM So, mKNM .Find m2 if m1 75 and mABC 140.m2 mABC m1A SubstitutionD2 1BC So, m2 .Find mJKL and mLKM if mJKM 140.L(2x 10)© Glencoe/McGraw-HillMmJKL mLKM mJKM (2x 10) 1404xJ KSubstitutionCombine like terms.6x 10 140 Add to each side.6x x Divide each side by .Geometry: Concepts and Applications 51

3–3Replace x within each expression.mJKL 4x mLKM 2x 10 4 2 10 10 Therefore, mJKL and mLKM .Your Turna. If mABC 95 and mCBD 65, find mABD.CD65B95Ab. If mXYZ 110 and mXYW 22, find mWYZ.ZWYXc. Find mRSZ and mZST if mRST 135.R9xSZ(4x 5)T© Glencoe/McGraw-Hill52 Geometry: Concepts and Applications

3–3BUILD YOUR VOCABULARY (page 44)A that divides an angle into angles ofequalis called the angle bisector.REVIEW ITDF is bisected at point E,and DF 8. What doyou know about thelengths DE and EF?(Lesson 2-3)If FD bisects CFE and mCFE 70, findm1 and m2.Since FD bisects CFE, m1 m2.m1 m mCFE Postulate 3–3FC12DEm1 m2 Replace mCFE with .m1 70 Replace m2 with .2 m 70 Combine like terms.2(m1) 70 Divide each side by .m1 Since m1 m2, m2 .Your TurnIf EG bisects FEH and mFEH 98, findm1 and m2.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTHE21GFPage(s):Exercises:Geometry: Concepts and Applications 53

3–4Adjacent Angles and Linear Pairs of AnglesBUILD YOUR VOCABULARY (page 44)WHAT YOU’LL LEARN• Identify and useadjacent angles andlinear pairs of angles.Adjacent angles share a common side and a vertex, buthave nopoints in common.When the noncommon sides of adjacent angles form a, the angles are said to form a linear pair.Determine whether 1 and 2 are adjacent angles.1 2. They have the samebut no .1 2. They have the sameand a commonpoints in common.with no interior12. They have abut no common .Your TurnDetermine whether 1 and 2 are adjacentangles.a. b.2 112© Glencoe/McGraw-Hillc.1254 Geometry: Concepts and Applications

3–4CM and CE are opposite rays.Name the angle that forms a linearpair with TCM.AT12H34CETCE and TCM have a commonMside, the samevertex , and opposite rays and .So, TCE forms a linear pair with TCM.WRITE ITList the differences andsimilarities betweenlinear pairs of anglesand adjacent angles.Do 1 and TCE form a linear pair? Justify your answer., they are not angles.Your TurnRefer to Examples 4 and 5.a. Name the angle that forms a linear pairwith HCE.b. Determine if TCA and TCH form a linear pair. Justifyyour answer.List at least two models of linear pairs in yourclassroom or home.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Your TurnList at least two models of adjacent angles on aschool playground.Geometry: Concepts and Applications 55

3–5Complementary and Supplementary AnglesBUILD YOUR VOCABULARY (pages 44–45)WHAT YOU’LL LEARN• Identify and usecomplementary andsupplementary angles.Complementary angles are two angles whose degreemeasures total 90.Supplementary angles are two angles whose degreemeasures total 180.Use the figure to name a pair of nonadjacentsupplementary angles.ABFG55803510CDEmAGB , and they have the samevertex , but sides. Therefore, AGBandare nonadjacent supplementary angles.Use the above figure to find the measure of an anglethat is supplementary to BGC.Let x measure of angle supplementary to BGC.mBGC x 180 x 180Defn. of SupplementaryAnglesmBGC © Glencoe/McGraw-Hill35 x 180 Subtract from eachside.x 56 Geometry: Concepts and Applications

3–5Your Turna. In the figure, name apair of nonadjacentsupplementary angles.b. In the figure, find an angle with a measure supplementaryto BAF.D90E 5435 A75F365515HGCBREMEMBER ITSupplementaryangles must be adjacentangles to form a linearpair.Angles C and D are supplementary. If mC 12x andmD 4(x 5), find x. Then find mC and mD.mC mD 180Defn. of Supplementary Angles 4(x 5) 180Substitution12x 4x 180 Distributive Property 160Combine like terms. 1 6x16 1 601 Divide each side by 16.6x Replace x within each expression.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:mC 12x 12mD 4(x 5) 4or 5 orYour TurnAngles X and Y arecomplementary. If mX 2x andmY 8x, find x. Then find mXand mY.Geometry: Concepts and Applications 57

3–6Congruent AnglesBUILD YOUR VOCABULARY (pages 44–45)WHAT YOU’LL LEARN• Identify and usecongruent and verticalangles.Congruent angles have the same measure.When two lines , angles areformed. There are two pairs of nonadjacent angles.These pairs are vertical angles.Theorem 3-1 Vertical Angle TheoremVertical angles are congruent.Find the value of x in each figure.The angles areangles.x 100So, x .75(x 18)Since the angles are vertical angles, theyare congruent.REMEMBER ITThe notationA B is read asangle A is congruent toangle B.x 18 x 18 75 Add to each side.x Your TurnFind the value of x in each figure.a. b.11538 (3x 10)x© Glencoe/McGraw-Hill58 Geometry: Concepts and Applications

3–6Theorem 3–2 If two angles are congruent, then theircomplements are congruent.Theorem 3–3 If two angles are congruent, then theirsupplements are congruent.Theorem 3–4 If two angles are complementary to thesame angle, then they are congruent.Theorem 3–5 If two angles are supplementary to the sameangle, then they are congruent.Theorem 3–6 If two angles are congruent andsupplementary, then each is a right angle.Theorem 3–7 All right angles are congruent.Suppose A B and mB 47. Find the measure ofan angle that is supplementary to A.Since A B, their supplements are congruent.The supplement of B is 180 47 or. So, themeasure of an angle that is supplementary to A is .In the figure, 1 is supplementary to2, 3 is supplementary to 2, andm2 is 105. Find m1 and m3.21 31 and 2 are supplementary.So, m1 105 or . 3 and 2 aresupplementary. So, m3 105 or .© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Your Turna. Suppose X Y and mY 82.Find the measure of an angle that issupplementary to X.b. In the figure, 1 is supplementary to 2 and 4.If m4 54, find m1, m2, and m3.2134Geometry: Concepts and Applications 59

3–7Perpendicular LinesBUILD YOUR VOCABULARY (page 45)WHAT YOU’LL LEARN• Identify, use propertiesof, and constructperpendicular linesand segments.Lines that at an angle of degreesare said to be perpendicular lines.Theorem 3–8If two lines are perpendicular, then they form right angles.Refer to the figure todetermine whether each ofthe following is true or false.M7PQS OP form. QS and OP do notangles.OQ1 5 8 32R6 S 4NTherefore, theyperpendicular.7 is an obtuse angle.. 7 forms a with an acute angle.REMEMBER ITRead p q as line pis perpendicular to line q.Your TurnIn the figureWY ZT. Determine whethereach of the following is trueor false.a. mWZU mUZT 90TUWZX© Glencoe/McGraw-Hillb. SZY is obtuse.SYR60 Geometry: Concepts and Applications

3–7Find m1 and m2 if AC BD ,m1 8x 2 and m2 16x 4.DFA21ECBSince AC BD, AED is a right angle.mAED 90Definition of perpendicularlines1 AED Angle Addition Postulatem1 m mAEDm1 m2 Substitution(8x 2) (16x 4) 90 Substitution24x 6 9024x 6 6 90 6Combine like terms.Add 6 to each side.24x 96 2 4x 9 6 Divide each side by 24.2424 x Replace x withto find m1 and m2.ml 8x 2 m2 16x 4 8 2 16 4© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises: 32 2 or 30 64 4 or 60Therefore, m1 30 and m2 60.Your TurnFind m3 and m4if AC BF , m3 7x 6 andm4 12x 27.BD43AECFGeometry: Concepts and Applications 61

C H A P T E R3BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 3 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 3, go to:www.glencoe.com/sec/math/t.resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder (pages 44–45)to help you solve the puzzle.3-1AnglesIndicate whether the statement is true or false.1. XY and YZ are the sides of XYZ.2. The vertex of an angle is a point where two rays intersect.3. A straight angle is also a line.3-2Angle MeasureUse a protractor to measure the specified angles. Then,classify them as acute, right, or obtuse angles.4. BACC5. CAEED6. DAE3-3The Angle Addition Postulate7. If mQPR 30 and mRPS 51,find mQPS.STAXB© Glencoe/McGraw-Hill8. If mQPX 137 and mQPR 30,find mRPX.RPYQ62 Geometry: Concepts and Applications

Chapter3BRINGING IT ALL TOGETHER3-4Adjacent Angles and Linear Pairs of Angles9. In the figure QN and QP are opposite rays.Name the angles that form a linear pair.N15Q4 2 3P3-5Complementary and Supplementary Angles10. If m1 36, what is the measure of itscomplement?11. What is the measure of an angle supplementaryto m1 36?3-6Congruent AnglesLines m and n intersect at point P. What is themeasure of each of the four angles formed?12. 13.mn14. 15.(3x 25)P(4x 5)3-7Perpendicular LinesIf WZ is constructed perpendicular to XY ,list six terms that describe XWZ and YWZ.Z© Glencoe/McGraw-Hill16.17.18. 20.19. 21.XWYGeometry: Concepts and Applications 63

C H A P T E R3ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geoconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 3.• You may want to take the Chapter 3 Practice Test on page 137of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 3 Study Guide and Reviewon pages 134–136 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 3 Practice Test onpage 137.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 3 Foldable.• Then complete the Chapter 3 Study Guide and Review onpages 134–136 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 3 Practice Test onpage 137.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature64 Geometry: Concepts and Application

C H A P T E R4ParallelsUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with three sheets of plain 8 1 " 11" paper.2FoldFold in half along the width.OpenOpen and fold thebottom to form apocket. Glue edges.RepeatRepeat steps 1 and 2three times and glueall three pieces together.© Glencoe/McGraw-HillLabelLabel each pocket with thelesson names. Place anindex card in each pocket.ParallelsNOTE-TAKING TIP: When taking you take notes, notes, it is listen often orread a good for idea main to ideas. write Then in your record own words those ideas a summary in asimplified of the lesson. form Be for sure future to paraphrase reference. key points.Chapter 4Geometry: Concepts and Applications 65

C H A P T E R4BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 4.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplealternate exterioranglesalternate interioranglesconsecutive interioranglescorresponding anglesexterior anglesfinitegreat circleinterior angles© Glencoe/McGraw-Hillline66 Geometry: Concepts and Applications

Chapter4BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleline of latitudeline of longitudelinear equationparallel lines[PARE-uh-lel]parallel planesskew lines[SKYOO]slope© Glencoe/McGraw-Hillslope-intercept formtransversaly-interceptGeometry: Concepts and Applications 67

4–1Parallel Lines and PlanesWHAT YOU’LL LEARN• Describe relationshipsamong lines, parts oflines, and planes.BUILD YOUR VOCABULARY (pages 66–67)Parallel lines are two lines in the samethat donot intersect.Parallel planes are the sameapart at allpoints andintersect.Lines that do notand are not in theplane are said to be skew lines.ORGANIZE ITUse the index cardlabeled Parallel Linesand Planes to record thedefinitions in this lesson,along with examples tohelp you remember themain idea.Name the parts of the prism shown below. Assumesegments that look parallel are parallel.all planes parallel to plane SKLMTSNLKRQParallelsPlaneis parallel to plane SKL.all segments that intersect MTintersect MT.all segments parallel to MTis parallel to MT.© Glencoe/McGraw-Hillall segments skew to MTare skew to MT.68 Geometry: Concepts and Applications

4–1Your TurnName the parts of the prism shown below.Assume segments that look parallel are parallel.REMEMBER ITA plane that passesthrough points A, B, Cand D can be namedusing any three of thepoints.RYZXTSa. all segments parallel to RSb. all segments that intersect RSc. a pair of parallel planesd. all segments skew to XT© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 69

4–2Parallel Lines and TransversalsBUILD YOUR VOCABULARY (pages 66–67)WHAT YOU’LL LEARN• Identify therelationships amongpairs of interior andexterior angles formedby two parallel linesand a transversal.A line, line segment, or ray that intersects two or morelines at differentis known as a transversal.Interior angles lie in between the two lines.Alternate interior angles are onsides ofthe transversal.Consecutive interior angles are on theof the transversal.sideExterior angles liethe two lines.Alternate exterior angles are onthe transversal.sides ofORGANIZE ITUse the index cardlabeled Parallel Linesand Transversals torecord the definitionsand theorems in thislesson. Draw picturesand examples to helpyou remember them.ParallelsIdentify each pair of angles as alternate interior,alternate exterior, consecutive interior, or vertical.3 and 53 and 5 are interior angles on the same side as thetransversal, so they are1 and 81 23 45 67 81 and 8 are exterior angles on opposite sides of theangles.© Glencoe/McGraw-Hilltransversal, so they areangles.70 Geometry: Concepts and Applications

Your TurnIdentify each pair of angles as alternateinterior, alternate exterior, consecutive interior, orvertical.a. 3 and 54–2b. 3 and 61 24 35 68 7nmaREVIEW ITThe sum of the degreemeasures of threeangles is 180. Are thethree anglessupplementary? Explain.(Lesson 3-5)Theorem 4-1 Alternate Interior AnglesIf two parallel lines are cut by a transversal, then each pairof alternate interior angles is congruent.Theorem 4-2 Consecutive Interior AnglesIf two parallel lines are cut by a transversal, then each pairof consecutive interior angles is supplementary.Theorem 4-3 Alternate Exterior AnglesIf two parallel lines are cut by a transversal, then each pairof alternate exterior angles is congruent.In the figure, pq, and r is atransversal. If m6 115,find m7.6 and 7 are alternate5 12 76 3 4angles, so by Theorem 4-3, they are .qp8r© Glencoe/McGraw-HillTherefore, m7 .Your TurnIf m1 50, find m8.1 23 45 67 8Geometry: Concepts and Applications 71

4–2REMEMBER ITIn figures with twopairs of parallel lines,arrowheads indicate thefirst pair and doublearrowheads indicate thesecond pair.In the figure, AB CD, and t isa transversal. If m6 128,find m7, m8, and m9.6 and 8 are consecutiveinterior angles, so by Theorem 4-2they are supplementary.ACt6789BDm6 m8 180 m8 180Replace m6.REVIEW ITIf angles P and Q arevertical angles andmP 47, what ismQ? (Lesson 3-6)128 m8 180 Subtract 128from each side.m8 7 and 8 are alternate interior angles, so by Theorem 4-1they are congruent. Therefore, m7 .6 and 9 areangles, so byTheorem 4-1 they are congruent. Therefore, m9 .Your TurnIn the figure, n m, and a is a transversal.If m6 73, find m1, m4, and m7.1 24 3nHOMEWORKASSIGNMENTa5 68 7m© Glencoe/McGraw-HillPage(s):Exercises:72 Geometry: Concepts and Applications

4–3Transversals and Corresponding AnglesBUILD YOUR VOCABULARY (page 66)WHAT YOU’LL LEARN• Identify therelationships amongpairs of correspondingangles formed by twoparallel lines and atransversal.When acrosses two lines, an interiorangle and an exterior angle that are on theside of the transversal and have different verticies arecalled corresponding angles.Lines a and b are cut by transversal c. Name two pairsof corresponding angles.ORGANIZE ITUse the index cardlabeled Transversals andCorresponding Angles torecord the postulates,theorems, and othermain ideas in this lesson.Draw pictures andexamples as needed.ParallelsCorresponding angles lie on the sametransversal and havec1 2 5 634 78aof thevertices. Two pairs ofcorresponding angles are .bPostulate 4-1 Corresponding AnglesIf two parallel lines are cut by a transversal, then each pairof corresponding angles is congruent.© Glencoe/McGraw-HillIn the figure, a b, and k isa transversal.Which angle is congruent to 1?Explain your answer.ab1234k1 , since angles arecongruent Postulate .Geometry: Concepts and Applications 73

4–3KEY CONCEPTSTypes of angle pairsformed when atransversal cuts twoparallel lines.1. Congruenta. alternate interiorb. alternate exteriorc. corresponding2. Supplementarya. consecutive interiorFind the measure of 1 if m4 60.m1 m33 and 4 are a linear pair, so they are supplementary.m3 m4 180m3 180 Replace m4with .m3 60 180 Subtract 60 fromeach side.m3 m1 SubstitutionYour Turna. Refer to the figure in Example 1. Name two different pairsof corresponding angles.b. Refer to the figure in Example 2. Which angle is congruentto 2? Explain your answer.c. Refer to the figure in Example 2. Find the measure of 2 ifm3 145.© Glencoe/McGraw-HillTheorem 4-4 Perpendicular TransversalIf a transversal is perpendicular to one of two parallel lines,it is perpendicular to the other.74 Geometry: Concepts and Applications

4–3REMEMBER ITThere are alwaysfour pairs ofcorresponding angleswhen two lines are cutby a transversal.In the figure, pq, and transversal ris perpendicular to q. Ifm2 3(x 2), find x.pr Theorem 4-42 is a right angle.Definition ofperpendicularlinesm2 Definition ofright angles2prqm2 Given 3(x 2) Replace m2 with .Distributive Property90 3x 6 Subtract 6 from each side.84 3x 8 4 3 x Divide each side by .3 3 xYour TurnIn the figure, a b and r is a transversal.If m1 3x 5 and m2 2x 35, find x.r1a© Glencoe/McGraw-HillHOMEWORKASSIGNMENT2bPage(s):Exercises:Geometry: Concepts and Applications 75

4–4Proving Lines ParallelWHAT YOU’LL LEARN• Identify conditions thatproduce parallel linesand construct parallellines.ORGANIZE ITUse the index cardlabeled Proving LinesParallel to record thepostulates, theorems,and important conceptsin this lesson. Recordexamples to help youremember the mainidea.Postulate 4-2 In a plane, if two lines are cut by a transversalso that a pair of corresponding angles is congruent, then thelines are parallel.aIf m1 5x 10 and m2 6x 4,2find x so that a b.bcFrom the figure, you know that 1 and 2 are correspondingangles. According to Postulate 4-2, if m1 m2, then a b.m1 m2Substitution5x 5x 10 6x 5x 4 Subtract 5x from each side.10 x 410 4 x 4 4 Add 4 to each side. x1ParallelsYour TurnFind c so that r s.r(3c 11)(4c 10)sTheorem 4-5 In a plane, if two lines are cut by a transversalso that a pair of alternate interior angles is congruent, thenthe two lines are parallel.Theorem 4-6 In a plane, if two lines are cut by a transversalso that a pair of alternate exterior angles is congruent, thenthe two lines are parallel.Theorem 4-7 In a plane, if two lines are cut by a transversalso that a pair of consecutive interior angles is supplementary,then the two lines are parallel.Theorem 4-8 In a plane, if two lines are perpendicular tothe same line, then the two lines are parallel.© Glencoe/McGraw-Hill76 Geometry: Concepts and Applications

Identify the parallel segments inthe letter E.4–4FEC and DCA are corresponding angles.EFmFEC mDCA Both angles measure 68°.EF CD Postulate 4-2BAC and DCE are corresponding angles.mBAC mDCE Both angles measure 112°.AB CD Postulate 4-2AB CD EFTransitive PropertyCA1126811268DBYour TurnIdentify the parallellines in the figure.122 58 59 pqrsREVIEW ITWhat is the relationshipbetween Theorem 4-1and Theorem 4-5?(Lesson 4-2)Find the value of x so thatQ KL MN.KPQ is a transversal for KL and MN .MIf (9x)° (10x 8)°, then KL MN by Theorem 4-6.9x 10x 89x 9x 10x 9x 8Subtract 9x from each side.0 x 89xN(10x 8)PL0 8 x 8 8 Add to each side.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises: xThus, if x , then .Your TurnFind c so that r s.rs(15c 12)132 Geometry: Concepts and Applications 77

4–5SlopeBUILD YOUR VOCABULARY (page 67)WHAT YOU’LL LEARN• Find the slopes of linesand use slope toidentify parallel andperpendicular lines.Slope is the ratio of the vertical change to the horizontalchange, or the to the , as you movefrom one point on the line to another.KEY CONCEPTDefinition of SlopeThe slope m of a linecontaining two pointswith coordinates (x 1 , y 1 )and (x 2 , y 2 ) is thedifference in they-coordinates dividedby the difference inthe x-coordinates.Find the slope of each line.y(0, 2)(2, 0)Oxm 0 2 2 2 0 2Use theindex card labeled Slopeto record the definitionsand postulates in thislesson.O(3, 2) (2, 2)yxm 2 (2 ( 2)3) 2 25 0 5 Your TurnFind the slope of each line.a. b. yy(4, 2)OxO(4, 2)x© Glencoe/McGraw-Hill(–2, –3)(4, –3)78 Geometry: Concepts and Applications

4–5WRITE ITExplain how you candetermine whether aline has a positive ornegative slope byobserving its graph.Postulate 4-3Two distinct nonvertical lines are parallel if and only if theyhave the same slope.Postulate 4-4Two nonvertical lines are perpendicular if and only if theproduct of their slopes is 1.Given A2, 1 2 , B2, 1 2 , C(5, 0), and D(4, 4), provethat AB CD.First, find the slopes of AB and CD .slope of AB 1 1 2 2 1 2 1 2 2 2slope of CD 4 0 4 4 5 1The product of the slopes for AB and CD is(4)or . Therefore, .2 (2)Geometry: Concepts and Applications 79Your TurnGiven A(3, 4), B(1, 7), C(2, 5), and D(4, 6),prove that AB CD.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:

4–6Equations of LinesWHAT YOU’LL LEARN• Write and graphequations of lines.BUILD YOUR VOCABULARY (pages 66–67)The graph of a linear equation is a straight line.The y-value of the point where the line crosses theis called the y-intercept.The slope-intercept form of a linear equation is written asy-intercept., where m is the slope and b is theKEY CONCEPTSlope-Intercept FormAn equation of the linehaving slope m andy-intercept b isy mx b.Name the slope and y-intercept of the graph of eachequation.y 2 x 6 The slope is . The y-intercept .3y 0 The slope is . The y-intercept .x 7 The graph is a line.The slope isundefined. There is no y-intercept.3y 12 6xRewrite the equation in slope-intercept form by solving for y.3y 12 6x3y 12 12 6x 12 Subtract 12 from each side.3y 6x 12 3 y 6x 12 Divide each side by 3.3 3© Glencoe/McGraw-Hilly 2x 4Simplify. This is written inslope-intercept form.The slope m . The y-intercept is .80 Geometry: Concepts and Applications

4–6ORGANIZE ITUse the index cardlabeled Equations ofLines to recordimportant formulas andideas in this lesson. Giveexamples that show themost important ideas inthe lesson.Your TurnName the slope and y-intercept of thegraph of each equation.a. y 6x 13b. y 8Parallelsc. x 7d. 4x 3y 5Graph 2x y 4 using the slope and y-intercept.First, rewrite the equation in slope-intercept form.2x y 42x y 4 Subtract 2x from each side.y y 4 2x Divide each side by 1. 1 1© Glencoe/McGraw-Hilly The y-intercept is 4. So, the point(0, 4) is on the line. Since the slopeis 2, or 2 , plot a point by using a rise1ofunits (up) and a run ofunit (right). Draw a line through thetwo points.Slope-intercept formOyx(1, 2)(0, 4)Geometry: Concepts and Applications 81

4–6Your Turny-intercept.Graph 3x 4y 8 using the slope andWRITE ITyExplain how you canfind the slope of a lineperpendicular to a givenline.OxWrite an equation of the line parallel to the graph ofy 2x 3 that passes through the point at (0, 1).Because the lines are parallel, they must have the sameslope. So, m .To find b, use the ordered pair (0, 1) and substitutefor m, x, and y in the slope-intercept form.y mx b1 (0) b m , (x, y) 1 0 b bThe value of b is. So, the equation of the line is.HOMEWORKASSIGNMENTPage(s):Exercises:Your Turna. Write an equation of the line parallel to the graph of5x y 6 that passes through the point (1, 3).b. Write an equation of the line perpendicular to the graph ofy 2x 1 that passes through the point (4, 5).© Glencoe/McGraw-Hill82 Geometry: Concepts and Applications

C H A P T E R4BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 4 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 4, go to:www.glencoe.com/sec/math/t_resources/free/index.php.BUILD YOURVOCABULARYYou can use your completedVocabulary Builder (pages 66–67)to help you solve the puzzle.4-1Parallel Lines and PlanesChoose the term that best completes each sentence.1. (Skew/Parallel) lines always lie on the same plane.2. (Perpendicular/Skew) lines never have any points in common.3. (Parallel/Perpendicular) lines never intersect.4-2Parallel Lines and TransversalsRefer to the figure and match theterm with the best representativeangle pair. Angle pairs cannot bematched more than once.5 67 81 24 3pq© Glencoe/McGraw-Hill4. consecutive interior angles5. exterior angles6. alternate interior anglesra. 2 and 7b. 3 and 6c. 4 and 6d. 1 and 77. alternate exterior anglese. 3 and 4Geometry: Concepts and Applications 83

Chapter4BRINGING IT ALL TOGETHER4-3Transversals and Corresponding AnglesIn the figure, m, andtransversal r is perpendicularto m. Name all angles congruentto the given angle.1 328. 4475 68 9m9. 3rs10. 9Refer to the above figure to find the measure of thespecified angle if m3 40.11. 4 12. 513. 8 14. 24-4Proving Lines ParallelFind the values of a, b, andc so that mn.15. a (5b 6)12c a m36 39 36 n16. b 17. c 18. Name the parallel lines.© Glencoe/McGraw-Hillk56 88 mn84 Geometry: Concepts and Applications

Chapter4BRINGING IT ALL TOGETHER4-5SlopeA wheelchair access ramp must be added to a home. Oneplan showed a ramp that started 30 feet away from theentrance. The entrance was 3 feet higher than ground level.The second plan started the ramp 15 feet from the same3-foot high entrance.19. What is the slope of each ramp?20. Which slope is steeper?21. Given A(0, 4), B(3, 6), C(1, 2), and D(3, 1), determinewhether AB and CD are parallel, perpendicular, or neither.4-6Equations of LinesIdentify the slope and y-intercept of each equation.22. y 6x 1 2 23. 5x 4y 724. y 2© Glencoe/McGraw-Hill25. x 526. Write an equation of a line parallel to y 3x 2 that passesthrough the point (1, 4).Geometry: Concepts and Applications 85

C H A P T E R4ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study are given witheach item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 4.• You may want to take the Chapter 4 Practice Test onpage 183 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 4 Study Guide and Reviewon pages 180–182 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 4 Practice Test onpage 183.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 4 Foldable.• Then complete the Chapter 4 Study Guide and Review onpages 180–182 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 4 Practice Test onpage 183.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature86 Geometry: Concepts and Applications

C H A P T E R5Triangles and CongruenceUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Chapter 5Begin with a sheet of plain 8 1 " 11" paper.2FoldFold in half lengthwise.FoldFold the top to the bottom.OpenOpen and cut along thesecond fold to make twotabs.© Glencoe/McGraw-HillLabelLabel each tab as shown.Trianglesclassifiedby AnglesTrianglesclassifiedby SidesNOTE-TAKING TIP: When you take notes, definenew terms and write about the new concepts youare learning in your own words. Then, write yourown examples that use the new terms andconcepts.Geometry: Concepts and Applications 87

C H A P T E R5BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 5.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExampleacute trianglebasebase anglescongruent trianglescorresponding partsequiangular triangle[eh-kwee-AN-gyu-lur]equilateral triangle[EE-kwuh-LAT-ur-ul]imageincluded angleincluded sideisometry[eye-SAH-muh-tree]© Glencoe/McGraw-Hillisosceles triangle[eye-SAHS-uh-LEEZ]88 Geometry: Concepts and Applications

Chapter5BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExamplelegsmappingobtuse trianglepreimagereflectionright trianglerotationscalene triangle[SKAY-leen]transformation© Glencoe/McGraw-Hilltranslationvertexvertex angleGeometry: Concepts and Applications 89

5–1Classifying TrianglesBUILD YOUR VOCABULARY (pages 88–89)WHAT YOU’LL LEARN• Identify the parts oftriangles and classifytriangles by their parts.The side that is opposite the vertex angle in antriangle is called the base.In an isosceles triangle, the two angles formed by theand one of the congruentare calledORGANIZE ITDraw examples of acute,obtuse, right, scalene,isosceles, and equilateraltriangles in your notes.Trianglesclassifiedby AnglesTrianglesclassifiedby Sidesbase angles.The congruent sides in an isosceles triangle are the legs.The vertex of each angle of ais a vertex ofthe triangle.The angle formed by thesides in antriangle is called the vertex angle.Classify each triangle by its angles and by its sides.The triangle is a30triangle.© Glencoe/McGraw-Hill12030The triangle is antriangle.90 Geometry: Concepts and Applications

5–1Your TurnClassify each triangle by its angles and byits sides.a. b.44 20 74 140 62 20 REMEMBER ITThe vertex of eachangle is a vertex of thetriangle.Find the measures of XY and YZ of isosceles triangleXYZ if X is the vertex angle.2n 2X10Y2n 2ZSince X is the vertex angle, .So, XY . Write and solve an equation.XY Substitution© Glencoe/McGraw-Hill2n 2 10 Subtract fromeach side.Divide each sideby .n The value of n is .Geometry: Concepts and Applications 91

5–1To find the measures of XY and YZ, replace n withthe expression for each measure.inXY 2n 2 2 2 2YZ 2n 2 2 2 2Therefore, XY and YZ .Your TurnTriangle DEF is an isosceles triangle with baseEF. Find DE and EF.F7x 2D3x 8EHOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill92 Geometry: Concepts and Applications

5–2Angles of a TriangleWHAT YOU’LL LEARNTheorem 5-1 Angle Sum TheoremThe sum of the measures of the angles of a triangle is 180.• Use the Angle SumTheoremFind mP in MNP if mM 80 and mN 45.mP mM mN 180Angle Sum TheoremmP 180 SubstitutionmP 125 180mP 125 180 Subtract.mP Find the value of each variable in ABC.ABC is a vertical angle to the given anglemeasure of 75. Since vertical angles arecongruent, mABC 75 x.Ay75Bx58CmABC mBCA mCAB 180Angle Sum Theorem© Glencoe/McGraw-HillREVIEW ITWhat does it meanwhen two angles arecomplementary?(Lesson 3-5) 58 180 Substitution133 y 180133 y 133 180 133 Subtract.y Therefore, x and y .Your TurnFind the value of each variable.x65 45 y Geometry: Concepts and Applications 93

5–2Theorem 5-2The acute angles of a right triangle are complementary.Theorem 5-3The measure of each angle of an equiangular triangle is 60.J(x 15)WRITE ITIs it possible to have tworight angles in atriangle? Justify youranswer.Find mJ and mK in righttriangle JKL.LmJ mK 90 Theorem 5-2(x 15) (x 9) 90 Substitution 24 902x 24 90 Subtract.(x 9)KCombine like terms.2x2x 66 66Divide.x Replace x withmJ mK 15 or 9 orin each angle expression.Therefore, mJ and mK .HOMEWORKASSIGNMENTYour TurnFind the value of a, b, and c.a c b 35 © Glencoe/McGraw-HillPage(s):Exercises:BUILD YOUR VOCABULARY (page 88)When all three angles in a triangle are congruent, thetriangle is said to be equiangular.94 Geometry: Concepts and Applications

5–3Geometry in MotionWHAT YOU’LL LEARN• Identify translations,reflections, androtations and theircorresponding parts.BUILD YOUR VOCABULARY (page 89)a figure from one position to anotherwithout turning it is called a translation.a figure over a line creates the mirror imageof the figure, or a reflection.a rotation.a figure around a fixed point createsIdentify each motion as a translation, reflection, orrotation.© Glencoe/McGraw-HillYour TurnIdentify each motion as a translation,reflection, or rotation.a. b.Geometry: Concepts and Applications 95

5–3BUILD YOUR VOCABULARY (pages 88–89)Pairing each point on the original figure, or, with exactly one point on theis called mapping.The moving of eachof a preimage to a newfigure called the image is a transformation.The new figure in aimage.In a transformation, thepreimage.is called thefigure is called theIn the figure, RSTXYZ by a translation.Name the image of T.RST XYZ T corresponds to.TRSZXYName the side that corresponds to XY .RSTXYZPoint R corresponds to point .Point S corresponds to point .So, corresponds to .Your TurnIn the figure,QRS DEF by a rotation.a. Name the angle that corresponds to R.RFD© Glencoe/McGraw-HillQb. Name the side that corresponds to QR.SE96 Geometry: Concepts and Applications

5–3BUILD YOUR VOCABULARY (page 88)Translations, reflections, and rotations are all isometriesand do not change the or of thefigure being moved.Identify the type(s) of transformations that were usedto complete the work below.Some figures can be moved toanother withoutturning or flipping. Other figures have been turned aroundapoint with respect to the original.Therefore, the transformations areand.Your TurnIdentify the type(s) of transformations thatwere used to complete the work below.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 97

5–4Congruent TrianglesWHAT YOU’LL LEARN• Identify correspondingparts of congruenttriangles.BUILD YOUR VOCABULARY (page 88)If a triangle can be translated, rotated, or reflected ontoanother triangle so that all of thecorrespond, the triangles are said to be congruent.The parts of congruent triangles thatcorresponding parts.are calledKEY CONCEPTDefinition of CongruentTriangles If thecorresponding parts oftwo triangles arecongruent, then the twotriangles are congruent.Likewise, if two trianglesare congruent, then thecorresponding parts ofthe two triangles arecongruent.If ABC FDE, name the congruent angles and sides.Then draw the triangles, using arcs and slash marks toshow congruent angles and sides.ABName the three pairs of congruent angles by looking at theorder of the vertices in the statement ABC FDE.CFDEA , B ,and C .Since A corresponds to , and B corresponds to , .Since B corresponds to D, and C corresponds to E, .© Glencoe/McGraw-HillSince corresponds to F, and corresponds to E, FE.98 Geometry: Concepts and Applications

5–4REMEMBER ITThe order of thevertices in a congruencestatement shows thecorresponding parts ofthe congruent triangles.The corresponding parts oftwo congruent trianglesare marked on the figure.Write a congruencestatement for the twotriangles.List the congruent angles and sides.TMSRLNL M R N LN ST TR SRThe congruence statement can be written by matching theof theangles. Therefore,LMN .Your Turna. If ACB ECD, name thecongruent angles and sides.Then draw the triangles,using arcs and slash marksto show congruent anglesand sides.b. Write another congruence statement for the two trianglesother than the one given above.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 99

5–5SSS and SASWHAT YOU’LL LEARN• Use the SSS and SAStests for congruence.Postulate 5-1 SSS PostulateIf three sides of one triangle are congruent to threecorresponding sides of another triangle, then the trianglesare congruent.REMEMBER ITThe letterdesignating theincluded angle appearsin the name of bothsegments that form theangle.In two triangles, DF UV , FE VW , and DE UW .Write a congruence statement for the two triangles.Draw a pair ofcongruent parts withone triangle.Ftriangles. Identify the. Label the vertices ofVDEUWUse the given information to label thesecond triangle.in theBy SSS, .Your TurnIn two triangles, CB EF, CA ED, andBA FD. Write a congruence statement for the two triangles.BUILD YOUR VOCABULARY (page 88)© Glencoe/McGraw-HillIn a triangle, theformed by two givenis the included angle.100 Geometry: Concepts and Applications

5–5Postulate 5-2 SAS PostulateIf two sides and the included angle of one triangle arecongruent to the corresponding sides and included angle ofanother triangle, then the triangles are congruent.WRITE ITExplain the SSS and SAStests for congruence inyour own words. Givean example of each.Determine whether thetriangles shown at the rightare congruent. If so, write acongruence statement andexplain why the trianglesare congruent. If not,explain why not.RSTXYZThere are three pairs ofsides,RS , ZX and RT .Therefore, by .Your TurnDetermine whetherthe triangles to the right arecongruent. If so, write a congruencestatement and explain why thetriangles are congruent. If not,explain why not.EABDC© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 101

5–6ASA and AASWHAT YOU’LL LEARN• Use the ASA and AAStests for congruence.BUILD YOUR VOCABULARY (page 88)Theof the triangle that falls between two givenis called the included side and is the onecommon side to both angles.Postulate 5-3 ASA PostulateIf two angles and the included side of one triangle arecongruent to the corresponding angles and included side ofanother triangle, then the triangles are congruent.In DEF and ABC, D C, E B, and DE CB .Write a congruence statement for the two triangles.Draw a pair oftriangles. Mark the congruentparts with and . Label the vertices ofone triangle D, E, and F.FADLocate C and B on the unlabeled triangle in the samepositions as and . The unassigned vertex is. Therefore, .EYour TurnIn RST and XYZ, ST XZ, S X, andT Z. Write a congruence statement for the two triangles.CB© Glencoe/McGraw-Hill102 Geometry: Concepts and Applications

5–6Theorem 5-4 AAS TheoremIf two angles and a nonincluded side of one triangle arecongruent to the corresponding two angles andnonincluded side of another triangle, then the trianglesare congruent.XYZ and QRS each have one pair of sides and onepair of angles marked to show congruence. What otherpair of angles needs to be marked so the two trianglesare congruent by AAS?If Q and X are marked, and , then and would haveto be congruent for the triangles to be congruent by .YQSX Z RYour TurnACB and FED each have one pair of sidesand one pair of angles marked to show congruence. What otherpair of angles needs to be marked so the two triangles arecongruent by AAS?CB© Glencoe/McGraw-HillHOMEWORKASSIGNMENTADEFPage(s):Exercises:Geometry: Concepts and Applications 103

C H A P T E R5BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 5 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 5, go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder (pages 88–89)to help you solve the puzzle.5–1Classifying TrianglesComplete each statement.1. The sum of the measures of a triangle’s interior angles is .2. The angle is the angle formed by two congruent sides of anisosceles triangle.3. The angles of a right triangle are complementary.4. A triangle with no congruent sides is .5–2Angles of a TriangleFind the value of each variable.5. 6. ya61 a a© Glencoe/McGraw-Hill104 Geometry: Concepts and Applications

Chapter5BRINGING IT ALL TOGETHER5–3Geometry in MotionSuppose SRN CDA.7. Which angle corresponds to S?RD8. Name the preimage of AD.SCNA9. Identify the transformation that occurred in the mapping.5–4Congruent TrianglesIf ABC QRS, name the corresponding congruent parts.10. B 11. AC12. RQ 13. C5–5SSS and SAS14. The pairs of triangles at the rightare congruent. Write a congruencestatement and the reason thetriangles are congruent.G LJFHK© Glencoe/McGraw-Hill5-6ASA and AASUnderline the best term to make the statement true.15. [Mapping/Congruence] of triangles is explained by SSS, SAS, ASA,and AAS.16. AAS indicates two angles and their [included/nonincluded] side.Geometry: Concepts and Applications 105

C H A P T E R5ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 5.• You may want to take the Chapter 5 Practice Test onpage 223 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 5 Study Guide and Reviewon pages 220–222 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 5 Practice Test onpage 223 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 5 Foldable.• Then complete the Chapter 5 Study Guide and Review onpages 220–222 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 5 Practice Test onpage 223 of your textbook.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature106 Geometry: Concepts and Applications

C H A P T E R6More About TrianglesUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with four sheets of lined 8 1 " 11" paper.2FoldFold each sheet of paper inhalf along the width. Thencut along the crease.Chapter 6StapleStaple the eight half-sheetstogether to form abooklet.CutCut seven lines from thebottom of the top sheet,six lines from the secondsheet, and so on.© Glencoe/McGraw-HillLabelLabel each tab with alesson number. The lasttab is for vocabulary.6-16-26-36-46-56-66-7VocabularyNOTE-TAKING TIP: As you read a lesson, takenotes on the materials. Include definitions,concepts, and examples. After you finish eachlesson, make an outline of what you learned.Geometry: Concepts and Applications 107

C H A P T E R6BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 6.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplealtitudeangle bisectorcentroidcircumcenter[SIR-kum-SEN-tur]concurrentEuler linehypotenuse[hi-PA-tin-oos]© Glencoe/McGraw-Hill108 Geometry: Concepts and Applications

Chapter6BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleincenterlegmediannine-point circleorthocenter[OR-tho-SEN-tur]perpendicular bisector© Glencoe/McGraw-HillPythagorean Theorem[puh-THA-guh-REE-uhn]Pythagorean tripleGeometry: Concepts and Applications 109

6–1MediansWHAT YOU’LL LEARN• Identify and constructmedians in triangles.BUILD YOUR VOCABULARY (page 109)A median is a segment that joins a vertex of a triangle andthe midpoint of the side opposite that vertex.ORGANIZE ITUnder the tab forLesson 6-1, draw anexample of a median.Label the congruentparts. Under the tab forVocabulary, write thevocabulary words forthis lesson.6-16-26-36-46-56-66-7VocabularyIn ABC, CE and AD are medians.If CD 2x 5, BD 4x 1, andAE 5x 2, find BE.Since CE and AD are medians,D and E are midpoints. Solve for x.CD BDDefinition of medianSubstitution2x 5 4x 1 Subtract.5 2x 15 2x 1 Add.6 2x Divide.CADEB xUse the values for x and AE to find BE.AE BEDefinition of median5x 2 BESubstitution5( ) 2 BE Substitution15 2 BE BEYour TurnIn OPS, ST and QP aremedians. If PT 3x 1, OT 2x 1, andOQ 4x 2, find SQ.OQTP© Glencoe/McGraw-HillS110 Geometry: Concepts and Applications

6–1BUILD YOUR VOCABULARY (page 108)The threeof a triangle intersect at acommon point known as the centroid.When three or more lines or segments meet at the samepoint, they are said to be concurrent.Theorem 6-1The length of the segment from the vertex to the centroidis twice the length of the segment from the centroid to themidpoint.YIn XYZ, XP, ZN, and YM aremedians.XNQPFind YQ if QM 4.MSince QM , YQ 2 or .ZIf QZ 18, what is ZN?Since QZ 18 and QZ 2 3 18 2 ZN for ZN.318 2 3 ZN ZN, solve the equation 3 2 (18) 3 2 2 3 ZN Multiply each side by . ZN© Glencoe/McGraw-HillHOMEWORKASSIGNMENTYour TurnIn EFG, FA, GB,and EC are medians.a. Find EO if CO 3.AEOBPage(s):Exercises:b. If FA 18, what are the measuresof AO and OF?GCFGeometry: Concepts and Applications 111

6–2Altitudes and Perpendicular BisectorsWHAT YOU’LL LEARN• Identify and constructaltitudes andperpendicular bisectorsin triangles.BUILD YOUR VOCABULARY (page 108)An altitude of a triangle is a perpendicular segment withone endpoint at aand the other endpoint ontheopposite that vertex.KEY CONCEPTIs AD an altitude of the triangle?AAltitudes of TrianglesAD isa perpendicularAcute Triangle Thealtitude is inside thetriangle.Right Triangle Thealtitude is a side of thetriangle.Obtuse Triangle Thealtitude is outside thethe triangle.segment. So, ADaltitude of the triangle.anIs GJ an altitude of the triangle?GJ FH,is a vertex, andis on the side opposite G.BFGJDCHSo, GJan altitude of the triangle.Your Turna. Is BD an altitude of the triangle?Bb. Is XY an altitude of the triangle?ZADC© Glencoe/McGraw-HillREMEMBER ITEvery triangle hasthree altitudes—onethrough each vertex.YX112 Geometry: Concepts and Applications

6–2BUILD YOUR VOCABULARY (page 109)Aline or segment thata side of a triangle is called the perpendicular bisector ofthat side.ORGANIZE ITUnder the tab forLesson 6-2, drawone example of analtitude and one of aperpendicular bisector.Label congruent partsand right angles.Is MN a perpendicular bisector of aside of the triangle?Since N is the midpoint of KL, MN is abisector of side KL. MNso MN isperpendicular to KL,a perpendicular bisector in KLM.LNKM6-16-26-36-46-56-66-7VocabularyIs AD a perpendicular bisector ofa side of the triangle?AAD BC but DtheBDCmidpoint of BC. So, ADside BC in ABC.a perpendicular bisector ofYour Turna. Is BD a perpendicular bisector ofthe triangle?B© Glencoe/McGraw-Hillb. Is LM a perpendicular bisectorof the triangle?ADKCJMLGeometry: Concepts and Applications 113H

6–2Tell whether MN is an altitude, a perpendicularbisector, both, or neither.NKLMWRITE IT; MN KL but N the midpoint ofHow is a perpendicularbisector different from amedian?KL. So, MNof side KL in KLM.aYour TurnTell whether XO is an altitude, a perpendicularbisector, both, or neither.XZOYHOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill114 Geometry: Concepts and Applications

6–3Angle Bisectors of TrianglesWHAT YOU’LL LEARN• Identify and use anglebisectors in triangles.BUILD YOUR VOCABULARY (page 108)An angle bisector of a triangle is a segment that separatesan angle of the triangle into twoangles.ORGANIZE ITUnder the tab forLesson 6-3, draw anexample of an anglebisector. Label thecongruent parts.6-16-26-36-46-56-66-7VocabularyIn ABD, AC bisects BAD. If m1 41, find m2.BSince AC bisects BAD, m1 .Since m1 , m2 .In KMN, NL bisects KNM. If KNM is a right angle,find m2.MA1 2LCK12NDm2 1 2 (mKNM)m2 1 2 m2 In WYZ, ZX bisects WZY. If m1 55, find mWZY.© Glencoe/McGraw-HillW12ZXYour Turna. In XYZ, YW bisects XYZ.If m2 33, find m1.YmWZY 2(m1)mWZY 2mWZY XW12ZYGeometry: Concepts and Applications 115

6–3b. In NOM, OP bisects NOM.If NOM 85, find m4.O3 4NPMc. In RST, SU bisects RST.If m6 36.5, find mRST.RU56STIn FHI, IG is an angle bisector.Find mHIG.mHIG mFIGGF5(x 1)(4x 1)IH4x 1 5x 5Distributive Property4x 1 4x 5x 5 4xSubtract.1 x 51 5 x 5 5 Add. xmHIG 4x 1 4 1 1 HOMEWORKASSIGNMENTPage(s):Exercises:Your TurnIn JKL, KM is anangle bisector. Find mJKM.KJ5x 24x 7M© Glencoe/McGraw-HillL116 Geometry: Concepts and Applications

6–4Isosceles TrianglesWHAT YOU’LL LEARN• Identify and useproperties of isoscelestriangles.BUILD YOUR VOCABULARY (page 109)A leg of an isosceles triangle is one of the twosides.Theorem 6-2 Isosceles Triangle TheoremIf two sides of a triangle are congruent, then the anglesopposite those sides are congruent.Theorem 6-3The median from the vertex angle of an isosceles trianglelies on the perpendicular bisector of the base and the anglebisector of the vertex angle.ORGANIZE ITUnder the tab forLesson 6-4, draw anexample of an isoscelestriangle. Label thecongruent parts, andthe special names forsides and angles.Find the values of the variables.In the top triangle, find the value ofbase angle x. Since the triangle isisosceles, and one base angle 35,x .3545In the bottom triangle, find the value of base angle y. Since thexy© Glencoe/McGraw-Hill6-16-26-36-46-56-66-7Vocabularyother base angle 45, y .Your TurnFind the valuesof the variables.y60JKx12LTheorem 6-4 Converse of Isosceles Triangle TheoremIf two angles of a triangle are congruent, then the sidesopposite those angles are congruent.Geometry: Concepts and Applications 117

6–4In DEF, 1 2 and m1 28.Find mF, DF, and EF.First, find mF. You know thatm1 28. Since 1 2,m2 28.m1 m2 mF 180DE23x 31F2x 2Angle Sum Theorem mF 180 Replace m1 andm2. mF 18056 mF 56 180 56 Subtract.WRITE ITCan an isosceles trianglebe an equiangulartriangle?mF Next, find DF. Since 1 2, Theorem 6-4 states thatDF EF.DF EFCongruent segments2x 2 3x 3Replace DF and EF.2x 2 3x 3 Subtract.2 x 32 x 3 Add. xBy replacing x with 5, you find that DF 2x 2 2(5) 2 10 2 or . EF 3x 3 3(5) 3 15 3 or .HOMEWORKASSIGNMENTYour Turnthe variables.Find the values ofXy5x 2Z3x 8yY© Glencoe/McGraw-HillPage(s):Exercises:Theorem 6-5A triangle is equilateral if and only if it is equiangular.118 Geometry: Concepts and Applications

6–5Right TrianglesWHAT YOU’LL LEARN• Use tests for congruenceof right triangles.BUILD YOUR VOCABULARY (pages 108–109)In atriangle the side opposite theangle is known as the hypotenuse.The two sides that form theangle are called legs.Theorem 6-6 LL TheoremIf two legs of one right triangle are congruent to thecorresponding legs of another right triangle, then thetriangles are congruent.Theorem 6-7 HA TheoremIf the hypotenuse and an acute angle of one right triangleare congruent to the hypotenuse and corresponding angleof another right triangle, then the triangles are congruent.Theorem 6-8 LA TheoremIf one leg and an acute angle of one right triangle arecongruent to the corresponding leg and angle of anotherright triangle, then the triangles are congruent.Postulate 6-1 HL PostulateIf the hypotenuse and a leg of one right triangle arecongruent to the hypotenuse and corresponding leg ofanother right triangle, then the triangles are congruent.© Glencoe/McGraw-HillORGANIZE ITUnder the tab forLesson 6-5, draw anexample of a righttriangle. Label thespecial names for thesides of the triangle.Under the tab forVocabulary, write thevocabulary words forthis lesson.6-16-26-36-46-56-66-7VocabularyDetermine whether each pair of right triangles iscongruent by LL, HA, LA, or HL. If it is not possible toprove that they are congruent, write not possible.DEGFThere is one pair of congruentangles, DFE GFE. Thehypotenuses are congruent, DF GF.Therefore, DEF GEF by .Geometry: Concepts and Applications 119

6–5XThere is one pair ofYJKLZacute angles, Z L. There is one pair of, XZ KL.Therefore, YXZ JKL by .WRITE ITWhich test forcongruence is used toestablish the LATheorem? Explain.Your TurnDetermine whether each pair of righttriangles is congruent by LL, HA, LA, orHL. If it isnot possible to prove that they are congruent, writenot possible.a.SRTUb.PQABCRHOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill120 Geometry: Concepts and Applications

6–6The Pythagorean TheoremBUILD YOUR VOCABULARY (page 109)WHAT YOU’LL LEARN• Use the PythagoreanTheorem and itsconverse.The Pythagorean Theorem can be used to determine thelengths of the sides of a right triangle. It states that theof the squares of theequals the square of the hypotenuse.of a right triangleTheorem 6-9 Pythagorean TheoremIn a right triangle, the square of the length of thehypotenuse c is equal to the sum of the squares of thelengths of the legs a and b.ORGANIZE ITUnder the tab forLesson 6-5, write thePythagorean Theorem.Draw a right triangleand label the legs a andb, and the hypotenuse c.6-16-26-36-46-56-66-7VocabularyFind the length of the hypotenuseof the right triangle.Use the Pythagorean Theorem to find thelength of the hypotenuse.c 2 a 2 b 2c 2 c 2 c 2 4002216 ftReplace a and b.12 ftTake the square root ofeach side.© Glencoe/McGraw-Hillc The length is .Your Turna. Find the length of the hypotenuseof the right triangle.c9 in.40 in.Geometry: Concepts and Applications 121

6–6REMEMBER ITAlways check to seethat c represents thelength of the longestside.b. Find the length of one leg of a right triangle if the length ofthe hypotenuse is 25 cm and the length of the other leg is23 cm.Theorem 6-10 Converse of the Pythagorean TheoremIf c is the measure of the longest side of a triangle, a and bare the lengths of the other two sides, and c 2 a 2 b 2 ,then the triangle is a right triangle.The lengths of three sides of a triangle are 4, 5, and6 meters. Determine whether this triangle is a righttriangle.Since the longest side is meters, use as c, themeasure of the hypotenuse.c 2 a 2 b 2Pythagorean Theorem6 2 4 2 5 2 Replace c with , a with36 16 2536 41, and b with .HOMEWORKASSIGNMENTPage(s):Exercises:Since c 2 a 2 b 2 , the triangle a righttriangle.Your TurnThe lengths of three sides of a triangle are 5,12, and 13 yards. Determine whether this triangle is a righttriangle.© Glencoe/McGraw-Hill122 Geometry: Concepts and Applications

6–7Distance on the Coordinate PlaneWHAT YOU’LL LEARN• Find the distancebetween two points onthe coordinate plane.Theorem 6-11 Distance FormulaIf d is the measure of the distance between two points withcoordinates (x 1 , y 1 ) and (x 2 , y 2 ), thend (x 2 x 1 y ) 2 ( 2 y 1 . ) 2ORGANIZE ITUnder the tab forLesson 6-7, write theDistance Formula. Thenshow an example tohelp you remember themain idea.Use the Distance Formula to find the distance betweenA(6, 2) and B(4, 4). Round to the nearest tenth, ifnecessary.Use the Distance Formula. Replace (x 1 , y 1 ) with (6, 2) and(x 2 , y 2 ) with .d (x x 2 1 ) 2 (y 2 y 1 ) 2Distance Formula6-16-26-36-46-56-66-7VocabularyAB 4 2 2 2 SubstitutionAB (2) 2 (6 ) 2 AB AB 40AB © Glencoe/McGraw-HillYour Turna. Use the Distance Formula to find the distance betweenM(2, 2) and N(6, 4). Round to the nearest tenth, ifnecessary.Geometry: Concepts and Applications 123

6–7b. Determine whether TRI with vertices T(4, 1), R(2, 5),and I(2, 2) is isosceles.RTIAkio took a ride in a hot-air balloon. The flight began4 miles north of his house. The balloon landed 3 milessouth and 2 miles east of his house. If the balloontraveled in a straight line between the starting andending points of the flight, what was the length ofAkio’s balloon ride?REMEMBER ITOnly use thepositive square rootssince distance is notnegative.Suppose Akio’s house is located at the origin (0, 0). If theballoon ride began 4 miles north of his house, it began at(x 1 , y 1 ), or (0, 4). Since the balloon landed 3 miles south and2 miles east of his house, it landed at (x 2 , y 2 ) at (2, 3). Usethe Distance Formula to find the length of the balloon ride.d (x x 2 1 ) 2 (y 2 y 1 ) 2 (2 ) 03 2 ( 4) 2 27) 2 ( 2RStart(0, 4)Akio’s house 4 9 4 End(2, 3)Akio’s balloon ride was approximatelymiles.HOMEWORKASSIGNMENTPage(s):Exercises:Your TurnMarcelle went to a friend’s house to complete ahomework project after school instead of going directly home.The school lies 2 blocks north of her home. Her friend’s houseis located 3 blocks west and 1 block north of her home. IfMarcelle traveled in a straight path from school to her friend’shome, what was the length of her walk?© Glencoe/McGraw-Hill124 Geometry: Concepts and Applications

C H A P T E R6BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 6 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 6, go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 108–109) to help yousolve the puzzle.6-1MediansComplete the sentence.1. The midpoint of a side of a triangle and the vertex of the oppositeangle are endpoints of a .2. A triangle’s medians are at the centroid.3. In ABC, BD is a median and BD 6.What is BE?BEADC© Glencoe/McGraw-Hill6-2Altitudes and Perpendicular BisectorsFor the triangles shown, state whether AB is an altitude, aperpendicular bisector, both, or neither.4. B5. B6.AAABGeometry: Concepts and Applications 125

Chapter6BRINGING IT ALL TOGETHER6-3Angle Bisectors of Triangles7. In JKL, KH bisects JKL. If m1 12, find mJKL.JHL12K8. What is the value of x so that BD is an angle bisector?AD(5x 2)C(6x 5)B6-4Isosceles TrianglesIndicate whether the statement is true or false.9. The vertex angle of an isosceles triangle is opposite one of thecongruent sides.10. An isosceles triangle must be equiangular.For each triangle, find the values of the variables.11. B 12.63A xF28xG5 y28H© Glencoe/McGraw-HillyC126 Geometry: Concepts and Applications

Chapter6BRINGING IT ALL TOGETHER6-5Right TrianglesDetermine whether each pair of right triangles iscongruent by LL, HA, LA, or HL. If it is not possible to provethat they are congruent, write not possible.13. 14.6-6The Pythagorean TheoremFind the missing measure in each right triangle. Round tothe nearest tenth, if necessary.15. 516.c ft812 10a cm© Glencoe/McGraw-Hill6-7Distance on the Coordinate PlaneUse the Distance Formula to find the distance between eachpair of points. Round to the nearest tenth, if necessary.17. G(3, 1), H(4, 5) 18. R(1, 2), S(5, 6) 19. A(12, 0), B(0, 5)20. Andre walked 2 blocks west of his home to school. After school,he walked to the store which is 1 block east and 1 block northof his home. About how far apart are the school and the store?Geometry: Concepts and Applications 127

C H A P T E R6ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 6.• You are probably ready for the Chapter Test.• You may want to take the Chapter 6 Practice Test onpage 271 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 6 Study Guide and Reviewon pages 268–270 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 6 Practice Test onpage 271.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 6 Foldable.• Then complete the Chapter 6 Study Guide and Review onpages 268–270 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 6 Practice Test onpage 271.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature128 Geometry: Concepts and Applications

C H A P T E R7Triangle InequalitiesUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with a sheet of sheet of notebook paper.FoldFold lengthwise to the holes.CutCut along the top line andthen cut 4 tabs.© Glencoe/McGraw-HillLabelLabel each tab withinequality symbols. Store theFoldable in a 3-ring binder.NOTE-TAKING TIP: When you take notes, definenew vocabulary words, describe new ideas, andwrite examples that help you remember themeanings of the words and ideas.Chapter 7Geometry: Concepts and Applications 129

C H A P T E R7BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 7.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExampleexterior angleinequality[IN-ee-KWAL-a-tee]remote interior angles© Glencoe/McGraw-Hill130 Geometry: Concepts and Applications

7–1Segments, Angles, and InequalitiesWHAT YOU’LL LEARN• Apply inequalities tosegment and anglemeasurements.BUILD YOUR VOCABULARY (page 130)Statements that contain the symbols orcompare quantities or measures that do not have the samevalue and are called inequalities.ORGANIZE ITWrite words under eachtab to describe eachsymbol on your foldable.Postulate 7-1 Comparison PropertyFor any two real numbers a and b, exactly one of thefollowing statements is true: a b, a b, ora b.Refer to the number line and replace in DR LN with, ,or to make a true sentence.DRLN 0 2 S R N D L6 5 4 3 21 0 1 2 3 4343 43 4© Glencoe/McGraw-HillYour TurnRefer to the number line and replacePR QS with , , or to make a true sentence.123inP Q R S5 4 3 2 104 5 6Theorem 7-1If point C is between points A and B, and A, C, and B arecollinear, then AB AC and AB CB.Theorem 7-2If EP is between ED and EF , then mDEF mDEP andmDEF mPEF.Geometry: Concepts and Applications 131

7–1Refer to the figure. Determinewhether each statement istrue or false.JN12 in.36 in.AB JKAB and JK 48 SubstitutionL 45˚KQH135˚A12 in.This is because isC48 in.greater than .mAHC mHKLBmAHC and mHKL 45 SubstitutionThis is because is not greater thanor equal to .Your TurnRefer to the figure. Determine whethereach statement is true or false.Xa. XY XZYb. mXYZ mZXYZ© Glencoe/McGraw-Hill132 Geometry: Concepts and Applications

7–1KEY CONCEPTSIn the figure, mC mA. If each of these measureswere divided by 5, would the inequality still be true?Transitive PropertyFor any numbers a, b,and c,1. If a b and b c, thena c.2. If a b and b c, thena c.A42 cmB4357 cm4739 cmCAddition and SubtractionProperties For anynumbers a, b, and c,1. If a b, then a c b c and a c b c.2. If a b, then a c b c and a c b c.Multiplication andDivision PropertiesFor any numbers a, b,and c,1. If c 0, and a b thenac bc and a c b c .2. If c 0 and a b thenac bc and a c b c .mC mA47 Replace mC withand mA with .47 43 Divide each side by .The inequality still holds because is greaterthan .Your TurnIn XYZ, mX mZ. If each of thesemeasures doubled, would this inequality still hold true?X75© Glencoe/McGraw-HillHOMEWORKASSIGNMENTZ5055YPage(s):Exercises:Geometry: Concepts and Applications 133

7–2Exterior Angle TheoremWHAT YOU’LL LEARN• Identify exterior anglesand remote interiorangles of a triangle anduse the Exterior AngleTheorem.BUILD YOUR VOCABULARY (page 130)An exterior angle of a triangle is an angle that forms apair with one of the angles of the triangle.Remote interior angles of a triangle are theangles that do not form a linear pair with theORGANIZE ITIn your notes, recordexamples of each typeof inequality under theappropriate tab. Be sureto write about therelationships betweensides and angles of atriangle.Name the remote interior angleswith respect to 4.Angleangle.forms awith 2. Therefore,and 3 areremote angles with respect to 4.12 34Your Turnto 2.Name the remote interior angles with respect12 4311 125 610 987© Glencoe/McGraw-HillTheorem 7-3 Exterior Angle TheoremThe measure of an exterior angle of a triangle is equal tothe sum of the measures of its two remote interior angles.134 Geometry: Concepts and Applications

7–2ORGANIZE ITUnder the tab labeledwith a greater than sign,summarize Theorem 7-4.Theorem 7-4 Exterior Angle Inequality TheoremThe measure of an exterior angle of a triangle is greaterthan the measure of either of its two remote interior angles.In the figure, if m1 145 andm5 82, what is m3?m1 m5 5 61 2 3 4Exterior Angle Theorem145 m3 Replace m1 with 145and m5 with 82.145 82 m3 Subtract m3from each side.In the figure, if m6 8x, m3 12, and m2 4(x 5),find the value of x.m6 m3 Exterior Angle Theorem 4(x 5) Replace m6 with 8x,m3 with 12 and m2with 4(x 5).8x 12 © Glencoe/McGraw-Hill8x 4x Combineterms.8x 32 4x Subtract4x 32 32from each side.Divide each sideby .x Geometry: Concepts and Applications 135

7–2REMEMBER ITYour Turna. Find the measure of 1in the figure.Q70The measures of theangles in any trianglehave a sum of 180degrees.1S40Rb. In the figure, if m6 10x 3, m3 6x 6,and m12 49, find thevalue of x.12 4311 125 610 987Name the two angles in CDEthat have measures less than 82.The measure of the exterior angle withCE 8213 2Drespect to 1 is . Angles and are itsremote interior angles. By Theorem, 82 mand 82 m . Therefore, and havemeasures less than 82.HOMEWORKASSIGNMENTPage(s):Exercises:Your TurnName the two angles in JKL that havemeasures less than 117.JL117KTheorem 7-5If a triangle has one right angle, then the other two anglesmust be acute.© Glencoe/McGraw-Hill136 Geometry: Concepts and Applications

7–3Inequalities Within a TriangleWHAT YOU’LL LEARN• Identify the relationshipsbetween the sides andangles of a triangle.Theorem 7-6If the measures of three sides of a triangle are unequal,then the measures of the angles opposite those sides areunequal in the same order.Theorem 7-7If the measures of three angles of a triangle are unequal,then the measures of the sides opposite those angles areunequal in the same order.In KLM, list the angles in order from least to greatestmeasure.15 mK10 mL21 mMWrite the segment measures in order fromtogreatest. Then, use Theorem to write the measuresof the angles opposite those sides in the same order.KM KL LMm m m© Glencoe/McGraw-HillTherefore, the angles in order from least to greatest are , , and .Your TurnIn QPS, list the anglesin order from least to greatest measure.P9.5 10Q11SGeometry: Concepts and Applications 137

7–3Identify the side of KLM with the greatest measure.K67L5954MWrite the angle measures in order from least to .Then, use Theorem to write the measures of the sidesopposite those angles in the same order.mM mL mKORGANIZE ITUnder the tab labeledwith a greater than sign,summarize Theorem 7-8using the words “greaterthan”.Therefore,has the greatest measure.Your TurnIn XYZ, list the sidesin order from least to greatest measure.Y60HOMEWORKASSIGNMENTX59 61Theorem 7-8In a right triangle, the hypotenuse is the side with thegreatest measure.Z© Glencoe/McGraw-HillPage(s):Exercises:138 Geometry: Concepts and Applications

7–4Triangle Inequality TheoremWHAT YOU’LL LEARN• Identify and use theTriangle InequalityTheorem.Theorem 7-9 Triangle Inequality TheoremThe sum of the measures of any two sides of a triangleis greater than the measure of the third side.Determine if the three numbers can be the measures ofthe sides of a triangle.ORGANIZE ITUnder the tab labeledwith a greater thansign, summarizeTheorem 7-9.6, 7, 96 7 96 9 77 9 6All possible casesform a triangle.1, 7, 87 8 18 1 71 7 8true. Sides with these measures© Glencoe/McGraw-HillAll possible casesmeasuresform a triangle.true. Sides with theseYour TurnDetermine if the three numbers can be themeasures of the sides of a triangle.a. 15, 40, 19b. 4, 18, 21Geometry: Concepts and Applications 139

7–4What are the greatest and least possible whole-numbermeasures for a side of a triangle whose other two sidesmeasure 4 feet and 6 feet?Let x be the measure of the third side of the triangle. x isgreater than the difference of the measures of the twoother sides.x 6 x x is less than the sum of the measures of the two other sides.x 6 WRITE ITIn your own words,explain why two sides ofa triangle, when addedtogether, cannot equalthe length of the thirdside.x Therefore, x .If the measures of two sides of a triangle are 12 metersand 14 meters, find the range of possible measures ofthe third side.Let x be the measure of the third side of the triangle. x isgreater than the difference of the measures of the twoother sides.x 14 x x is less than the sum of the measures of the two other sides.x 14 x © Glencoe/McGraw-HillTherefore, x .140 Geometry: Concepts and Applications

7–4Your Turna. What are the greatest and least possible whole-numbermeasures for a side of a triangle whose other two sidesmeasure 23 cm and 29 cm?b. If the measures of two sides of a triangle are 11 inchesand 3 inches, find the range of possible measures of thethird side.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 141

C H A P T E R7BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 7 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 7, go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder (page 130)to help you solve the puzzle.7-1Segments, Angles, and InequalitiesReplacewith , , or to make a true sentence.1. JK KX 2. LM JLJ K X L M3. KM JL 4. KL XM6 5 4 3 2 101234 55. mBCD mBDEAB6. mCBE mEDCE607938 50 1647DC7-2Exterior Angle Theorem7. Name the remote interior angles of ABCwith respect to 5.8. BD AC and m15 139.B1 23 4A 5 6 E 7 8 C9 10 11 12 13 14© Glencoe/McGraw-HillWhat is m10?9. If m1 19x, m16 6x, andmDAB 91, find the value of x.16 1715 18D142 Geometry: Concepts and Applications

Chapter7BRINGING IT ALL TOGETHER7-3Inequalities Within a TriangleIn each triangle, list the angles from least to greatest.10. A11.Z98.514BY10.1C87.6XIn each triangle, list the sides measuring least to greatest.12. B13.XC8296 Z 16Y46A7-4Triangle Inequality Theorem© Glencoe/McGraw-HillDetermine if the numbers given can be measures of the sides ofa triangle.14. 7.7, 16.8, 11.3 15. 36, 12, 2816. 7, 9, 16Find the range of possible values for the third side of the triangle.17. 16, 7 18. 12, 1019. 5, 9Geometry: Concepts and Applications 143

C H A P T E R7ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 7.• You may want to take the Chapter 7 Practice Test on page305 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 7 Study Guide and Reviewon pages 302–304 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 7 Practice Test onpage 305.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 7 Foldable.• Then complete the Chapter 7 Study Guide and Review onpages 302–304 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 7 Practice Test onpage 305.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature144 Geometry: Concepts and Applications

C H A P T E R8QuadrilateralsUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with three sheets of lined 8 1 " 11"2paper.FoldFold each sheet of paperin half from top tobottom.CutCut along the fold. Staplethe six sheets together toform a booklet.CutCut five tabs. The top tabis 3 lines wide, the next tabis 6 lines wide, and so on.© Glencoe/McGraw-HillLabelLabel each of the tabswith a lesson number.QuadrilateralsNOTE-TAKING TIP: When you read and learn newconcepts, help yourself remember these conceptsby taking notes, writing definitions andexplanations, and drawing models as needed.8–18–2Chapter 8Geometry: Concepts and Applications 145

C H A P T E R8BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 8.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in thesecond column for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplebase anglesbasesconsecutive[con-SEK-yoo-tiv]diagonalsisosceles trapeziodkitelegs© Glencoe/McGraw-Hillmedian146 Geometry: Concepts and Applications

Chapter8BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExamplemidsegmentnonconsecutiveparallelogramquadrilateralrectanglerhombus[ROM-bus]© Glencoe/McGraw-Hillsquaretrapezoid[TRAP-a-ZOYD]Geometry: Concepts and Applications 147

8–1QuadrilateralsBUILD YOUR VOCABULARY (pages 146–147)WHAT YOU’LL LEARN• Identify parts ofquadrilaterals andfind the sum of themeasures of theinterior angles of aquadrilateral.A quadrilateral is asides andgeometric figure withvertices.Any two sides, vertices, or angles of a quadrilateral areeither consecutive or nonconsecutive.ORGANIZE ITUnder the tab forLesson 8-1, write therules for classifyingquadrilaterals. Draw aquadrilateral and labelthe consecutive andnonconsecutive sides, aswell as the diagonals.Segments that joincalled diagonals.Refer to quadrilateral DEFG.DGvertices areQuadrilaterals8–18–2FEName all pairs of consecutive angles.and G, and F, and E, andand D are consecutive angles.Name all pairs of nonconsecutive vertices.D and are nonconsecutive vertices. and G arenonconsecutive vertices.Name all pairs of consecutive sides.DG and , FG and , EF and , and DE© Glencoe/McGraw-Hillandare pairs of consecutive sides.148 Geometry: Concepts and Applications

8–1REMEMBER ITConsecutive sidesshare a vertex;nonconsecutive sidesdo not.Your TurnRefer to quadrilateralWXYZ.a. Name all pairs of consecutive angles.ZWXYConsecutive vertices arethe endpoints of a sidewhile nonconsecutivevertices are not.Consecutive anglesshare a side of thequadrilateral whilenonconsecutive anglesdo not.b. Name all pairs of nonconsecutive vertices.c. Name all pairs of consecutive sides.Theorem 8-1The sum of the measures of the angles of a quadrilateralis 360.REMEMBER ITIn a quadrilateral,nonconsecutive sides,vertices, or angles arealso called oppositesides, vertices, or angles.Find the missing measure if three of the four anglemeasures in quadrilateral ABCD are 90, 120, and 40.mA mB mC mD 360 Theorem 8-1 mD 360 Substitution© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises: mD 360250 mD 250 360 250 Subtract.mD Your TurnFind the missing measure if three of the fourangle measures in quadrilateral RMSQ are 115, 75, and 50.Geometry: Concepts and Applications 149

8–2 ParallelogramsWHAT YOU’LL LEARN• Identify and use theproperties ofparallelograms.BUILD YOUR VOCABULARY (page 147)A parallelogram is awith two pairs ofsides.ORGANIZE ITUnder the tab forLesson 8-2, write thedefinitions andtheorems to help youclassify parallelograms.Draw a parallelogramand label the congruentsides and angles, as wellas properties of thediagonals.Quadrilaterals8–18–2Theorem 8-2Opposite angles of a parallelogram are congruent.Theorem 8-3Opposite sides of a parallelogram are congruent.Theorem 8-4The consecutive angles of a parallelogram aresupplementary.In parallelogram KLMN,KL 23, KN 15, andmK 105.Find LM and MN.KL MN and KN LM Theorem 8-315NK10523MLKL and KN Definition of congruentsegments MN and LM Replace KL withand KN with .Find mM.M K Theorem 8-2mM Definition of congruent angles© Glencoe/McGraw-HillmM Replace mK with .150 Geometry: Concepts and Applications

8–2Find mL.mL mK 180 Theorem 8-4mL 180 Replace mK with .mL 105 – 105 180 – 105Subtract.mL Your TurnIn parallelogramABCD, AB 8, BC 3, andmC 115.DA8CB115 3a. Find AD and CD.b. Find mA.c. Find mB.Theorem 8-5The diagonals of a parallelogram bisect each other.Theorem 8-6A diagonal of a parallelogram separates it into twocongruent triangles.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:In parallelogram PQRS,if PR 32, find PL.Theorem 8-5 states that the diagonalsof a parallelogram bisect each other.Therefore, PL LR or PL 1 2 (PR).PL 1 2 (PR)PL 1 (32) or Replace PR with 32.2Your TurnIn parallelogramPARL, if LA 48, find LO.SPPLORAQLRGeometry: Concepts and Applications 151

8–3Tests for ParallelogramsWHAT YOU’LL LEARN• Identify and use tests toshow that aquadrilateral is aparallelogram.ORGANIZE ITUnder the tab forLesson 8-3, write thetests for parallelograms.Remember to includethe definition of aparallelogram. Drawpictures to accompanyeach theorem.Quadrilaterals8–18–2Theorem 8-7If both pairs of opposite sides of a quadrilateral arecongruent, then the quadrilateral is a parallelogram.In quadrilateral WXYZ, ifWYZ YWX, how couldyou prove that WXYZ is aparallelogram?Show that both pairs of opposite sides are congruent.StatementReason1. WYZ YWX 1. Given2. YZ WX 2.3. WZ YX 3. CPCTC4. WXYZ is a parallelogram. 4.ZWYXYour TurnIn quadrilateralABCD, CAB ACD andAB CD. Show that ABCD is aparallelogram by providinga reason for each step.ADBCREVIEW ITWhat does CPCTCrepresent? (Lesson 5-4)StatementReason1. CAB ACD 1. Given2. AB CD 2. Given3. AC AC 3.4. CAB ACD 4. SAS© Glencoe/McGraw-Hill5. BC AD 5.6. ABCD is a parallelogram. 6.152 Geometry: Concepts and Applications

8–3Theorem 8-8If one pair of opposite sides of a quadrilateral is paralleland congruent, then the quadrilateral is a parallelogram.Theorem 8-9If the diagonals of a quadrilateral bisect each other, thenthe quadrilateral is a parallelogram.WRITE ITExplain how alternateinterior angles could beused to show thatopposite angles in aparallelogram arecongruent.Determine whether each quadrilateral is aparallelogram. If the figure is a parallelogram, give areason for your answer.The figure has one pair of opposite sidesthat areand congruent.Therefore, the quadrilateral is aby Theorem 8-8.One pair of opposite sides is congruentbutopposite sides is. The other pair ofbut not. Therefore, the© Glencoe/McGraw-HillREMEMBER ITThere is usuallymore than one way toprove a statement.HOMEWORKASSIGNMENTPage(s):Exercises:quadrilateralYour TurnDetermine whether the figure is aparallelogram. Justify your answer.a.30 mb.15 m30 m15 ma parallelogram.Geometry: Concepts and Applications 153

8–4Rectangles, Rhombi, and SquaresBUILD YOUR VOCABULARY (page 147)WHAT YOU’LL LEARN• Identify and useproperties ofrectangles, rhombi, andsquares.A rectangle is aangles.A parallelogram withrhombus.with fourcongruent sides is aA parallelogram withsides and fourORGANIZE ITUnder the tab forLesson 8-4, draw thediagram for classifyingrectangles, rhombi, andsquares. Write notes andtheorems to help youremember the mainidea.angles is a square.Identify the parallelogram shown.The parallelogram has fourQuadrilaterals8–18–2sides andright angles. It is a.Your TurnIdentify theparallelogram shown.REMEMBER ITRhombi is the pluralof rhombus.Theorem 8-10The diagonals of a rectangle are congruent.Theorem 8-11The diagonals of a rhombus are perpendicular.Theorem 8-12Each diagonal of a rhombus bisects a pair ofopposite angles.© Glencoe/McGraw-Hill154 Geometry: Concepts and Applications

8–4A12 34BRefer to rhombus ABCD.Which angles are congruent to 1?D8756CTheorem 8-12 states the diagonals of a rhombusopposite . Therefore, is congruent to2, , and 6.If m7 35, find mADC.Theorem 8-12 states the diagonals of abisectangles.Therefore, m7 1 2 (mADC). 1 (mADC) m7 .2WRITE IT 1 (mADC) Multiply each side.2Explain how squares canbe rhombi, rectangles,and parallelograms.Your Turn mADCRefer to the figure.PQA© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:SRa. Which angles are congruent to PSQ in square PQRS?b. If AP 7, find QS.Geometry: Concepts and Applications 155

8–5TrapezoidsWHAT YOU’LL LEARN• Identify and useproperties of trapezoidsand isoscelestrapezoids.BUILD YOUR VOCABULARY (pages 146–147)A trapezoid is a quadrilateral with exactlysides.pair ofThesides are the bases of the trapezoid.ORGANIZE ITUnder the tab forLesson 8-5, write thedefinition of a trapezoidand of an isoscelestrapezoid. Draw atrapezoid and label thebases, legs, and baseangles. Label thecongruent angles, anddraw the median.Thelegs.Each trapezoid hasIn trapezoid ABCD, name the bases,legs, and the base angles.sides of the trapezoid are known aspairs of base angles.ABCQuadrilaterals8–18–2Bases: AB and areparallel segments.DLegs: AD and are nonparallel segments.Base Angles: A andform one pair of base angles,while C andangles.are the other pair of baseYour TurnIn trapezoid WXYZ, name the bases, legs, andthe base angles.WX© Glencoe/McGraw-HillZY156 Geometry: Concepts and Applications

8–5BUILD YOUR VOCABULARY (pages 146–147)The median of a trapezoid is the segment that joins theof the legs.Another name for the median is the midsegment.If the legs of the trapezoid aretrapezoid is an isosceles trapezoid., then theTheorem 8-13The median of a trapezoid is parallel to the bases, and thelength of the median equals one-half the sum of thelengths of the bases.Find the length of the median KLin trapezoid EFGH if EF 35 andGH 40.EK35FLREVIEW ITThe word “isosceles” isused for classifyingtriangles and trapezoids.What similarities doisosceles triangles andisosceles trapezoidshave? (Lesson 6-4)KL 1 (EF GH) Theorem 8-132KL 1 2( ) Replace EF and GH.KL 1 2() orH40G© Glencoe/McGraw-HillYour TurnFind the length ofthe median NO in trapezoid JKLM ifJK 22 and LM 26.NMJ2622KLOTheorem 8-14Each pair of base angles in an isosceles trapezoid iscongruent.Geometry: Concepts and Applications 157

8–5REMEMBER ITTrapezoids andparallelograms are bothquadrilaterals, but noquadrilateral can beboth a trapezoid and aparallelogram.The measure of one angle in an isosceles trapezoidis 55. Find the measures of the other three angles.Let 1 be the given angle, and let 2 be the base anglecongruent to 1.2 1 Theorem 8-14m2 m2 Replace m1 by 55.Let 3 and 4 be the other pair of base angles, with 3adjacent to 1.m3 m1 Consecutive interiorangles aresupplementary.m3 180 Replace m1with .m3 55 180 – Subtract 55 from eachside.m3 Since 3 and 4 are congruent base angles, m4 .The measures of the three missing angles are , ,and .HOMEWORKASSIGNMENTPage(s):Exercises:Your TurnThe measure of one angle in an isoscelestrapezoid is 76. Find the measures of the other three angles.© Glencoe/McGraw-Hill158 Geometry: Concepts and Applications

C H A P T E R8BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 8 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 8, go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 146–147) to helpyou solve the puzzle.8-1Quadrilaterals1. Name the side opposite WZ .WX2. Name two diagonals.Z91 95 2a a 3. Name the vertex opposite Z.4. Name all consecutive sides.Y5. Find mX and mY.© Glencoe/McGraw-Hill8-2ParallelogramsGiven that JKLM is a parallelogram, find the missingmeasures.6. mL 7. mJ8. LM 9. mKM10. If the measure of one angle of parallelogram PQRS is 79, whatare the measures of the other three interior angles?J66 10KLGeometry: Concepts and Applications 159

Chapter8BRINGING IT ALL TOGETHERState whether each figure is a parallelogram. Justify yourreason.11.8-3Tests for Parallelograms12.13. Explain why quadrilateral ABCDis a parallelogram.ABDC8-4Rectangles, Rhombi, and SquaresUnderline the best term to complete the statement.14. A parallelogram with four congruent sides is a[rhombus/rectangle].Identify each figure with as many terms as possible.Indicate if no term applies.Quadrilateral Parallelogram Square Rhombus Rectangle15. 16.© Glencoe/McGraw-Hill160 Geometry: Concepts and Applications

Chapter8BRINGING IT ALL TOGETHER8-5TrapezoidsComplete each statement.17. The segment that joins the midpoints of each leg ofa trapezoid is the .18. A is a quadrilateral with exactly one pair ofparallel sides.19. The nonparallel sides of a trapezoid are its .20. The parallel sides of a trapezoid are its .Refer to trapezoid ABCD with medianJK . Name each of the following.A29BJKD23C21. bases22. legs23. base angle pairs24. If AB 29 and DC 23, what is JK?© Glencoe/McGraw-Hill25. If AD 18, find JD.26. If WXYZ is an isosceles trapezoid and one base anglemeasures 66, what are the remaining angle measures?Geometry: Concepts and Applications 161

C H A P T E R8ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 8.• You may want to take the Chapter 8 Practice Test onpage 345 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 8 Study Guide and Reviewon pages 342–344 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 8 Practice Test onpage 345.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 8 Foldable.• Then complete the Chapter 8 Study Guide and Review onpages 342–344 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 8 Practice Test onpage 345.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature162 Geometry: Concepts and Applications

C H A P T E R9Proportions and SimilarityUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Chapter 9Begin with a sheet of notebook paper.FoldFold lengthwise tothe holes.CutCut along the top lineand then cut 10 tabs.LabelLabel each tab withimportant terms.Store the Foldablein a 3-ring binder.© Glencoe/McGraw-HillNOTE-TAKING TIP: You can design visuals such asgraphs, diagrams, pictures, charts, and conceptmaps to help you organize information so thatyou can remember what you are learning.Geometry: Concepts and Applications 163

C H A P T E R9BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 9.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplecross productsextremesgolden ratiomeanspolygon[PA-lee-gon]© Glencoe/McGraw-Hill164 Geometry: Concepts and Applications

Chapter9BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleproportion[pro-POR-shun]ratio[RAY-she-oh]scale drawingscale factorsides© Glencoe/McGraw-Hillsimilar polygonsGeometry: Concepts and Applications 165

9–1Using Ratios and ProportionsBUILD YOUR VOCABULARY (page 165)WHAT YOU’LL LEARN• Use ratios andproportions tosolve problems.A comparison ofratio.numbers by division is called aWrite each ratio in simplest form.754 0075 400 Divide the numerator anddenominator by .ORGANIZE ITLabel the first tab ratio.Under the tab, writethe definition and givean example.24 inches to 3 feetThe units of measure must be the same in a ratio. There areinches in one foot, so 24 inches equals feet.The ratio is .Your TurnWrite each ratio in simplest form.a. 1 6939b. 30 minutes to 2 1 2 hours© Glencoe/McGraw-Hill166 Geometry: Concepts and Applications

9–1BUILD YOUR VOCABULARY (pages 164–165)ORGANIZE ITLabel the next four tabsproportion, crossproducts, extremes, andmeans. Under each tab,write the definition andgive an example.An equation that shows two equivalent ratios is aproportion.The cross products are the product of theand the product of the .In a proportion, theof the first ratio andtheextremes.In a proportion, theand themeans.of the second ratio are theof the first ratioof the second ratio are theTheorem 9-1 Property of ProportionsFor numbers a and c and any nonzero numbers b and d, ifa c a c d , then ad bc. Conversely, if ad bc, then b d .bSolve 2 4 6x 4.3035 2 4 6x 43035© Glencoe/McGraw-Hill(35) (6x 4)840 180x 120 Distributive Property840 180x 120 Subtractfrom each side.720 180x720180xDivide each sideby . xGeometry: Concepts and Applications 167

9–1REMEMBER ITThe denominatorcan never equal zero.Your Turn15Solve 4 x 1 5 .The ratio of children to adults at a holiday parade is2.5 to 1. If there are 1440 adults at the parade, howmany children are there?children 2.5 x1 1440adults2.5(1440) Cross products xYour TurnThe ratio of Republicans to Democrats castingtheir votes in the local election was 73 to 27. If 135 Democratsvoted, how many Republicans cast their votes?HOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill168 Geometry: Concepts and Applications

9–2Similar PolygonsBUILD YOUR VOCABULARY (pages 164–165)WHAT YOU’LL LEARN• Identify similarpolygons.A polygon is asegments called sides.figure in a plane formed bySimilar polygons are the samebut notnecessarily the same .KEY CONCEPTSimilar Polygons Twopolygons are similarif and only if theircorresponding anglesare congruent and themeasures of theircorresponding sidesare proportional.Determine if the polygons are similar. Justifyyour answer.The polygons are32 254.53 37.5. The corresponding angles are© Glencoe/McGraw-HillLabel the next threetabs polygon, sides,and similar polygons.Under each tab, writethe definition and givean example.congruent andYour TurnDetermine if thepolygons are similar. Justify youranswer.232 .3 7.530 2040 40 40 4050 40Geometry: Concepts and Applications 169

9–2Find the values of x and y if ABC FED.A5Bx8CD1512FyEUse the corresponding order of the vertices to write proportions.ABFEFDBCDefinition of similar polygons5yx8SubstitutionWrite the proportion to solve for x.x1512 (8) 1 212 x 1 20 Divide each side by .12x Now write the proportion that can be solved for y.5y8(12) y(8) Cross products60 8y 6 0 8 y Divide each side by .8 8© Glencoe/McGraw-Hill ySo, x and y .170 Geometry: Concepts and Applications

9–2Your TurnThe triangles aresimilar. Find the values of x and y.6y817x15BUILD YOUR VOCABULARY (page 165)Scale drawings are used to represent something either tooor tooto be drawn at its actual size.1 in.114in.12 in.ORGANIZE ITLabel the next tab scaledrawings. Under thetab, write the definitionand give an example.In the blueprint, 1 inchrepresents an actuallength of 16 feet. Usethe blueprint to findthe actual dimensionsof the dining room.34 in.114in.DININGROOMLIVING ROOMKITCHENUTILITYROOMGARAGE112in.blueprintactual1 in.16 ftx ftin.blueprintactualx © Glencoe/McGraw-HillHOMEWORKASSIGNMENT1 in.in.16 ft y ft(y) 16 3 4 Cross productsy The dimensions of the dining room are ft by ft.Page(s):Exercises:Your Turnthe kitchen.Refer to Example 3. Find the dimensions ofGeometry: Concepts and Applications 171

9–3Similar TrianglesWHAT YOU’LL LEARN• Use AA, SSS, and SASsimilarity tests fortriangles.Postulate 9-1 AA SimilarityIf two angles of one triangle are congruent to twocorresponding angles of another triangle, then the twotriangles are similar.Theorem 9-2 SSS SimilarityIf the measures of the sides of a triangle are proportionalto the measures of the corresponding sides of anothertriangle, then the triangles are congruent.Theorem 9-3 SAS SimilarityIf the measures of two sides of a triangle are proportionalto the measures of two corresponding sides of anothertriangle and their included angles are congruent, then thetriangles are similar.Determine whether thetriangles are similar, If so, tellwhich similarity test is usedand write a similaritystatement.F37K43GL43H37MSince mF and mH ,WRITE ITWhy must only two pairsof corresponding anglesbe congruent for twotriangles to be similarrather than three?FGH MLK by .Your TurnDetermine whether the triangles are similar,If so, tell which similarity test is used and write a similaritystatement.ACB© Glencoe/McGraw-HillED172 Geometry: Concepts and Applications

9–3Find the value of x.8Since 1 2, the triangles are1 2 18 similar by SAS similarity.8 1 Definition of similar polygons1 2 x481214x1812x (12)(14)8x 1688x 168Cross productsDivide each side by .x Your TurnFind the value of x.5x721© Glencoe/McGraw-HillGeometry: Concepts and Applications 173

9–3The shadow of a flagpole is 2 meters long at the sametime that a person’s shadow is 0.4 meters long. If theperson is 1.5 meters tall, how tall is the flagpole?flagpole’s shadowperson’s shadow20.4xflagpole’s heightperson’s heightx (2)Cross products0.4x 0.4x 3Divide.x The flagpole ismeters tall.Your TurnA diseased tree must be cut down before it falls.Which direction the fall is directed depends on the height ofthe tree. The man who will cut the tree down is 74-in. tall andcasts a shadow 60-in. long. If the tree’s shadow measures20 feet from its base, how tall is the tree?HOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill174 Geometry: Concepts and Applications

9–4Proportional Parts and TrianglesWHAT YOU’LL LEARN• Identify and use therelationships betweenproportional parts oftriangles.Theorem 9-4If a line is parallel to one side of a triangle and intersectsthe other two sides, then the triangle formed is similar tothe original triangle.Using the figure, complete the?proportion V W ST .SWSince VW RT, SVW SRT.VRTherefore,VWST .SWSWTYour TurnUse the figure to completethe proportion X YAY ?.BYAYBZX© Glencoe/McGraw-HillIn the figure, MN KL . Find the value of x.JMN JKLMNKLx9x (6)9x 6JNJLDefinition ofsimilar polygonsSubstitutionCross productsKMJx126N3Lx Divide each side by 9.Geometry: Concepts and Applications 175

9–4Your TurnFind the value of b.bD6HG25F15ETheorem 9-5If a line is parallel to one side of a triangle and intersectsthe other two sides, then it separates the sides intosegments of proportional lengths.In the figure, AB DE .Find the value of x.CEEBCDTheorem 9.5AB8D6 Exx + 5Cxx CE x, EB 6,CD x 5, DA 8x(8) (x 5) Cross products8x 6x Distributive Property8x 6x 6x 30 6x2x 30Subtract 6x from each side. 2 x 3 0 Divide each side by 2.2 2HOMEWORKASSIGNMENTx Your TurnFind the value of a.TRa© Glencoe/McGraw-HillPage(s):Exercises:10SA W8 2a 3176 Geometry: Concepts and Applications

9–5Triangles and Parallel LinesWHAT YOU’LL LEARN• Use proportions todetermine whetherlines are parallel tosides of triangles.Theorem 9-6If a line intersects two sides of a triangle and separates thesides into corresponding segments of proportional lengths,then the line is parallel to the third side.Determine whether DE BC .Determine whether B D andDA C E form a proportion.EAD3B6A8E4CBDCEEA6 8(8) (4) Cross products24 ✔Therefore, DE BC by Theorem 9-6.© Glencoe/McGraw-HillYour TurnDetermine whether HJ KM.LxH 23xJK6MGeometry: Concepts and Applications 177

9–5Theorem 9-7If a segment joins the midpoint of two sides of a triangle,then it is parallel to the third side, and its measure equalsone-half the measure of the third side.REMEMBER ITThe midsegment’sendpoints are themidpoints of the legs oftwo sides of a triangle.For Examples 2 and 3, refer to thefigure shown.In the figure, X, Y, and Z aremidpoints of the sides of UVW.If XZ 7c, then what does UW equal?XZ 1 Theorem 9-72 1 UW Replace XZ with .2VXUZYW7c 1 2 UW Multiply each side by . UWIn the figure, if mUYX d, then what is mYWZ?By Theorem 9-6, XY VW. Since XY and VW are parallelsegments cut by transversal UW, UYX andarecongruentangles.Therefore, mYWZ .HOMEWORKASSIGNMENTYour TurnN, O, and Pare the midpoints of thesides of EFG.a. If EF 25, then what does NP equal?ENGOPF© Glencoe/McGraw-HillPage(s):Exercises:b. If mEGF 85, then what is mENO?178 Geometry: Concepts and Applications

9–6Proportional Parts and Parallel LinesWHAT YOU’LL LEARN• Identify and use therelationships betweenparallel lines andproportional parts.Theorem 9-8If three or more parallel lines intersect two transversals, thelines divide the transversals proportionally.STComplete the proportion N P.R T ?REVIEW ITWhat is the definitionof a transversal?(Lesson 4-2)RMSNTPSince NS PT,the transversals are dividedST NP. Therefore, .R TIn the figure, a b c.Find the value of x.a b cR S T15 9P QN 12TS S RNMMx© Glencoe/McGraw-Hill 195 xTS 9, SR 15,PN , NM x9(x) 15 Cross products9x x Divide each side by .Geometry: Concepts and Applications 179

9–6REVIEW ITExplain how to constructparallel lines.(Lesson 4-4)Your TurnZPa. Complete the proportion N B.Q P ?QZNAlmPBnb. In the figure, a b c. Find the value of x.xFD J8.5 E K 618LabcTheorem 9-9If three or more parallel lines cut off congruent segmentson one transversal, then they cut off congruent segmentson every transversal.HOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill180 Geometry: Concepts and Applications

9–7Perimeters and SimilarityWHAT YOU’LL LEARN• Identify and useproportionalrelationships ofsimilar triangles.Theorem 9-10If two triangles are similar, then the measures of thecorresponding perimeters are proportional to the measuresof the corresponding sides.The perimeter of DEF is 90 units, and ABC DEF.Find the value of each variable.D26 A xy10B E F24 CzORGANIZE IT• Write each vocabularyword from the lesson.• Explain its meaning inyour own words.• Use diagrams toclarify. D E perimeter of DEF Theorem 9-10AB perimeter of ABCx 2 6 9 06026 10 24 (60) (90) Cross products60x 2340x Divide.© Glencoe/McGraw-HillBecause the triangles are similar, find y and z.D FD E A CA B EF B C D E AB 3y9 26y 39026z26z 2 426 y z Geometry: Concepts and Applications 181

9–7Your TurnThe perimeter of ABC is 20 units, andABC XYZ. Find the value of each variable.X1014AdgZ6YCfBBUILD YOUR VOCABULARY (page 165)The scale factor, also known as the constant of, is the found bycomparing the measures of corresponding sides ofsimilar triangles.ORGANIZE ITLabel the next tab scalefactor. Under the tab,write the definition andgive an example.Determine the scale factorof ABC to DEF.AB D E243DA3624 2416B C E F20 30 B C EF302HOMEWORKASSIGNMENTPage(s):Exercises:AC The scale factor is .D F 24 3Your TurnDetermine the scale factor of RST to XYZ.Y3.2 412.8 16 RT5S© Glencoe/McGraw-HillX20Z182 Geometry: Concepts and Applications

C H A P T E R9BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 9 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 9, go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 164–165)to help you solve the puzzle.9-1Using Ratios and ProportionsIndicate whether the statement is true or false.1. Every proportion has two cross products.2. A ratio is a comparison of two numbers by division.3. The two cross products of a ratio are the extremesand the means.4. Cross products are always equal in a proportion.5. Simplify 2 27. 006. Solve: 8 46 3 1211 x© Glencoe/McGraw-Hill9-2Similar PolygonsComplete the sentence.7. In measures of corresponding sides areproportional, and corresponding angles are congruent.8. represent something either too large ortoo small to be drawn at actual size.Geometry: Concepts and Applications 183

Chapter9BRINGING IT ALL TOGETHER9. Given that the rectangles are similar, find the values of x andy to show similarity.155 515x 73 3y 49-3Similar TrianglesDetermine whether the pair of triangles is similar. Justifyyour reasons.10. 11.36299-4Proportional Parts and TrianglesComplete the proportions.BAD12. D ECBE13. EAEB 9-5ADTriangles and Parallel LinesVertices A, B, and C are midpoints.14. AC .ASDC© Glencoe/McGraw-Hill15. If BC 6, then RT .BC16. If SB 4, AC .RAT184 Geometry: Concepts and Applications

Chapter9BRINGING IT ALL TOGETHER9-6Proportional Parts and Parallel LinesA tract of land bordering school property was divided intosections for five biology classes to plant gardens. The fencesseparating the plots are parallel, and the plots’ frontmeasures are shown. The entire back border measures254 feet. What are the individual border lengths, to thenearest tenth of a foot?22 ft.20 ft.25 ft.28 ft. 16 ft.frontABCDEback17. A 18. D 19. E 9-7Perimeters and SimilarityComplete the sentence.20. The scale factor is also called the constant of21. Find the scale factor..BY© Glencoe/McGraw-HillJKL MNO. The perimeter of JKL is 54. What are thevalues for the variables?22. a K10A15C16XN24Z23. b ab912M15O24. c JcLGeometry: Concepts and Applications 185

C H A P T E R9ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 9.• You may want to take the Chapter 9 Practice Test onpage 397 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 9 Study Guide and Reviewon pages 394–396 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 9 Practice Test onpage 397.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 9 Foldable.• Then complete the Chapter 9 Study Guide and Review onpages 394–396 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 9 Practice Test onpage 397.© Glencoe/McGraw-HillStudent SignatureParent/Guardian SignatureTeacher Signature186 Geometry: Concepts and Applications

C H A P T E R10Polygons and AreaUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notesBegin with a sheet of 8 1 2 " 11" paper. PrefixFoldFold the short side infourths.Chapter 10DrawDraw lines along the foldsand label each columnPrefix, Number of Sides,Polygon Name, and Figure.Numberof SidesPolygonNameFigure© Glencoe/McGraw-HillNOTE-TAKING TIP: When you take notes, it isimportant to record major concepts and ideas.Refer to your journal when reviewing for tests.Geometry: Concepts and Applications 187

C H A P T E R10BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 10.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplealtitudeapothem[a-pa-thum]centercomposite figure[kahm-PA-sit]concaveconvexirregular figureline of symmetry[SIH-muh-tree]© Glencoe/McGraw-Hill188 Geometry: Concepts and Applications

Chapter10BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleline symmetrypolygonal regionregular polygonregular tessellationrotational symmetrysemi-regular tessellationsignificant digits© Glencoe/McGraw-Hillsymmetrytessellation[tes-a-LAY-shun]turn symmetryGeometry: Concepts and Applications 189

10– 1Naming PolygonsWHAT YOU’LL LEARN• Name polygonsaccording to the numberof sides and angles.BUILD YOUR VOCABULARY (page 189)A regular polygon has allcongruent and allcongruent.ORGANIZE ITUnder the tabs labeledPrefix, Number of Sides,and Polygon Name,write the informationgiven in the table onpage 402. Under the tablabeled Figure, draw apicture of each polygon.Include regular andirregular polygons, aswell as convex andconcave polygons.PrefixNumberof SidesPolygonNameFigureRefer to the figure for Examples 1–2.a. Identify polygon VWXYZ.The polygon has sides. It is a .b. Determine whether the polygon VWXYZ appears tobe regular or not regular. If not regular, explain why.Theappear to be the same length, and theappear to have the same measure. The polygon is regular.VZWYXName two nonconsecutive vertices of polygon VWXYZ.W and Z, W and Y, V and X, V and Y, X and Z are examplesofvertices.Your TurnRefer to the figure forparts a, b, and c.a. Identify polygon DEFGHIJ by its sides.b. Determine whether the polygon DEFGHIJ appears to beregular or not regular. If not regular, explain why.DJEFGHI© Glencoe/McGraw-Hillc. Name two nonconsecutive vertices of polygon DEFGHIJ.190 Geometry: Concepts and Applications

10–1BUILD YOUR VOCABULARY (page 188)REMEMBER ITMost polygons havemore than one diagonal.As the number of sidesincreases, so does thenumber of diagonals.All of the diagonals of a convex polygon lie in theof the polygon.If any part of a diagonal liesof the polygon,the polygon is concave.Classify each polygon as convex or concave.a. E D When all the diagonals are drawn,FABCpoints lie outside of thepolygon. So polygon ABCDEFis .b. QDiagonal QS lies outside the polygon,R Sso PQRSTU is .P UTYour TurnClassify each polygon as convex or concave.a. b.KIV XJU© Glencoe/McGraw-HillHOMEWORKASSIGNMENTLMHZYPage(s):Exercises:Geometry: Concepts and Applications 191

10– 2Diagonals and Angle MeasureWHAT YOU’LL LEARN• Find measures ofinterior and exteriorangles of polygons.Theorem 10-1If a convex polygon has n sides, then the sum of themeasures of the interior angles is (n 2)180.Refer to the regular pentagon for Examples 1–2.KORGANIZE ITOn the back of yourFoldable, you may wishto write the interiorangle sum for each ofthe different polygonslisted on your Foldable.JLN MFind the sum of the measures of the interior angles.Sum of measures of interior angles (n 2)180 Theorem 10-1PrefixNumberof SidesPolygonNameFigure ( 2)180 Substitution ( )180The sum of the measures of the interior angles of a pentagonis .Find the measure of one interior angle.Each interior angle of a regular polygon has the same measure.Divide theof the measures by theof angles.measure of one interior angle or© Glencoe/McGraw-HillThe measure of one interior angle of a regular pentagon is.192 Geometry: Concepts and Applications

10–2WRITE ITHow do you find themeasure of an interiorangle of an n-sidedregular polygon?Your Turna. Find the sum of the measures of the interior angles of aregular 15-sided polygon.b. Find the measure of one interior angle of a regular15-sided polygon.REMEMBER ITTheorems 10-1 and10-2 only apply toconvex polygons.Theorem 10-2In any convex polygon, the sum of the measures of theexterior angles, one at each vertex, is 360.Find the measure of one exterior angle of a regularoctagon.By Theorem 10-2, the sum of the measures of exterior anglesis . An octagon has exterior angles.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:measure of one exterior angle 36 0 8Your TurnFind the measure of one exterior angle of aregular 15-sided polygon.Geometry: Concepts and Applications 193

10– 3Areas of PolygonsWHAT YOU’LL LEARN• Estimate the areas ofpolygons.Postulate 10-1 Area PostulateFor any polygon and a given unit of measure, there is aunique number A called the measure of the area of thepolygon.Postulate 10-2Congruent polygons have equal areas.Postulate 10-3 Area Addition PostulateThe area of a given polygon equals the sum of the areas ofthe nonoverlapping polygons that form the given polygon.BUILD YOUR VOCABULARY (pages 188–189)Any polygon and itsare called a polygonalregion.A composite figure is a figure made fromthat have been placed together.REVIEW ITWhat formulas for areahave you learned?(Lesson 1-6)Find the area of the polygon.Each square represents1 square centimeter.Since the area of each square represents one squarecentimeter, the area of each triangular half square represents0.5 square centimeter. There are 8 squares and 4 half squares.A 8(1) cm 2 4(0.5) cm 2A cm 2 cm 2A cm 2Your TurnFind the area of the polygon. Eachsquare represents 1 square inch.© Glencoe/McGraw-Hill194 Geometry: Concepts and Applications

10–3BUILD YOUR VOCABULARY (page 188)Irregular figures are not polygons and cannot be madefrom combinations of polygons. Their areas can beapproximated using combinations of polygons.Estimate the area of the polygon.Each square represents 20 squaremiles.WRITE ITHow can you determinethe area of a polygon bydividing it into familiarshapes?Count each square as one unit and eachpartial square as a half unit regardlessof size. There areandpartial squares.whole squaresnumber of squares (1) (0.5)Area 20 Each square represents20 square miles.The area of the polygon is aboutsquare miles, or.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Your TurnA swimming pool ata resort is shaped as shown on thegrid. Each square on the gridrepresents 16 square meters.Estimate the area of the pool.Geometry: Concepts and Applications 195

10– 4Areas of Triangles and TrapezoidsWHAT YOU’LL LEARN• Find the areas oftriangles andtrapezoids.Theorem 10-3 Area of a TriangleIf a triangle has an area of A square units, a base of b units,and a corresponding altitude of h units, then A 1 2 bh.Find the area of each triangle.6 ft15 ftA 1 bh Theorem 10-32 1 2 Replace b with and h with . 1 2 REVIEW ITWhat is an altitude of atriangle? (Lesson 6-2)9 cm8 cmA 1 bh Theorem 10-32 1 2 Replace b with and h with . 1 2 Your Turna.4Find the area of each triangle.© Glencoe/McGraw-Hill11196 Geometry: Concepts and Applications

10–4b.5 m12 mc.12 ft235 ft4ORGANIZE ITDraw and label the baseand altitude for eachtriangle on yourFoldable. In addition,draw a trapezoidas an example of aquadrilateral. Drawand label the basesand altitude for thetrapezoid.BUILD YOUR VOCABULARY (page 188)The altitude of a trapezoid is a segment perpendicular toeach .Theorem 10-4 Area of a TrapezoidIf a trapezoid has an area of A square units, bases of b 1 andb 2 units, and an altitude of h units, then A 1 2 h(b 1 b 2 ).PrefixNumberof SidesPolygonNameFigureFind the area of the trapezoid.6 mA 1 2 h(b 1 b 2 ) Theorem 10-44 m18 m© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:A 1 2 Replace h with 6, b 1 with 4,and b 2 with 18.A 1 2 A orYour TurnFind the area of the trapezoid.4.5 cm1.5 cm2 cmGeometry: Concepts and Applications 197

10– 5Areas of Regular PolygonsBUILD YOUR VOCABULARY (page 188)WHAT YOU’LL LEARN• Find the areas ofregular polygons.The center of a regular polygon is an interior point that isequidistant from all .The segment drawn from the center andto a side of a regular polygon is an apothem.ORGANIZE ITDraw an apothem foreach of the regularpolygons drawn onyour Foldable.Theorem 10-5 Area of a Regular PolygonIf a regular polygon has an area of A square units, anapothem of a units, and a perimeter of P units, thenA 1 2 aP.PrefixNumberof SidesPolygonNameFigureA regular octagon has a side lengthof 9 inches and an apothem that isabout 10.9 inches long. Find the areaof the octagon.10.9 in.9 in.First, find the perimeter of the octagon.P 8sAll sides of a regular octagon are congruent. 8(9) or 72 Replace s with 9.REMEMBER ITOnly regular polygonshave apothems.Now find the area.A 1 aP Theorem 10-52 1 (10.9)(72) or2Replace a with 10.9 andP with 72.The area of the octagon is about in 2 .Your TurnA regular pentagon has a sidelength of 8 inches and an apothem that is about5.5 inches long. Find the area of the pentagon.8 in.© Glencoe/McGraw-Hill5.5 in.198 Geometry: Concepts and Applications

10–5A regular octagon has a side lengthof 12 inches and an apothem that isabout 14.5 inches long. Find the areaof the shaded region of the octagon.Find the area of the octagon minus thearea of the unshaded region.Area of an octagon:A 1 aP Theorem 10-5214.5 in.12 in. 1 2 Replace a with 14.5 andP with 96. in 2Area of a Triangle:A 1 bh Theorem 10-32 1 2 (12)(14.5) in 2Replace b with 12 andh with 14.5.The area of one triangular section is 87 in 2 . There are5 triangular sections in the unshaded region.The area of the unshaded region is 5 in2 .Subtract the area of the unshaded region from the area of theoctagon.Area of shaded region or in 2© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Your TurnFind the area of theshaded region of the regular hexagon.BUILD YOUR VOCABULARY (page 189)Significant digits represent the precision of a.17.3 cm20 cmGeometry: Concepts and Applications 199

10– 6SymmetryWHAT YOU’LL LEARN• Identify figures withline symmetry androtational symmetry.BUILD YOUR VOCABULARY (pages 188–189)Symmetry is when a figure has balanced proportions acrossa reference, line, or plane.When a line is drawn through theof a figureand one half is theimage of the other, thefigure is said to have line symmetry.The reference line is known as the line of symmetry.Find all lines of symmetry for equilateral triangle ABC.ABCFold along all possible lines to see if the sides match. Therearelines of symmetry along the lines shown inthe figure.REMEMBER ITFigures that haverotational symmetry donot necessarily haveline symmetry.Your TurnJKXYZ.JZDraw all lines of symmetry for regular pentagonJKZK© Glencoe/McGraw-HillYXYX200 Geometry: Concepts and Applications

10–6BUILD YOUR VOCABULARY (page 189)A figure that can be turned or rotated less than 360ºabout a fixed point and that looks exactly as it does intheis said to have turn symmetryor rotational symmetry.WRITE ITWhich of the figures have rotational symmetry?a.Draw a polygon that hasline symmetry but doesnot have rotationalsymmetry. Do you thinkit is possible to drawa figure with more than1 line of symmetry, butthat does not haverotational symmetry?Explain.b.The figure can be turned 120° and 240° to look like theoriginal. The figure hassymmetry.The figure must be turned 360° about its center to look likethe original. Therefore, itsymmetry.have rotationalYour Turnsymmetry?a.Which of the figures has rotational© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:b.Geometry: Concepts and Applications 201

10– 7TessellationsBUILD YOUR VOCABULARY (page 189)WHAT YOU’LL LEARN• Identify tessellationsand create them byusing transformations.REMEMBER ITRegular and semiregulartessellationsare created using onlyregular polygons.Tessellations are tiled patterns created byfigures to fill a plane without gaps or overlaps. They can bemade by translating, rotating, or reflecting polygons.A pattern is a regular tessellation when onlytype of regular polygon is used to form the pattern.When two or more regular polygons are used in thesame order at every vertex to form a pattern, it is asemi-regular tessellation.Identify the figures used to create each tessellation.Then identify the tessellation as regular, semi-regular,or neither.Only squares are used. A square is a regularpolygon. The tessellation is .Hexagons are used and there are no gaps in thepattern, but the hexagons are not .The tessellation issemi-regular tessellation.a regular nor aHOMEWORKASSIGNMENTPage(s):Exercises:Your TurnIdentify the tessellation as regular, semiregular,or neither.a. b.© Glencoe/McGraw-Hill202 Geometry: Concepts and Applications

C H A P T E R10BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 10 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 10, go to:www.glencoe.com/sec/math/t_resources/free/index.php.BUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 188–189) to help yousolve the puzzle.10-1Naming PolygonsIndicate whether the statement is true or false.1. All the diagonals of a concave polygon lieon the interior.2. A regular polygon is both equilateral andequiangular.Identify each figure by its sides. Indicate if the polygon appearsto be regular or not regular. If not regular, justify your reason.3. 4.© Glencoe/McGraw-Hill10-2Diagonals and Angle MeasureFind the sum of the measures of the interior angles.5. 6.Geometry: Concepts and Applications 203

Chapter10BRINGING IT ALL TOGETHERFind the measure of one interior angle and oneexterior angle of the regular polygon.7. dodecagon 8. decagon9. The sum of the measures of four exterior angles of a pentagonis 280. What is the measure of the fifth exterior angle?10-3Areas of PolygonsIndicate whether the statement is true or false.10. A polygon and its interior are known as apolygonal region.Find the area of the polygon in square units.11. 12.10-4Areas of Triangles and TrapezoidsIndicate whether the statement is true or false.13. The segment perpendicular to the parallelbases of a trapezoid is a median.Find the area of the triangle or trapezoid.14. 15.12.218259© Glencoe/McGraw-Hill5204 Geometry: Concepts and Applications

Chapter10BRINGING IT ALL TOGETHER16. Find the area of a trapezoid whosealtitude measures 4.5 cm and hasbases measuring 6.2 and 8.8 cm.17. What is the area of a triangle withbase length 6 1 3 in. and height 2 in.?10-5Areas of Regular Polygons18. Find the area of a regular 11-sided polygon with each sidemeasuring 7 cm and an apothem length of 11.9 cm.19. Find the area of the shaded region.610-6Symmetry5.2Underline the best term to make the statement true.20. When a line is drawn through a figure and makes each half a mirrorimage of the other, the figure has [line/rotational] symmetry.21. When a figure looks exactly as it does in its original position after beingturned less than 360º around a fixed point, it has [line/rotational] symmetry.Determine whether the figure has line symmetry, rotational symmetry,both, or neither.22. 23.© Glencoe/McGraw-Hill10-7TessellationsIdentify the tessellation as regular, semi-regular, or neither.24. 25.Geometry: Concepts and Applications 205

C H A P T E R10ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 10.• You may want to take the Chapter 10 Practice Test onpage 449 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 10 Study Guide and Reviewon pages 446–448 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 10 Practice Test onpage 449 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 10 Foldable.• Then complete the Chapter 10 Study Guide and Review onpages 446–448 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 10 Practice Test onpage 449 of your textbook.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature206 Geometry: Concepts and Applications

C H A P T E R11CirclesUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with seven sheets of plain paper.DrawDraw and cut a circle fromeach sheet. Use a small plateor a CD to outline the circle.StapleStaple the circles together toform a booklet.LabelLabel the chapter name onthe front. Label the inside sixpages with the lesson titles.CirclesChapter 11© Glencoe/McGraw-HillNOTE-TAKING TIP: When you take notes, writeconcise definitions in your own words. Addexamples that illustrate the concepts.Geometry: Concepts and Applications 207

C H A P T E R11BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 11.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExampleadjacent arcsarcscentercentral anglechordcirclecircumference[sir-KUM-fur-ents]circumscribedconcentric© Glencoe/McGraw-Hilldiameter208 Geometry: Concepts and Applications

Chapter11BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleexperimental probability[ek-speer-uh-MEN-tul]inscribedlocilocusmajor arcminor arcpi ()© Glencoe/McGraw-Hillradius[RAY-dee-us]sectorsemicircletheoretical probability[thee-uh-RET-i-kul]Geometry: Concepts and Applications 209

11–1Parts of a CircleWHAT YOU’LL LEARN• Identify and use partsof circles.BUILD YOUR VOCABULARY (pages 208–209)A circle is the set of all points in a plane that are a givendistance from a given point in the plane, called theof the circle.In a circle, all points arefrom the center.A radius is a segment whose endpoints are theof the circle and aon the circle.A chord is a segment whosecircle.are on theA diameter is aof the circle.that contains theTwo circles are concentric if they lie in the same plane,have the same , and have ofORGANIZE ITUnder the tab forLesson 11-1, draw acircle with a radius, achord and a diameter.Label each specialsegment.Circlesdifferent lengths.Use circle P to determine whether eachstatement is true or false.RT is a diameter of circle P.; RT go through thecenter P. Therefore, RT is not a diameter.RTQPS© Glencoe/McGraw-HillPS is a radius of circle P.; the endpoints of PS are on the P anda point on the circle S. Therefore, PS is a radius.210 Geometry: Concepts and Applications

11–1WRITE ITYour TurnUse circle T to determine whether eachstatement is true or false.a. AB is not a diameter.ADescribe the differencesbetween a radius, adiameter, and a chord.b. TD is not a radius.TECDBTheorem 11-1All radii of a circle are congruent.Theorem 11-2The measure of the diameter d of a circle is twice themeasure of the radius r of the circle.In circle R, QT is a diameter.If QR 7, find QT.QR is a radius, and d 2r.QT 2(QR) Replace d and r.RTQ7SQT 2 Replace QR with .QT Your TurnIn circle A, FC is a diameter. If FC 25, find AB.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTFEBADCPage(s):Exercises:Geometry: Concepts and Applications 211

11–2Arcs and Central AnglesBUILD YOUR VOCABULARY (pages 208–209)WHAT YOU’LL LEARN• Identify major arcs,minor arcs, andsemicircles and find themeasures of arcs andcentral angles.When two sides of an angle meet at the center of a circle,a central angle is formed.Each side of the central angle intersects a point on thecircle, dividing it intolines called arcs.A minor arc is formed by the intersection of the circle andsides of a central angle with interior degree measure lessthan 180.A major arc is the part of the circle in theofthe central angle that measures greater than 180.Semicircles arearcs whose endpoints lieon the diameter of the circle.Adjacent arcs are arcs of a circle with exactly one pointin common.KEY CONCEPTSThe degree measure ofa minor arc is the degreemeasure of its centralangle.The degree measure ofa major arc is 360 minusthe degree measure ofits central angle.The degree measure ofa semicircle is 180.Under the tabfor Lesson 11-2, draw acircle with a centralangle. Label the centralangle, the major andminor arcs and giveexamples of degreemeasurements for each.In circle J, find mLM , mKJM,and mLK .mLM mLJMmLM 125mKJM mKM mKJM mLK 360 mLJM mKJMmLK 360 125 130mLK Measure of minor arcLJ125Measure of central angleMeasure of major arcSubstitutionKM130© Glencoe/McGraw-Hill212 Geometry: Concepts and Applications

11–2Postulate 11-1 Arc Addition PostulateThe sum of the measures of two adjacent arcs is themeasure of the arc formed by the adjacent arcs.REMEMBER ITA circle contains360°.In circle A, CE is a diameter. Find mBC , mBE,and mBDE.mBC mBACmBC mBC mEBCCBD48 AMeasure of minor arcSubstitution82Arc AdditionPostulateEmBE 180 SubstitutionmBE 132mBDE mBESubtract.Measure major arcmBDE 132 SubstitutionmBDE Your TurnIn circle X,mAXB 70, mDC 45,and BE and AD are diameters.EXAB© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:a. Find mEA , mBXC,and mED .DC b. Find mAC, mDAE,and mABE .Theorem 11-3 In a circle or in congruent circles, twominor arcs are congruent if and only if their correspondingcentral angles are congruent.Geometry: Concepts and Applications 213

11–3Arcs and ChordsWHAT YOU’LL LEARN• Identify and use therelationships amongarcs, chords, anddiameters.Theorem 11-4In a circle or in congruent circles, two minor arcs arecongruent if and only if their corresponding chordsare congruent.Theorem 11-5In a circle, a diameter bisects a chord and its arc if andonly if it is perpendicular to the chord.ORGANIZE ITUnder the tab forLesson 11-3, drawdiagrams and givedescriptions tosummarizeTheorems 11-4 and 11-5.CirclesIn circle R, if PR QT , find PQ.PSQ is a right angle.Definition of perpendicularPSQ is a triangle. Definition of right triangle 2 (SQ) 2 (PQ) 2 Pythagorean TheoremYour TurnFind PC.2SQ 24 Theorem 11-5 24 2 (PQ) 2100 576 (PQ) 2676 (PQ) 2 PQ10TReplace PS withand SQ with 24.In circle P, AB 8 and PD 3.QPS24 RTake the square rootof each side.APCDB© Glencoe/McGraw-Hill214 Geometry: Concepts and Applications

11–3In circle W, find XV if UW XV , VW 35, and WY 21.XUY21W35VVYW is a angle. Definition of perpendicularVYW is a right triangle.Definition of right triangle(WY) 2 (YV) 2 2 Pythagorean Theorem21 2 (YV) 2 35 2 Replace WY and VW. (YV) 2 1225(YV) 2 Subtract.(YV) 2 Take the square root ofeach side.YV XY Theorem 11-5XV YV XYSegment additionXV 28 SubstitutionXV © Glencoe/McGraw-HillHOMEWORKASSIGNMENTYour TurnIn circle G, if CG AE, EG 20, CG 12, find AE.A C12 EG 20Page(s):Exercises:Geometry: Concepts and Applications 215

11–4Inscribed PolygonsWHAT YOU’LL LEARN• Inscribe regularpolygons in circles andexplore the relationshipbetween the length ofa chord and its distancefrom the center of thecircle.BUILD YOUR VOCABULARY (pages 208–209)A polygon is inscribed in a circle if and only if everyof the polygon lies on the circle.A circumscribed polygon is a polygon with each sideto a .ORGANIZE ITUnder the tab forLesson 11-4, draw apolygon inscribed ina circle and anothercircumscribed aboutthe circle. Label eachdrawing appropriately.Construct a regular octagon.Construct aquadrilateralby connecting the consecutiveof twodiameters.CirclesBisect adjacent. Extend the bisectors through theof the circle to the edges of the circle. The otherfourare where the other two perpendicularintersect the circle. Connect all of theconsecutive to form the regular .Your TurnConstruct a regular hexagon.© Glencoe/McGraw-Hill216 Geometry: Concepts and Applications

11–4Theorem 11-6In a circle or in congruent circles, two chords are congruentif and only if they are equidistant from the center.In circle O, point O is the midpoint of A B .If CR 2x 1 and ST x 10, find x.RSAOBCTOA Definition of midpointWRITE ITExplain Theorem 11-6 inyour own words. TS Theorem 11-62x 1 x Substitution2x x Add to each side.x Subtract fromeach side.Your TurnIn circle Y, NY YO. If AX 2x 15 andBZ 3x 6, what is the value of x?NXAZY© Glencoe/McGraw-HillHOMEWORKASSIGNMENTOBPage(s):Exercises:Geometry: Concepts and Applications 217

11–5Circumference of a CircleWHAT YOU’LL LEARN• Solve problemsinvolving circumferenceof circles.BUILD YOUR VOCABULARY (pages 208–209)The perimeter of ais known as thecircumference. It is thearound the circle.The ratio of theof a circle to itsORGANIZE ITUnder the tab forLesson 11-5, give theformulas for finding thecircumference of a circleand give an example ofhow they are used.Circlescalled pi.is always equal to the irrational numberTheorem 11-7 Circumference of a CircleIf a circle has a circumference of C units and a radius ofr units, then C 2r or C d.The radius of a circle is 8 feet. Find the circumferenceof the circle to the nearest tenth.C 2 Theorem 11-7C 2 Replace r with .C 16 feetThe diameter of a plastic pipe is 5 cm. Find thecircumference of the pipe to the nearest centimeter.C Theorem 11-7© Glencoe/McGraw-HillC SubstitutionC 5 cm218 Geometry: Concepts and Applications

11–5Your TurnREVIEW ITHow do you find theperimeter of a polygon?(Lesson 1-6)a. Find the circumference of circleA to the nearest tenth.A5 ftb. The diameter of a CD is 4.5 inches.Find its circumference to the nearest tenth.REMEMBER ITPi () is an exactconstant. The decimalapproximation 3.14… isonly an estimate.A circular garden has a radius of 20 feet. There is apath around the garden that is 3 feet wide. Jasminestands on the inside edge of the path, and Hitesh standson the outside edge. They each walk around the gardenexactly once while staying along their edge of the path.To the nearest foot, how much farther does Hitesh walkthan Jasmine?Jasmine:Hitesh:C 2r Theorem 11-7 C 2rC 2 Substitution C 2 C C © Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:So, Hitesh walked or approximatelyfeet more than Jasmine.Your TurnA circle has a circumference of 20.5 meters.Find the radius of the circle to the nearest tenth.Geometry: Concepts and Applications 219

11–6Area of a CircleWHAT YOU’LL LEARN• Solve problemsinvolving areas andsectors of circles.Theorem 11-8 Area of a CircleIf a circle has an area of A square units and a radius ofr units, then A r 2 .ORGANIZE ITUnder the tab forLesson 11-6, give theformula for finding thearea of a circle and givean example of how itis used.Find the area of circle G.A r 2 Theorem 11-82A Replace r.A 100 cm 210 cmGCirclesYour TurnFind the area of a circle to the nearest tenthwhose diameter is 10 cm.If circle S has a circumference of 16 inches, find thearea of the circle to the nearest hundredth.C 2r Theorem 11-7 2r Replace C with .16 2r2 2Divide each side by . rA r 2 Theorem 11-8© Glencoe/McGraw-HillA 2Replace r with .A 64 in 2220 Geometry: Concepts and Applications

11–6Your TurnFind the area of the circle to the nearesthundredth whose circumference is 84 cm.BUILD YOUR VOCABULARY (pages 208–209)Theoretical probability is the chance for a successfuloutcome based on .Experimental probability is calculated from actualobservations and recording. It is the chance fora successful outcome based on observing patterns ofoccurrences.A pond has a radius of 10 meters. In the center of thepond is a square island with a side length of 5 meters.The seeds of a nearby maple tree float down randomlyover the pond. What is the probability that a randomlychosenseed will land in the water rather than on theisland? Assume that the seed will land somewherewithin the circular edge of the pond.A of pond 2© Glencoe/McGraw-HillA of island m 2P(landing in pond) 2 m 2A of pond A of islandA of pond314.2 314.2314.2Geometry: Concepts and Applications 221

11–6Your TurnAssume that all dartswill land on the dartboard. Find theprobability that a randomly-throwndart will land in the shaded region.44 22020BUILD YOUR VOCABULARY (page 209)A sector of a circle is a region bounded by a centraland its corresponding .Theorem 11-9 Area of a Sector of a CircleIf a sector of a circle has an area of A square units, a centralangle measurement of N degrees, and a radius of r units,then A N360 r 2 .Find the area of a 45° sector of a circle whose radius is8 in. Round to the nearest hundredth.A N360 r 2 Theorem 11-9A 45360 8 2SubstitutionHOMEWORKASSIGNMENTPage(s):Exercises:A (0.125)(64)A in 2Your TurnFind the area of a 30° sector of a circle whoseradius is 7.75 feet. Round to the nearest hundredth.© Glencoe/McGraw-Hill222 Geometry: Concepts and Applications

C H A P T E R11BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 11 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle, wordsearch, or jumble puzzle of thevocabulary words in Chapter 11,go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 208–209) to help yousolve the puzzle.11-1Parts of a CircleUnderline the term that best completes the statement.1. A chord that contains the center of the circle is the[diameter/radius].2. A [chord/radius] is a segment with endpoints of the circle.3. Two circles are [circumscribed/concentric] if they lie on thesame plane, have the same center, and have radii of differentlengths.11-2Arcs and Central Angles© Glencoe/McGraw-HillIn circle C, BD is a diameter and mGCF 63. Find eachmeasure.4. mFG DAE F5. mAD 6. mABCGB7. mGEFGeometry: Concepts and Applications 223

Chapter11BRINGING IT ALL TOGETHER11-3Arcs and ChordsComplete each statement.8. If two chords are congruent in the same circle, the interceptedare also congruent.9. When the diameter of the circle bisects a chord of the circle,then it isto the chord andthe corresponding arc.10. In a circle, if two arcs are , theirare congruent.11-4Inscribed Polygons11. Construct an equilateraltriangle inscribed in acircle with radius 1 inch.12. Draw a circle inscribedin the triangle from theprevious problem.Which segment of thetriangle equals the radiusof the inscribed circle?© Glencoe/McGraw-Hill13. What is the approximate length of the segment in Exercise 12?224 Geometry: Concepts and Applications

Chapter11BRINGING IT ALL TOGETHER11-5Circumference of a CircleFind the circumference of each circle.14. r 1 2 yd15. d 4.2 in.Find the radius of the circle whose circumference is given.16. 47 ft17. 22.7 in.11-6Area of a CircleUnderline the term that best completes the statement.18. A region of a circle bounded by a central angle and itscorresponding arc is a(n) [arc/sector].© Glencoe/McGraw-Hill19. The segment with endpoints at the center and on the circle isa [sector/radius].20. Find the area of the shaded region incircle B to the nearest hundredth.B6 ft.57Geometry: Concepts and Applications 225

C H A P T E R11ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 11.• You may want to take the Chapter 11 Practice Test onpage 491 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 11 Study Guide and Reviewon pages 488–490 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 11 Practice Test onpage 491.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 11 Foldable.• Then complete the Chapter 11 Study Guide and Review onpages 488–490 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 11 Practice Test onpage 491.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature226 Geometry: Concepts and Applications

C H A P T E R12Surface Area and VolumeUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with a plain piece of 11" 17" paper.FoldFold the paper inthirds lengthwise.OpenOpen and fold a 2"tab along the shortside. Then fold therest in fifths.DrawDraw lines alongfolds and label asshown.SurfaceArea Volume Ch. 12PrismsCylindersPyramidsConesSpheres© Glencoe/McGraw-HillNOTE-TAKING TIP: When taking notes, explaineach new idea or concept in words and give oneor more examples.Chapter 12Geometry: Concepts and Applications 227

C H A P T E R12BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 12. Asyou complete the study notes for the chapter, you will see Build Your Vocabularyreminders to complete each term’s definition or description on these pages.Remember to add the textbook page number in the second column for referencewhen you study.Vocabulary TermFoundon PageDefinitionDescription orExampleaxiscomposite solidconecubecylinder[SIL-in-dur]edgefacelateral area[LAT-er-ul]lateral edgelateral facenetoblique cone[oh-BLEEK]oblique cylinder© Glencoe/McGraw-Hilloblique prism228 Geometry: Concepts and Applications

Chapter12BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleoblique pyramidPlatonic solidpolyhedron[pa-lee-HEE-drun]prism[PRIZ-um]pyramid[PEER-a-MID]regular pyramidright coneright cylinderright prismright pyramidsimilar solidsslant height© Glencoe/McGraw-Hillsolid figuressphere[SFEER]surface areatetrahedronvolumeGeometry: Concepts and Applications 229

12–1Solid FiguresWHAT YOU’LL LEARN• Identify solid figures.BUILD YOUR VOCABULARY (pages 228–229)Solid figures enclose a part of space.Solids with flat surfaces that areas polyhedrons.are knownThe two-dimensional polygonal surfaces of a polyhedronare its faces.Two faces of a polyhedroncalled an edge.in a segmentA prism is awith two faces, called bases, whichare formed by congruent polygons that lie in parallel planes.Faces in a prism that are not bases are parallelograms andare called lateral faces.The intersection of twolateral faces in aprism are called lateral edges and are parallel segments.A pyramid is a solid with all faces but one intersecting at acommon point called the vertex. The face not intersectingat the vertex is the base. The base of a pyramid is apolygon. The faces meeting at the vertex are lateral facesand are triangles.REMEMBER ITEuclidean solids arealso called solid figures.Name the faces, edges, and vertices ofthe polyhedron.The faces are quadrilaterals ABCD,, DCGH, ADHE, ABFE, .DHAECGBF© Glencoe/McGraw-HillThe edges are ,BC, CD, , BF, AE, DH, CG,EF, FG, GH, EH.The vertices are A,B, ,D,E,F, ,H.230 Geometry: Concepts and Applications

12–1Your Turnpolyhedron.Name the faces, edges, and vertices of theYWRITE ITXZGive three real-worldexamples ofpolyhedrons.WBUILD YOUR VOCABULARY (pages 228–229)A Platonic solid is apolyhedron.A cube is a special rectangular prism where all the facesare .A triangular pyramid is known as a tetrahedron because allof its faces are .A cylinder is a solid that is not abases are two congruentand its lateral surface is curved.A cone is a solid that is not a. Itsin parallel planes,. Its base isa, and the lateral surface is curved.© Glencoe/McGraw-HillA composite solid is a solid made byor more solids.twoGeometry: Concepts and Applications 231

12–1MREMEMBER ITCylinders and conesare terms referring tocircular cylinders andcircular cones.Is the pyramid in the figure atetrahedron or a rectangularpyramid?The pyramid has aandbaselateral faces. It is aNRPQpyramid.Your TurnDescribe the Washington Monument in termsof solid figures.HOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill232 Geometry: Concepts and Applications

12–2Surface Areas of Prisms and CylindersBUILD YOUR VOCABULARY (pages 228–229)WHAT YOU’LL LEARN• Find the lateral areasand surface areas ofprisms and cylinders.In a right prism, a lateral edge is also an altitude.In an oblique prism, a lateral edge is not an altitude.The lateral area of a solid figure is theof all theareas of its lateral faces.The surface area of a solid figure is theareas of all its surfaces.A net is a two-dimensional figure thata solid.of theto formTheorem 12-1 Lateral Area of a PrismIf a prism has a lateral area of L square units and a heightof h units and each base has a perimeter of P units, thenL Ph.Theorem 12-2 Surface Area of a PrismIf a prism has a surface area of S square units and a heightof h units and each base has a perimeter of P units and anarea of B square units, then S Ph 2B.© Glencoe/McGraw-HillORGANIZE ITIn the box for SurfaceArea of Prisms, make asketch of a prism. Thenwrite the formula forfinding the surface areaof a prism.SurfaceArea Volume Ch. 12PrismsCylindersPyramidsConesSpheresFind the lateral area and total surface area of a cubewith side length 6 inches.Perimeter of BaseArea of BaseP 4s B s 2 4(6) or 6 2 orLateral AreaSurface AreaL PhS L 2B (24)(6) or 144 2(36) 144 72 orThe lateral area of the cube isin 2 , and the surfacearea is in 2Geometry: Concepts and Applications 233

12–2WRITE ITWhat is the differencebetween lateral areaand surface area?Your TurnFind the lateral area and the surface area ofthe rectangular prism.15 ft9 ft7 ftFind the lateral area and the surfacearea of the triangular prism.Use the Pythagorean Theorem to find thelength of side b.Perimeter of Base10 cm8 cm6 cmArea of Basec 2 a 2 b 2 P 10 6 b B 1 2 bhREVIEW ITWhat is the length ofthe hypotenuse of aright triangle with legs5 cm and 12 cm long?(Lesson 6-6)10 2 6 2 b 2 10 6 8 1 2 (6)(8)100 36 b 2 b 264 b2 bFind the lateral and surface areas.L PhS L 2B (24)(8) 192 2(24) cm 2 192 48 cm 2Your TurnFind the lateral area andthe surface area of the triangular prism.15cm© Glencoe/McGraw-Hill12cm5cm13cm234 Geometry: Concepts and Applications

12–2BUILD YOUR VOCABULARY (pages 228–229)ORGANIZE ITIn the box for SurfaceArea of Cylinders, makea sketch of a cylinder.Then write the formulafor finding the surfacearea of a cylinder.The axis of a cylinder is the segment whoseare centers of the circular bases.In a right cylinder, the axis is also an .In an oblique cylinder, the axis is not an altitude.SurfaceArea Volume Ch. 12PrismsCylindersPyramidsConesSpheresTheorem 12-3 Lateral Area of a CylinderIf a cylinder has a lateral area of L square units and aheight of h units and the bases have radii of r units, thenL 2rh.Theorem 12-4 Surface Area of a CylinderIf a cylinder has a surface area of S square units and aheight of h units and the bases have radii of r units, thenS 2rh 2r 2 .Find the lateral area and surface area ofthe cylinder to the nearest hundredth.L 2rh S 2rh 2r 2 2(8)(14) 703.72 2(8) 2 14 in.8 in.The lateral area is aboutin 2 , and the surface© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:area is about in 2 .Your TurnFind the lateral area and surface area of thecylinder to the nearest hundredth.15 m34 mGeometry: Concepts and Applications 235

12–3Volumes of Prisms and CylindersBUILD YOUR VOCABULARY (page 229)WHAT YOU’LL LEARN• Find the volumes ofprisms and cylinders.Volume measures the space contained within a solid.Theorem 12-5 Volume of a PrismIf a prism has a volume of V cubic units, a base with an areaof B square units, and a height of h units, then V Bh.ORGANIZE ITUse the box for Volumeof Prisms. Sketch andlabel a prism. Thenwrite the formula forfinding the volume ofa prism.SurfaceArea Volume Ch. 12PrismsCylindersPyramidsConesSpheresFind the volume of the triangular prism.Area of triangular baseB 1 (10)(24) or .2V Bh Theorem 12-5 Substitution m 3Find the volume of the rectangular prism.Area of base B (2)(5) or .V Bh Theorem 12-5 Substitution ft 3Your Turna. Find the volume of the triangular prism.10 mb. Find the volume of a rectangular prism with basedimensions of 8 cm by 9 cm and height 4.1 cm.20 ft20 in.16 in.12 in.5 ft24 m2 ft36 m© Glencoe/McGraw-Hill236 Geometry: Concepts and Applications

12–3Theorem 12-6 Volume of a CylinderIf a cylinder has a volume of V cubic units, a radius ofr units, and a height of h units, then V r 2 h.ORGANIZE ITUse the box for Volumeof Cylinders. Sketch andlabel a cylinder. Thenwrite the formula forfinding the volume of acylinder.SurfaceArea Volume Ch. 12PrismsCylindersPyramidsConesSpheresFind the volume of the cylinderto the nearest hundredth.V r 2 h Theorem 12-6(5) 2 (12)Substitution 300 cm 3Your TurnFind the volume ofthe cylinder to the nearest hundredth.12 cm21 ft5 cm45 ftLeticia is making a sand sculpture by filling a glasstube with layers of different-colored sand. The tube is24 inches high and 1 inch in diameter. How many cubicinches of sand will Leticia use to fill the tube?V r 2 h Theorem 12-6(0.5) 2 (24) (0.25)(24) 6Substitution© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Leticia will need aboutin 3 of sand.Your TurnSam fills the cylindrical coffee grind containers.One bag has 32 cubic inches of grinds. How many cylindricalcontainers can Sam fill with two bags of grinds if each cylinderis 4 inches wide and 4 inches high?Geometry: Concepts and Applications 237

12–4Surface Areas of Pyramids and ConesBUILD YOUR VOCABULARY (page 229)WHAT YOU’LL LEARN• Find the lateral areasand surface areas ofregular pyramidsand cones.In a right pyramid or a right cone, theisperpendicular to the base at its center.In a oblique pyramid or a oblique cone, the altitude isto the base at a point other thanits center.A pyramid is a regular pyramid if and only if it is apyramid and its base is apolygon.The height of eachis called the slant height of the pyramid.face of a regular pyramidORGANIZE ITUse the box for SurfaceArea of Pyramids. Sketchand label a pyramid.Then write the formulafor finding the surfacearea of a pyramid.SurfaceArea Volume Ch. 12PrismsCylindersPyramidsConesSpheresTheorem 12-7 Lateral Area of a Regular PyramidIf a regular pyramid has a lateral area of L square units, abase with a perimeter of P units, and a slant height of units, then L 1 2 P.Theorem 12-8 Surface Area of a Regular PyramidIf a regular pyramid has a surface area of S square units, aslant height of units, and a base with perimeter of P unitsand an area of B square units, then S 1 P B.2Find the lateral area andthe surface area of thesquare pyramid.First, find the perimeter and the area ofthe base.P 4s B s 2 4(15) or 15 2 or15 cm15 cm25 cm© Glencoe/McGraw-Hill238 Geometry: Concepts and Applications

12–4Lateral AreaL 1 2 PSurface AreaS L B 1 (60)(25) 750 2252 cm 2 cm 2Your TurnFind the lateral area andsurface area of the square pyramid.1066Find the lateral area and the surface area of a regulartriangular pyramid with a base perimeter of 24 inches,a base area of 27.7 square inches, and a slant height of8 inches.L 1 P Theorem 12-72 1 2 (24)(8)Substitution in 2S L B Theorem 12-7 96 27.7Substitution© Glencoe/McGraw-Hill in 2Your TurnFind the lateral area and the surface area of aregular triangular pyramid with a base perimeter of 18 inches,a base area of 15.6 square inches, and a slant height of11 inches.Geometry: Concepts and Applications 239

12–4ORGANIZE ITUse the box for SurfaceArea of Cones. Sketchand label a cone. Thenwrite the formula forfinding the surface areaof a cone.Theorem 12-9 Lateral Area of a ConeIf a cone has a lateral area of L square units, a slant heightof units, and a base with a radius of r units, thenL 1 (2r) or r.2Theorem 12-10 Surface Area of a ConeIf a cone has a surface area of S square units, a slant heightof units, and a base with a radius of r units, thenS r r 2 .SurfaceArea Volume Ch. 12PrismsCylindersPyramidsConesSpheresFind the lateral area and the surfacearea of the cone to the nearesthundredth.20 cm35 cmL r Theorem 12-9(20)(35)Substitution 700 cm 2S r r 2 Theorem 12-10 (35) (20)2 Substitution 2199.11 400 2199.11 1256.64The lateral area is aboutcm 2 , and the surfaceHOMEWORKASSIGNMENTPage(s):Exercises:area is about cm 2 .Your TurnFind the lateral area and the surface area ofthe cone to the nearest hundredth.3.2 yd5.2 yd© Glencoe/McGraw-Hill240 Geometry: Concepts and Applications

12–5Volumes of Pyramids and ConesWHAT YOU’LL LEARN• Find the volumes ofpyramids and cones.Theorem 12-11 Volume of a PyramidIf a pyramid has a volume of V cubic units and a height ofh units and the area of the base is B square units, thenV 1 3 Bh.Find the volume of the rectangularpyramid.B w (10)(4) orV 1 Bh Theorem 12-11310 cm12 cm4 cm 1 3 (40)(12)Substitution cm 3© Glencoe/McGraw-HillORGANIZE ITUse the boxes forVolume of Pyramids andCones. Sketch and labela pyramid and a cone.Then write the formulafor finding the volumesof a pyramid and acone.SurfaceArea Volume Ch. 12PrismsCylindersPyramidsConesSpheresFind the volume of the cone to the nearest hundredth.Find the height hh 2 21 2 35 2h 2 441 1225h 2 784h 2 784 h V 1 3 r2 h Theorem 12-11 1 3 (21)2 (28)Substitution in 3 42 in.35 in.Theorem 12-12 Volume of a ConeIf a cone has a volume of V cubic units, a radius of r units,and a height of h units, then V 1 3 r2 h.Geometry: Concepts and Applications 241

12–5Your Turna. Find the volume of the triangularpyramid.25 m30 m35 mb. Find the volume of the cone to thenearest hundredth.80 in.42 in.REMEMBER ITUse the altitude ofa solid, not the slantheight, to find thevolume of the solid.The sand in a cone with radius 3 cm and height 10 cm ispoured into a square prism with height of 29.5 cm andbase area of 4 cm 2 . How far up the side of the prism willthe sand reach when leveled?Volume of ConeV 1 3 r2 h 1 3 (3)2 (10)Volume of PrismV Bh94.25 4h 30h HOMEWORKASSIGNMENTPage(s):Exercises:The sand will level off at a height of aboutprism.Your TurnThe salt in a cone with radius 6 cm andheight 8 cm is poured into a square prism with height of20 cm and base area of 12 cm 2 . Will the prism be able tohold all of the salt?cm in the© Glencoe/McGraw-Hill242 Geometry: Concepts and Applications

12–6SpheresBUILD YOUR VOCABULARY (page 229)WHAT YOU’LL LEARNA sphere is the set of all points that are a fixed• Find the surface areasand volumes of spheres.from a givencalled the center.ORGANIZE ITUse the boxes forVolume and SurfaceArea of Spheres. Sketchand label a sphere. Thenwrite the formulas forfinding the surface areaand volume of a sphere.SurfaceArea Volume Ch. 12PrismsCylindersPyramidsConesSpheresTheorem 12-13 Surface Area of a SphereIf a sphere has a surface area of S square units and a radiusof r units, then S 4r 2 .Theorem 12-14 Volume of a SphereIf a sphere has a volume of V cubic units and a radius ofr units, then V 4 3 r 3 .Find the surface area and volume of a sphere withradius 5 cm.Surface AreaVolumeS 4r 2 V 4 3 r 3 4(5) 2 4 3 (5) 3 100 50 03 cm 2 cm 3© Glencoe/McGraw-HillYour TurnFind the surface area and volume of a spherewith diameter 15 in.Geometry: Concepts and Applications 243

12–6Some students build a snow sculpturefrom a cylinder and a sphere of snow.Both the sphere and the cylinder havea radius of 1 ft. and the height of thecylinder is 4 ft. Find the volume of thesnow used to build the sculpture.1 ft1 ft4 ftVolume of CylinderVolume of SphereV r 2 h V 4 3 r 3(1) 2 (4) 4 3 (1) 3 4 4 3 The volume of the snow used for the sculpture is about12.57 4.19, or ft 3 .Your TurnFelix and Brenda want to share an ice creamcone. Brenda wants half the scoop of ice cream on top, whileFelix wants the ice cream inside the cone. Assuming the halfscoop of ice cream on top is a perfect sphere, who will havemore ice cream? The cone and scoop both have radii of 1.5 inch;the cone is 3.25 inches long.HOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill244 Geometry: Concepts and Applications

12–7Similarity of Solid FiguresWHAT YOU’LL LEARN• Identify and use therelationship betweensimilar solid figures.BUILD YOUR VOCABULARY (page 229)Similar solids are solids that have the samenot necessarily the same .butKEY CONCEPTDetermine whether the pairof solids is similar.9 cm27 cm54 cmCharacteristics of SimilarSolids For similar solids,the correspondinglengths are proportional,and the correspondingfaces are similar.18 cm9 1 8 2 7 54 Definition of similarity(9)(54) (27)(18) Cross products486 486The corresponding lengths are in, so thesolidssimilar.Your Turnis similar.Determine whether each pair of solidsa. b.© Glencoe/McGraw-HillREMEMBER ITA scale factor is aone-dimensionalmeasure. Surface area isa two-dimensionalmeasure. Volume is athree-dimensionalmeasure.3 344162215Theorem 12-15If two solids are similar with a scale factor of a:b, then thesurface areas have a ratio of a 2 :b 2 and the volumes have aratio of a 3 :b 3 .12812181218Geometry: Concepts and Applications 245

12–7For the similar prisms, find thescale factor of the prism on theleft to the prism on the right.Then find the ratios of thesurface areas and the volumes.21 cm30 cm7 cm7 cm21 cm 10 cmThe scale factor is 2 1 3 0 or .7 10The ratio of the surface areas is 3 2 or .12The ratio of the volumes is 3 3 or .13Your TurnFind the scale factor ofthe prism on the left to the prism on theright. Then find the ratios of the surfaceareas and the volumes.18 ft30 ftSara made a scale model of the Great AmericanPyramid in Memphis, Tennessee, which has a base sidelength of 544 ft and a lateral area of 456,960 ft 2 . If thescale factor of the model to the original is 1:136, whatwill be the lateral area of the model?surface area of the model21 surface area of Great Amer. Pyr. 1 36 2L1HOMEWORKASSIGNMENTPage(s):Exercises:18,496L 456,960L ft 2Your TurnA scale model of a house is made using a scale1factor of . What fraction of the actual house material would1 12would the dollhouse need to cover all of its floors?© Glencoe/McGraw-Hill246 Geometry: Concepts and Applications

C H A P T E R12BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 12 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 12, go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 228–229) to help yousolve the puzzle.12-1Solid FiguresComplete each sentence.1. Two faces of a polyhedron intersect at a(n) .2. A triangular pyramid is called a .3. A is a figure that encloses a part of space.4. Three faces of a polyhedron intersect at a point called a(n).12-2Surface Areas of Prisms and Cylinders© Glencoe/McGraw-HillFind the lateral area and surface area ofeach solid to the nearest hundredth.5. a regular pentagonal prism with apothema 4, side length s 6, and height h 12a. L b. S 46126. a cylinder with radius r 42 and height h 10a. L b. S Geometry: Concepts and Applications 247

Chapter12BRINGING IT ALL TOGETHER12-3Volumes of Prisms and CylindersFind the volume of each solid and round to the nearesthundredth.7. the regular pentagonal prism from Exercise #58. How much water will a 24 in. by 15 in. by 10 in. fish tank hold?12-4Surface Areas of Pyramids and ConesFind the lateral and surface areas of each solid. Round tothe nearest hundredth if necessary.9. a rectangular pyramid with base dimensions 2 ft by 3 ft andlateral height h = 1 fta. L b. S 10. a cone with diameter 3.6 cm and lateral height 2.4 cma. L b. S 12-5Volumes of Pyramids and ConesFind the volume of each solid rounded to the nearesthundredth, if necessary.11. a cone with its height as three times the radius© Glencoe/McGraw-Hill12. the cone in Exercise #10248 Geometry: Concepts and Applications

Chapter12BRINGING IT ALL TOGETHER12-6SpheresComplete the sentence.13. The set of all points a given distance from the center is a.A beach ball will have a diameter of 30 in.14. How much material will be used to make the beach ball?15. How much air will be needed to fill it?12-7Similarity of Solid Figures16. Solids having the same shape but not always the same sizeare .If the radius of a sphere is doubled:17. How does the surface area change?18. How does the volume change?© Glencoe/McGraw-HillThe diameter of the moon is about 2160 miles. The diameterof the Earth is about 7900 miles.19. Assuming both are spheres, what is the scale factor of theEarth to the moon?20. Are they similar solid figures?Geometry: Concepts and Applications 249

C H A P T E R12ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 12.• You may want to take the Chapter 12 Practice Test on page543 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 12 Study Guide and Reviewon pages 540–542 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 12 Practice Test onpage 543.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 12 Foldable.• Then complete the Chapter 12 Study Guide and Review onpages 540–542 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 12 Practice Test onpage 543.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature250 Geometry: Concepts and Applications

C H A P T E R13Right Triangles and TrigonometryUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Chapter 13Begin with three sheets of lined 8 1 " 11" paper.2StackStack sheets of paper withedges 1 4 inch apart.FoldFold up bottom edges.All tabs should be thesame size.CreaseCrease and staple along fold.© Glencoe/McGraw-HillTurnTurn and label the tabswith the lesson names.Right Trianglesand Trigonometry13-1 Simplifying Square Roots13-2 45º -45º -90º Triangles13-3 30º -60º -90º Triangles13-4 Tangent Ratios13-5 Sine and Cosine RatiosNOTE-TAKING TIP: When taking notes, it is oftenhelpful to remember what you’ve learned if youcan paraphrase or summarize key terms andconcepts in your own words.Geometry: Concepts and Applications 251

C H A P T E R13BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 13.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in thesecond column for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExample30°-60°-90° triangle45°-45°-90° triangleangle of depressionangle of elevationcosinehypsometerperfect square© Glencoe/McGraw-Hillradical expression[RAD-ik-ul]252 Geometry: Concepts and Applications

Chapter13BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleradical signradicand[RAD-i-KAND]simplest formsinesquare roottangent[TAN-junt]© Glencoe/McGraw-Hilltrigonometric identity[TRIG-guh-no-MET-rik]trigonometric ratiotrigonometryGeometry: Concepts and Applications 253

13–1Simplifying Square RootsBUILD YOUR VOCABULARY (pages 252–253)WHAT YOU’LL LEARN• Multiply, divide, andsimplify radicalexpressions.Perfect squares are products of twowhen a number multiplies itself.The square root, therefore, is one offactors, orequal factors.Anumber has both positive () and negative ()square roots, indicated by the radical sign .A radical expression is an expression that contains a.The number under the radical sign is the radicand.WRITE ITWhat are the nextthree perfect squaresafter 16?Simplify each expression.3636 , because 6 2 36.8181 , because 9 2 81.KEY CONCEPTProduct Property ofSquare Roots The squareroot of a product isequal to the product ofeach square root.2424 2 2 2 3 2 2 2 3Prime factorizationProduct Property ofSquare Roots© Glencoe/McGraw-Hill 2 6 2 2 2254 Geometry: Concepts and Applications

13–16 30KEY CONCEPTQuotient Property ofSquare Roots The squareroot of a quotient isequal to the quotient ofeach square root.On the tabfor Lesson 13-1, writethe names of the twoproperties introduced inthis lesson. Then writeyour own example ofeach property on theback of the tab.6 30 6 6 6 5Prime factorizationProduct Property ofSquare Roots 5 Product Property ofSquare Roots 6 6 6Your TurnSimplify each expression.a. 25 b. 121 c. 18 d. 3 12Simplify each expression.16 8168 1 68Quotient Property© Glencoe/McGraw-HillREMEMBER ITSimplifying afraction with a radical inthe denominator iscalled rationalizing thedenominator. 1 2149 1 2149 121 49Quotient PropertyYour TurnSimplify each expression.144 a. 20b. 425Geometry: Concepts and Applications 255

13–110 Simplify . 7107 10 77 17 7710 7Product Property of7 7Square Roots70 7 7 77KEY CONCEPTRules for SimplifyingRadical Expressions1. There are no perfectsquare factors otherthan 1 in the radicand.2. The radicand is not afraction.3. The denominator doesnot contain a radicalexpression.We used the Identity Property and the Product Property ofSquare Roots to simplify the above radical expression. Thedenominator does not have a radical sign.16Simplify . 616 16 16 616 66 6Product Property ofSquare Roots166 6 6 616 66orWe used the Identity Property and the Product Property ofSquare Roots to simplify the above expression and eliminatethe radical in the denominator.HOMEWORKASSIGNMENTa.Your Turn72Simplify.© Glencoe/McGraw-HillPage(s):Exercises:b.45256 Geometry: Concepts and Applications

13–245°-45°-90° TrianglesWHAT YOU’LL LEARN• Use the properties of45°-45°-90° triangles.BUILD YOUR VOCABULARY (page 252)Theof a square separates the square intotwo 45°-45°-90° triangles.WRITE ITDescribe two differentways to find the lengthof the hypotenuse of a45°-45°-90° triangle.Theorem 13-1 45°-45°-90° Triangle TheoremIn a 45°-45°-90° triangle, the hypotenuse is 2 timesthe length of a leg.In a scale model of a town, a baseball diamond has sides36 inches long. What is the distance from first base tothird base on the model? Round to the nearest tenth.h s2 Theorem 13-1 2 SubstitutionThe distance from first to third base on the scale model isaboutinches.© Glencoe/McGraw-HillYour TurnFind the length of the diagonal of a squarewhose side measures 22 inches.Geometry: Concepts and Applications 257

13–2REMEMBER ITA 45°-45°-90°triangle is isosceles, sothe legs are alwayscongruent.If DJT is an isosceles right triangle and the measureof the hypotenuse is 200 , find the measure ofeither leg.h s2 Theorem 13-1 s2Substitution sThe length of each leg measures .ORGANIZE ITOn the tab forLesson 13-2, draw a45°-45°-90° triangleand label its parts.Then write your ownreal-world example thatcan be solved using thisinformation.Your TurnIf XYZ is an isoscelesright triangle and the measure of thehypotenuse is 25, find the measure ofeither leg.X25sZYsRight Trianglesand Trigonometry13-1 Simplifying Square Roots13-2 45º -45º -90º Triangles13-3 30º -60º -90º Triangles13-4 Tangent Ratios13-5 Sine and Cosine RatiosHOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill258 Geometry: Concepts and Applications

13–330°-60°-90° TrianglesBUILD YOUR VOCABULARY (page 252)WHAT YOU’LL LEARN• Use the properties of30°-60°-90° triangles.The median of an equilateral triangle separates it intotwo 30°-60°-90° triangles.REMEMBER ITThe shorter leg isalways opposite the30° angle, and thelonger leg is alwaysopposite the 60° angle.Theorem 13-2 30°-60°-90° Triangle TheoremIn a 30°-60°-90° triangle, the hypotenuse is twice thelength of the shorter leg, and the longer leg is 3 timesthe length of the shorter leg.In ABC, a 12. Find b and c.a b3 b3The longer leg is 3 timesthe length of the shorter leg.Replace a with .cB30˚a bA60˚bCc 2bThe hypotenuse is twicethe shorter leg.© Glencoe/McGraw-Hillc 2c Replace b with .Your TurnRefer to Example 1.a. If b 3.5, find a and c. b. If b 1 , find a and c.3Geometry: Concepts and Applications 259

13–3In DEF, DE 18. Find EF and DF.EUse Theorem 13–2.DE EF3D30˚The longer leg is 3 timesthe shorter leg.60˚FReplace DE with . EF3Divide each side by 3. EFDF 2(EF)The hypotenuse is twicethe shorter leg.ORGANIZE ITUnder the tab forLesson 13-3, draw a30°-60°-90° triangle andlabel its parts. Thenwrite a summary of howyou can find the lengthof the longer leg giventhe length of theshorter leg.DF 2DF Replace EF with .Associative PropertyYour TurnRefer to Example 2. If DE 11, find EF and DF.Right Trianglesand Trigonometry13-1 Simplifying Square Roots13-2 45º -45º -90º Triangles13-3 30º -60º -90º Triangles13-4 Tangent Ratios13-5 Sine and Cosine RatiosHOMEWORKASSIGNMENTFind the length, to the nearest tenth,of the median in the equilateral triangle.The median bisects one side into two 5-metersegments and is opposite the 60° angle.x 53 Theorem 13-2meters10 mx10 m10 m© Glencoe/McGraw-HillPage(s):Exercises:Your TurnFind the length, to the nearesttenth, of the median in the equilateral triangle.23x2323260 Geometry: Concepts and Applications

13–4Tangent RatioBUILD YOUR VOCABULARY (page 253)WHAT YOU’LL LEARN• Use the tangent ratio tosolve problems.Trigonometry is the study of the properties of.A trigonometric ratio is a ratio of the measures of two sidesof atriangle.The tangent is the ratio of oneto the other.If A is an acute angle of a right triangle,measure of leg opposite Atan A .measure of leg adjacent to AREMEMBER ITThe ratio of themeasures of the legs ofa right triangle can becompared to the ratioof rise to run in thedefinition of slopeof a line.Find tan K and tan M.tan K M LKL 2 1 or28K3528M21Lopposite adjacentSubstitution© Glencoe/McGraw-HillKLtan M M L 2 8 or21Your Turnopposite adjacentSubstitutionFind tan 30°, tan 45°, tan 60°.Geometry: Concepts and Applications 261

13–4PREMEMBER ITThe symbol tan B isread the tangent ofangle B.Find QR to the nearesttenth of a meter.tan Q RPQopposite adjacentQRtan Substitution5520 mQR QR Multiplication Propertyof Equality QRORGANIZE ITWrite the definition oftangent on the tab forLesson 13-4. Under thetab, sketch and label atriangle. Then expressthe tangent of one ofthe angles.Right Trianglesand TrigonometryA ranger sights the top of a tree at a 40° angle ofelevation. Find the height of the tree if it is 80 feet fromwhere the ranger is standing.tanheight of treetan height of tree Multiplication Propertyof Equality height of treeopposite adjacent13-1 Simplifying Square Roots13-2 45º -45º -90º Triangles13-3 30º -60º -90º Triangles13-4 Tangent Ratios13-5 Sine and Cosine RatiosThe height of the tree is aboutfeet.BUILD YOUR VOCABULARY (page 252)The line of sight and a horizontal line when looking upform the angle of elevation.Angles of elevation can be measureda hypsometer.The line of sight and a horizontal line when lookingform the angle of depression.using© Glencoe/McGraw-Hill262 Geometry: Concepts and Applications

13–4REMEMBER ITThe inverse tangentis also called thearctangent.Your Turna. Find YZ to the nearest tenth of a foot.X1771YZb. The ranger sights the top of another tree at a 52° angle ofelevation. Find the height of the tree if it is 20 feet fromwhere he stands.Find m1 to the nearest tenth.47 in.122 in.tan(1) opposite adjacenttan 1 22 47 Definition of arctangent© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:The measure of 1 is about .Your TurnFind m2 to the nearest tenth.R9K112JGeometry: Concepts and Applications 263

13–5Sine and Cosine RatiosBUILD YOUR VOCABULARY (pages 252–253)WHAT YOU’LL LEARNBoth the sine and the cosine ratios relate an angle measure• Use the sine and cosineratios to solve problems.to the ratio of the measures of a triangle’s.If A is an acute angle of a right triangle,measure of leg opposite Asin A , andmeasure of hypotenusemeasure of leg adjacent Acos A .measure of hypotenuseto itsORGANIZE ITWrite the definitions ofsine and cosine on thetab for Lesson 13-5. Onthe back of the tab,describe a similarity anda difference betweensine and cosine.Right Trianglesand Trigonometry13-1 Simplifying Square Roots13-2 45º -45º -90º Triangles13-3 30º -60º -90º Triangles13-4 Tangent Ratios13-5 Sine and Cosine RatiosFind sin K, cos K, sin M,and cos M.sin K KLMM sin M KLKM Substitution SubstitutionoppositehypotenuseK1517oppositehypotenuseL8Mcos KKL K Mcos M KLMM Substitution SubstitutionYour Turncos 60°.adjacenthypotenuseadjacenthypotenuseFind sin 30°, cos 30°, sin 45°, cos 45°, sin 60°,© Glencoe/McGraw-Hill264 Geometry: Concepts and Applications

13–5Find the value of x to thenearest tenth.200 m26x mxsin 26 20 0oppositehypotenuse200 sin 26 x Multiplication Propertyof Equality xLREMEMBER ITSin 1 and cos 1 arealso known as arcsinand arccos.Find the measure of K to thenearest degree.sin K KLMM sin K 6 882oppositehypotenuseSubstitutionmK sin 1 6 882 Inverse sineM6882KmK Your Turna. Find the value of x to the nearest tenth.58x33© Glencoe/McGraw-HillHOMEWORKASSIGNMENTb. Find the measure of A to the nearest degree.B86C32APage(s):Exercises:Theorem 13-3If x is a measure of an acute angle of a right triangle,sinxthen c os x tan x.Geometry: Concepts and Applications 265

C H A P T E R13BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 13 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 13, go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 252–253) to help yousolve the puzzle.13-1Simplifying Square RootsSimplify.1. 63 2. 1 3. 10 8 34. Find the value of x if 2 2x.x 313-245°-45°-90° TrianglesA fabric square is cut on the diagonal for a quilt.The perimeter of the square is 116 in.5. What is the length of each leg/side?6. What is the length of the hypotenuse/diagonal?© Glencoe/McGraw-Hill7. What is the measure of each leg of an isosceles right triangleif its hypotenuse measures 10?266 Geometry: Concepts and Applications

Chapter13BRINGING IT ALL TOGETHER13-330°-60°-90° Triangles8. The Gothic arch, similar to the figure, is basedon an equilateral triangle. Find the width of thebase of the triangle if the median is 4 ft long.4Find the missing measure. Simplify all radicals.9. 10.5.56011b60c66 313-4Tangent Ratio11. You spot a cat on the roof of a house 80 feet away from whereyou’re standing. Your eye level is 5 feet above ground level, andthe angle of elevation from eye level is 33. How tall is the house?13-5Sine and Cosine RatiosFind the missing measures.12. If y 20, find x and z.© Glencoe/McGraw-Hill13. If z 2.3, find x and y.Z30yxX60zY14. If x 9, find y and z.Geometry: Concepts and Applications 267

C H A P T E R13ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 13.• You may want to take the Chapter 13 Practice Test onpage 581 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 13 Study Guide and Reviewon pages 578–580 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 13 Practice Test onpage 581.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 13 Foldable.• Then complete the Chapter 13 Study Guide and Review onpages 578–580 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 13 Practice Test onpage 581.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature268 Geometry: Concepts and Applications

C H A P T E R14Circle RelationshipsUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with three sheets of plain 8 1 " 11"2paper.FoldFold in half alongthe width.Chapter 14OpenOpen and fold thebottom to form apocket. Glue edges.RepeatRepeat steps 1 and 2three times and glue allthree pieces together.© Glencoe/McGraw-HillLabelLabel each pocket withthe lesson names. Place anindex card in each pocket.CircleRelationshipsNOTE-TAKING TIP: When taking notes, definenew terms and write about the new ideas andconcepts you are learning in your own words.Write your own examples that use the new termsand concepts.Geometry: Concepts and Applications 269

C H A P T E R14BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 14.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExampleexternal secant segment[SEE-kant]externally tangent[TAN-junt]inscribed angleintercepted arcinternally tangent© Glencoe/McGraw-Hill270 Geometry: Concepts and Applications

Chapter14BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExamplepoint of tangencysecant anglesecant-tangent anglesecant segment© Glencoe/McGraw-Hilltangenttangent-tangent angleGeometry: Concepts and Applications 271

14–1Inscribed AnglesWHAT YOU’LL LEARN• Identify and useproperties of inscribedangles.BUILD YOUR VOCABULARY (page 270)An inscribed angle is an angle whoselies on acircle and whose sides containof the circle.An intercepted arc is an arc of a circle, formed by an angle,ORGANIZE ITUnder the tab forInscribed Angles, writethe definition of aninscribed angle anddraw a picture toillustrate the concept.Record the theoremsand other importantinformation from thislesson.CircleRelationshipssuch that theDetermine whether ABC is aninscribed angle. Name theintercepted arc for the angle.The vertex of ABC, point B, is oncircle Q. Therefore, ABC is anof the arc lie on the sidesof the angle and all other points of the arc lie on theof the angle.ABQCangle. The interceptedarc is AC .Your TurnDetermine whether JKL is an inscribedangle. Name the intercepted arc for the angle.PKJML© Glencoe/McGraw-HillTheorem 14-1The degree measure of an inscribed angle equals one-halfthe degree measure of its intercepted arc.272 Geometry: Concepts and Applications

14–1Refer to the figure.If mAB 76, find mADB.mADB 1 2 (mAB ) Theorem 14-1DCKABmADB 1 2 Replace mAB .mADB If mBDC 40, find mBC .mBDC 1 2 (mBC ) Theorem 14-140 1 2 (mBC )Replace mBDC.2 40 2 1 2 (mBC ) Multiply each side by 2. mBCYour TurnRefer to the figure.a. If mZW 124,find mWXZ.YZWRITE ITb. If mYXZ 49,find mYZ .XWWhat is the differencebetween a central angleand an inscribed angle?Theorem 14-2If inscribed angles intercept the same arc or congruent arcs,then the angles are congruent.© Glencoe/McGraw-HillREMEMBER ITThere are 360 in acircle and 180 in a semicircle.In circle A, suppose mTLN 6y 7and mTWN 7y. Find the value of y.TLN and TWN both intercept TN.TLN TWN Theorem 14-2LmTLN mTWN Definition of congruent angles yReplace mTLN and mTWN.Subtract 6y from each side.T1 A 2WNGeometry: Concepts and Applications 273

14–1WRITE ITYour TurnIn the circle, if mAHM 10x andmATM 20x 30, find the value of x.How does the measureof an inscribed anglerelate to the measure ofits intercepted arc?MHATTheorem 14-3If an inscribed angle of a circle intercepts a semicircle, thenthe angle is a right angle.In circle G, m1 6x 5 andm2 3x 4. Find the value of x.Inscribed angle DEF intercepts semicircleDF . DEF is a right angle by Theorem 14-3.Therefore, 1 and 2 are complementary.E1D2GFm1 m2 90Complementary angles(6x 5) (3x 4) 90 Substitution 90Combine like terms.9x 9 90 Add to each side. 9 x 9 9 Divide each side by 9.9 9HOMEWORKASSIGNMENTx Your TurnIn circle W, mRKP 1 2 x andmKRP 1 3 x 5. Find the value of x.© Glencoe/McGraw-HillPage(s):Exercises:KWRP274 Geometry: Concepts and Applications

14–2Tangents to a CircleWHAT YOU’LL LEARN• Identify and applyproperties of tangentsto circles.BUILD YOUR VOCABULARY (page 271)In a plane, a line is a tangent if and only if it intersects acircle in exactly point.The point of intersection is the point of tangency.Theorem 14-4In a plane, if a line is tangent to a circle, then it isperpendicular to the radius drawn to the point of tangency.Theorem 14-5In a plane, if a line is perpendicular to a radius of a circle atits endpoint on the circle, then the line is a tangent.AB is tangent to circle C at B. Find BC.AB CB by Theorem 14-4, making CBAa right angle by definition. Therefore, ABCis a right triangle.CA3528 B© Glencoe/McGraw-HillORGANIZE ITUnder the tab forTangents to a Circle,write the definition oftangent and draw apicture to illustrate theconcept. Record thetheorems and otherimportant informationfrom this lesson.CircleRelationships(BC) 2 (AB) 2 (AC) 2(BC) 2 22Pythagorean TheoremReplace AB and AC.(BC) 2 Square and .(BC) 2 784 784 1225 784(BC) 2 (BC) 2 441 BC Subtract 784 from each side.Take the square root ofeach side.Geometry: Concepts and Applications 275

14–2Your TurnAE is tangent to circle Cat E. Find AE.34 in.C16 in.EATheorem 14-6If two segments from the same exterior point are tangentto a circle, then they are congruent.HEF and EG are tangent to circle H.Find the value of x.F(3x + 10) mEG43 mEF EG Theorem 14-6Replace EF and EG .3x 10 43 Subtract 10 from each side.3x 33 3 x 3 3 Divide each side by .3 3x Your TurnAD , AC , and AB aretangents to circles Q and R, respectively.Find the value of x.DQCRBHOMEWORKASSIGNMENTPage(s):Exercises:6x 5 2x 37ABUILD YOUR VOCABULARY (pages 270–271)If two circles are tangent and one circle istheother, the circles are internally tangent.If two circles are tangent andcircle is insidethe other, the circles are externally tangent.© Glencoe/McGraw-Hill276 Geometry: Concepts and Applications

14–3Secant AnglesWHAT YOU’LL LEARN• Find measures of arcsand angles formed bysecants.BUILD YOUR VOCABULARY(page 271)A secant segment is a segment that contains aof a circle.A secant angle is the angle formed when twosegments intersect.ORGANIZE ITUnder the tab forSecant Angles, write thedefinition of a secantsegment. Draw a pictureof secant angles toillustrate the concept.Record the theoremsand other importantinformation from thislesson.CircleRelationshipsTheorem 14-7A line or line segment is a secant to a circle if and only if itintersects the circle in two points.Theorem 14-8If a secant angle has its vertex inside a circle, then itsdegree measure is one-half the sum of the degree measuresof the arcs intercepted by the angle and its vertical angle.Theorem 14-9If a secant angle has its vertex outside a circle, then itsdegree measure is one-half the difference of the degreemeasures of the intercepted arcs.A 40 BFind m1.The vertex of 1 is inside circle P.C76m1 1 2 (mAB mCD ) Theorem 14-81DP© Glencoe/McGraw-Hillm1 1 2 Replace mAB and mCD.m1 1 2 Your Turnfind m1. or If mMA 40 and mHT 50,SM1AHTGeometry: Concepts and Applications 277

14–3JREMEMBER ITThe diameter of acircle is also a secant.Find mJ.The vertex of J is outside circle Q.mJ 1 2 (mMN mKL)Theorem 14-9mJ 1 2 MK 50LQN100mJ 1 2 orFind the value of x. Then find mCD .The vertex lies inside circle P.(3x – 1)AB57 1 2 (mAB mCD )57 1 2 57 PC(6x + 7) D57 1 (9x 6) Combine like terms.22 57 2 1 (9x 6) Multiply each side by 2.2114 6 9x 6 6 Subtract 6 from each side. xSubtraction PropertyDivision PropertymCD 6x 7 6 7 7 HOMEWORKASSIGNMENTPage(s):Exercises:Your Turn a. If mCE 85 and mBD 40,find mA.b. Find the value of x. Then find mTH .CETA40x 2 39MEKBADT30x 10H© Glencoe/McGraw-Hill278 Geometry: Concepts and Applications

14–4Secant-Tangent AnglesWHAT YOU’LL LEARN• Find measures of arcsand angles formed bysecants and tangents.Theorem 14-10If a secant-tangent angle has its vertex outside the circle,then its degree measure is one-half the difference of thedegree measures of the intercepted arcs.Theorem 14-11If a secant-tangent angle has its vertex on the circle, thenits degree measure is one-half the degree measure of theintercepted arc.In the figure, AD is tangentto circle K at A.D1CA2K160© Glencoe/McGraw-HillSecant – Tangent AnglesVertex Outside the CircleSecant – tangent anglePQR intercepts PR andPS .PVertext on the CircleSecant - tangent angleABC intercepts AB .DRKEY CONCEPTABSCQUnder thetab for Secant-TangentAngles, write thedefinitions of secanttangentangles andtangent-tangent angles.Find m1.Vertex D of the secant-tangent angle is outside circle K.Apply Theorem 14-10.The degree measure of the whole circle is 360°. So, themeasure of AC is 360° 160° 110° 90°.m1 1 2 (mAB mAC ) Theorem 14-10m1 1 2 Substitutionm1 1 2 Find m2. orVertex A of the secant-tangent angle is on circle K.m2 1 2 (mACB ) Theorem 14-11m2 1 2 Substitutionm2 1 2 or110BGeometry: Concepts and Applications 279

14–4Your TurnREMEMBER ITThe vertex of asecant-tangent anglecannot be located insidethe circle.a. AZ is tangent to circle D at A.If mAB 150, find mZ.ZYA80DBREVIEW ITExplain the differencebetween a minor arcand a major arc of acircle. (Lesson 11-2)b. EF is tangent to circle D at E.If mEGC 230, find mFEC.BUILD YOUR VOCABULARY (page 271)CDEGFA tangent-tangent angle is formed by two .Its vertex is always outside the circle.Theorem 14-12The degree measure of a tangent-tangent angle is one-halfthe difference of the degree measures of the intercepted arcs.Find mG.G is a tangent-tangent angle. ApplyTheorem 14-12.By definition of a right angle, mFOH 90. So,mFH 90, because a minor arc is congruent to its central angle.Since the sum of the measures of a minor arc and its major arcis 360°, major arc FJH is 360° 90° 270°.JFOGHHOMEWORKASSIGNMENTPage(s):Exercises:mG 1 (major arc FJH minor arc FH)2mG 1 (270 90)2mG 1 2 orYour TurnFind mB.XEQ152Z© Glencoe/McGraw-HillB280 Geometry: Concepts and Applications

14–5Segment MeasuresBUILD YOUR VOCABULARY (page 270)WHAT YOU’LL LEARN• Find measures ofchords, secants, andtangents.A segment is an external secant segment if and only if itis the part of a secant segment that isa circle.ORGANIZE ITUnder the tab forSegment Measures,write the definition ofan external secantsegment. Record thetheorems and othermain ideas from thislesson.CircleRelationshipsTheorem 14-13If two chords of a circle intersect, then the product of themeasures of the segments of one chord equals the productof the measures of the segments of the other chord.Theorem 14-14If two secant segments are drawn to a circle from anexterior point, then the product of the measures of onesecant segment and its external secant segment equals theproduct of the measures of the other secant segment andits external secant segment.Theorem 14-15If a tangent segment and a secant segment are drawn to acircle from an exterior point, then the square of the measureof the tangent segment equals the product of the measuresof the secant segment and its external secant segment.© Glencoe/McGraw-HillIn circle A, find the value of x.PT TR QT TS Theorem 14-136 12 Substitution48 12x 4 8 1 2x Divide each side by .4 4 xDivision PropertyPQx6 TA812RSYour TurnFind the value of x in the circle.82x123xGeometry: Concepts and Applications 281

14–5WRITE ITFind the value of x to thenearest tenth.6xy5Explain the differencebetween Theorem 14-13and Theorem 14-14 inyour own words.(x 6) 6 (5 7) 5 Theorem 14-1457 60Distributive Property6x 36 36 60 36 Subtract from each side.6x 6 x 2 4 Divide each side by .6 6x Use the value of x to find the value of y.y 2 (x 5) 5 Theorem 14-15y 2 (4 5) 5Substitutiony 2 y 2 45Take the square root.y Your TurnHOMEWORKASSIGNMENTPage(s):Exercises:a. Find the value of x.b. Find the value of x.x8x1864© Glencoe/McGraw-Hill48282 Geometry: Concepts and Applications

14–6Equations of CirclesWHAT YOU’LL LEARN• Write equations ofcircles using the centerand the radius.Theorem 14-16 General Equation of a CircleThe equation of a circle with center at (h, k) and a radius ofr units is (x h) 2 (y k) 2 r 2 .Write the equation of a circle with center at (4, 0) anda radius of 5 units.ORGANIZE ITUnder the tab forEquations of Circles,write the GeneralEquation of a Circle, anddraw a picture, labelingthe center and radius.Record several examplesto help you rememberthe main idea.(x h) 2 (y k) 2 r 2 Equation of a Circlex 2 y 2 (h, k) (4, 0), r 5 The equation for the circle is .CircleRelationshipsFind the coordinates of the center and the measure ofthe radius of a circle whose equation is x 3 2 2 y 1 2 2 14 .Rewrite the equation.(x h) 2 (y k) 2 r 2© Glencoe/McGraw-Hillx 2 y 2 2Since h , k , and r , the centerof the circle is at . Its radius is .Geometry: Concepts and Applications 283

14–6Your Turna. Write the equation of a circle with center C(5, 3) anda radius of 6 units.b. Find the coordinates of the center and the measure of theradius of a circle whose equation is (x 2) 2 (y 7) 2 81.HOMEWORKASSIGNMENT© Glencoe/McGraw-HillPage(s):Exercises:284 Geometry: Concepts and Applications

C H A P T E R14BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 14 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 14, go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 270–271) to help yousolve the puzzle.14-1Inscribed AnglesIn circle P, AC is a diameter; mCD 68 andmBE 96. Find each of the following.1. mABCBAPCED2. mCED3. mAD © Glencoe/McGraw-HillIn circle A, HE is a diameter.4. If mHTC 52, find mCH .5. Find mHCE .ECTAUH6. If mHTC 52, find mCEH .Geometry: Concepts and Applications 285

Chapter14BRINGING IT ALL TOGETHER14-2Tangents to a CircleUnderline the best term to complete the statement.7. If a line is tangent to a circle, then it is perpendicular to theradius drawn to the [point of tangency/vertex].8. AB is tangent to circle C. Find the value of x.x12CA1616B9. Circle P is inscribed in right CTA.Find the perimeter of CTA if theradius of circle P is 5, CT 18, andJT 11.CPAJT14-3Secant AnglesUnderline the best term to complete the statement.10. A [radius/secant segment] is a line segment that intersects acircle in exactly two points.Find the value of x.11. 12.52xS8428Sx12© Glencoe/McGraw-Hill286 Geometry: Concepts and Applications

Chapter14BRINGING IT ALL TOGETHER14-4Secant–Tangent AnglesUnderline the best term to complete the statement.13. The measure of a(n) [tangent-tangent/inscribed] angle is alwaysone-half the difference of the measures of the intercepted arcs.Find the value of x. Assume that segments that appear to betangent are tangent.14. 15.B20x4xSA148D128C14-5Segment MeasuresFind the value of x.16. 17.2x68x 9x18© Glencoe/McGraw-Hill14-6Equations of Circles18. Write the equation of the circle with center (5, 9) and radius25.19. What are the coordinates of the center and length of theradius for the circle (x 4) 2 y 2 121.Geometry: Concepts and Applications 287

C H A P T E R14ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 14.• You may want to take the Chapter 14 Practice Test onpage 627 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 14 Study Guide and Reviewon pages 624–626 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 14 Practice Test onpage 627.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 14 Foldable.• Then complete the Chapter 14 Study Guide and Review onpages 624–626 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 14 Practice Test onpage 627.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature288 Geometry: Concepts and Applications

C H A P T E R15Formalizing ProofUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with four sheets of 8 1 " 11" grid paper.2FoldFold each sheet of paperin half along the width.Then cut along the crease.StapleStaple the eighthalf-sheets togetherto form a booklet.CutCut seven lines from thebottom of the top sheet,six lines from the secondsheet, and so on.Chapter 15© Glencoe/McGraw-HillLabelLabel each tab witha lesson number. The lasttab is for vocabulary.FormalizingProof15-115-215-315-415-515-6VocabularyNOTE-TAKING TIP: To help you organize data,create a study guide or study cards when takingnotes, solving equations, defining vocabularywords and explaining concepts.Geometry: Concepts and Applications 289

C H A P T E R15BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 15.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplecompound statementconjunctioncontrapositivecoordinate proofdeductive reasoning[dee-DUK-tiv]disjunctionindirect proofindirect reasoninginverse© Glencoe/McGraw-HillLaw of Detachment290 Geometry: Concepts and Applications

Chapter15BUILD YOUR VOCABULARYVocabulary TermFoundon PageDefinitionDescription orExampleLaw of Syllogism[SIL-oh-jiz-um]logically equivalentnegationparagraph proofproofproof by contradictionstatement© Glencoe/McGraw-Hilltruth tabletruth valuetwo-column proofGeometry: Concepts and Applications 291

15–1Logic and Truth TablesBUILD YOUR VOCABULARY (pages 290–291)WHAT YOU’LL LEARN• Find the truth values ofsimple and compoundstatements.A statement is any sentence that is either true or false, butnot both.Everyhas a truth value, true (T) or false (F).If a statement is represented by p, thennegation of the statement.p is theThe relationship between thestatement are organized on a truth table.of aWhen two statements arecompound statement., they form aA conjunction is astatement formed byjoining two statements with the word .A disjunction is astatement formed byORGANIZE ITUnder the tab forLesson 15-1, list anddefine the followingsymbols used in thelesson: , , , and →.Under the last tab, listthe vocabulary wordsand their definitionsfrom Lesson 15-1.FormalizingProof15-115-215-315-415-515-6Vocabularyjoining two statements with the word .Let p represent “An octagon has eight sides” and qrepresent “Water does not boil at 90°C.”Write the negation of statement p. p: An octagonWrite the negation of statement q.have eight sides.© Glencoe/McGraw-Hill q: Waterboil at 90°C.292 Geometry: Concepts and Applications

15–1REVIEW ITWrite in if-then form:All natural numbersare whole numbers.(Lesson 1-4)Your TurnLet p represent “Tofu is a protein source”and q represent “ is not a rational number.”a. Write the negation b. Write the negationof statement p. of statement q.Let p represent “9 2 99”, q represent “An equilateraltriangle is equiangular”, and r represent “A rectangularprism has six faces.” Write the statement for eachconjunction or disjunction. Then find the truth value.REMEMBER ITIn the Negationtruth table, p doesnot have to be a truestatement and p isnot necessarily a falsestatement. p q9 2 99 and an equilateral triangle is equiangular. Because pis , p is . Therefore, p q isbecause both p and q are .p r9 2 99 or a rectangular prism does not have six faces.Because r is , r is . Therefore, p r isbecause both p and r are .© Glencoe/McGraw-Hill q rAn equilateral triangle is not equiangular and a rectangularprism does not have six faces. Because q is , q is; and because r is , r is .Therefore, q r isbecause both q and rare .Geometry: Concepts and Applications 293

15–1Your TurnLet p represent “0.5 is an integer”, qrepresent “A rhombus has four congruent sides”, and rrepresent “A parallelogram has congruent diagonals.”Write the statement for each conjunction or disjunction.Then find the truth value.a. p q b. p r c. q rConstruct a truth tablefor the conjunction (p q).p q p q (p q)Make columns with theheadings p, q, ,and (p q). Then, listall possible combinationsof truth values for p and q.Use these truth values tocomplete the last two columns of theandits .REMEMBER ITA disjunction isfalse only when bothstatements are false.The converse of aconditional is falsewhen p is false and q istrue. A conditional isfalse only when p is trueand q is false.Your TurnConstructa truth table for thedisjunction (p q).HOMEWORKASSIGNMENTPage(s):Exercises:BUILD YOUR VOCABULARY (pages 290–291)The inverse of a conditional is formed byboth p and q.The contrapositive of a conditional statement is formed bynegating theof the statement.Two statements are logically equivalent if their truth tablesare the .© Glencoe/McGraw-Hill294 Geometry: Concepts and Applications

15–2Deductive ReasoningBUILD YOUR VOCABULARY (pages 290–291)WHAT YOU’LL LEARN• Use the Law ofDetachment and theLaw of Syllogism indeductive reasoning.Deductive reasoning is the process of using facts, rules,definitions, and properties in a logical order.The Law of Detachment allows us to reach logicalfromstatements.The Law of Syllogism is similar to the Transitive Propertyof Equality.KEY CONCEPTLaw of DetachmentIf p → q is a trueconditional and p istrue, then q is true.Under the tabfor Lesson 15-2,summarize the Law ofDetachment and theLaw of Syllogism in yourown words.Use the Law of Detachment to determine a conclusionthat follows from statements (1) and (2). If a validconclusion does not follow, then write no valid conclusion.(1) In a plane, if a line is perpendicular to one of twoparallel lines, then it is perpendicular to the other line.(2) AB CD and EF AB.p: AB CD and .q: EF Statement (1) indicates that p → q is, and statement(2) indicates that p is . So, is true. Therefore,© Glencoe/McGraw-Hill EF . CD(1) Two nonvertical lines have the same slope if andonly if they are parallel.(2) AB is a vertical line.p: Two lines are nonvertical and .q: Two lines have the same .Statement (2) indicates that p is. Therefore,there is no valid conclusion.Geometry: Concepts and Applications 295

15–2Your TurnUse the Law of Detachment to determinea conclusion that follows from statements (1) and (2).If a valid conclusion does not follow, then write novalid conclusion.a. (1) If a figure is an isosceles triangle, then it has twocongruent angles.(2) A figure is an isosceles triangle.REMEMBER ITIn the Law ofSyllogism, bothconditionals must betrue for the conclusionto be true.b. (1) If a hexagon is regular, each interior anglemeasures 120°.(2) The hexagon is regular.Use the Law of Syllogism to determine a conclusion thatfollows from statements (1) and (2).(1) If mK 90, then K is a right angle.(2) If K is a right angle, then JKL is a right triangle.p: mK q: K is a right angle.r: JKL is a triangle.Use the Law of Syllogism to conclude p → r.Therefore, iftriangle., then JKL is aHOMEWORKASSIGNMENTPage(s):Exercises:Your TurnUse the Law of Syllogism to determine aconclusion that follows from statements (1) and (2).(1) If it is rainy tomorrow, then Alan cannot play golf.(2) If Alan cannot play golf, then he will watch television.© Glencoe/McGraw-Hill296 Geometry: Concepts and Applications

15–3Paragraph ProofsBUILD YOUR VOCABULARY (page 291)WHAT YOU’LL LEARN• Use paragraph proofsto prove theorems.A proof is a logical argument in which each statement isbacked up by a that is accepted as .Statements and reasons are written inform in a paragraph proof.Write a paragraph proof for the conjecture.ORGANIZE ITUnder the tab forLesson 15-3, summarizewhat information islisted as “Given” and“Prove” in a paragraphproof.FormalizingProof15-115-215-315-415-515-6VocabularyIn RST, if TX RS and TX bisects RTS, then RX XS .SXTRGiven: TX RS; TX bisects RTS.Prove: RX XSProof:If TX RS, then RXT and TXS areangles and RXT andare right triangles.© Glencoe/McGraw-HillIf TX bisects RTS, then RTX STX by thedefinition of angle. Also, TX since congruence is. So, RTX STXby the Theorem. Therefore, RX XS becauseparts of congruent triangles arecongruent (CPCTC).Geometry: Concepts and Applications 297

15–3REVIEW ITWhat can you say aboutcorresponding anglesformed when parallellines are cut by atransversal? (Lesson 4-3)If 1 and 2 are congruent, then is parallel to m.Given: 1 2Prove: Proof: Vertical angles are congruent so 2 3. Since1 2, 1 13 m2tby substitution. If two linesin a are cut by a so thatcorresponding angles are, then thelines are . Therefore, .Your Turnconjecture.Write a paragraph proof for eachREMEMBER ITThere is more thanone way to plan aproof.a. If A is the midpoint of DCand EB,then DAE CAB.Given: A is the midpoint of DCand EBProve: DAE CABDEABCHOMEWORKASSIGNMENTb. If 3 4, then 5 6. XJ51 234Y6K© Glencoe/McGraw-HillPage(s):Exercises:298 Geometry: Concepts and Applications

15–4Preparing for Two-Column ProofsBUILD YOUR VOCABULARY (page 291)WHAT YOU’LL LEARN• Use properties ofequality in algebraicand geometric proofs.A two-column proof is a deductive argument withandorganized in twocolumns.Justify the steps for the proof of theconditional. If XWY XYW, thenAWX BYX.AWXYBStatementsReasonsORGANIZE ITUnder the tab forLesson 15-4, summarizethe Properties ofEquality fromLesson 2-2.FormalizingProof15-115-215-315-415-515-6Vocabulary1. 1. Given2. mXWY mXYW 2.3. mAWX mXWY 180; 3.mBYX mXYW 1804. mAWX mXWY 4. Substitution© Glencoe/McGraw-Hill5. mAWX 5. Subtraction property6. 6. Definition of congruentanglesGeometry: Concepts and Applications 299

15–4Your TurnJustify the stepsfor the proof of the conditional.If mAOC mBOD, thenmAOB mCOD.ABCDGiven: mAOC mBODProve: mAOB mCODOProof:REMEMBER ITYou cannot write astatement unlessyou give a reason tojustify it.StatementsReasons1. 1.2.2.3.3.4.5.4.5.WRITE ITWhat information isalways in the firststatement of a proof?What information canalways be found in thelast satement?Show that if A 1 2 bh, then b 2 A .hGiven: A 1 2 bhProve: b 2 A hProof:StatementsReasons1. A 1 bh 1. Given22. bh 2. Multiplication property3. 2 A b 3.h© Glencoe/McGraw-Hill4. b 4. Symmetric property300 Geometry: Concepts and Applications

15–4Your TurnShow that if PV nRT, then R P V .nTGiven:Prove:Proof:StatementsReasons1. 1.2.2.3.3.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 301

15–5Two-Column ProofsWHAT YOU’LL LEARN• Use two-column proofsto prove theorems.Write a two-column proof for the conjecture.If 1 2, then quadrilateral ABCD is a trapezoid.Given: 1 2Prove: ABCD is a trapezoid1ADProof:32 BnCtmStatementsReasonsORGANIZE ITUnder the tab forLesson 15-5, summarizethe process to write atwo-column proof.1. 1. Given2. 2 2.3. 3. SubstitutionFormalizingProof15-115-215-315-415-515-6Vocabulary4. 4. If two lines in a planeare cut by a transversalso that correspondingangles are congruent,then the lines areparallel.5. Quadrilateral ABCD is 5.a trapezoid.Your TurnWrite a two-columnproof. If XYZ is isosceles withXZ XY and OZ NY, thenOY NZ.Given: XYZ is isosceles withXZ XY and OZ NY Prove: OY NZ ZOXNY© Glencoe/McGraw-Hill302 Geometry: Concepts and Applications

15–5Proof:StatementsReasons1. 1.2.2.3.4.5.6.3.4.5.6.Write a two-column proof.DXACBGiven: X is the midpoint of both BD and AC.Prove: DXC BXAProof:StatementsReasons1. X is the midpoint of both 1. GivenBD and AC.© Glencoe/McGraw-Hill2. DX BX; 2.3. 3. Vertical angles arecongruent.4. 4.Geometry: Concepts and Applications 303

15–5Your TurnWrite a two-column proof.Given: AD and CE bisect each other.Prove: AE CDE1B2D43ACProof:StatementsReasons1. 1.2.2.3.3.4.5.6.4.5.6.HOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill304 Geometry: Concepts and Applications

15–6Coordinate ProofsBUILD YOUR VOCABULARY (page 290)WHAT YOU’LL LEARN• Use coordinate proofsto prove theorems.A proof that usescoordinate proof.on a coordinate plane is a© Glencoe/McGraw-HillKEY CONCEPTGuidelines for PlacingFigures on a CoordinatePlane1. Use the origin as avertex or center.2. Place at least oneside of a polygon onan axis.3. Keep the figurewithin the firstquadrant, if possible.4. Use coordinates thatmake computations assimple as possible.Under thetab for Lesson 15-6,summarize the Guidelinesfor Placing Figures on aCoordinate Plane.Position and label a rectanglewith length b and height d ona coordinate plane.• Use the origin as a .• Place one side on the x-axis and one side on the .• Label the A, B, C and D.yA(0, d) B(b, d)O D(0, 0) C(b, 0)• Label the coordinates D , C , 0 ,B , and A 0, .Your TurnPosition and label an isosceles triangle withbase m units long and height n units on a coordinate plane.xWrite a coordinate proof to prove that the oppositesides of a parallelogram are congruent.Given: parallelogram ABDCProve: AB CD and AC BDGeometry: Concepts and Applications 305

15–6REVIEW ITWhat is slope and howwould you determinethe slope of a line?(Lesson 4-6)Proof:Label the vertices A(0, 0), B(a, 0),D(a b, c), and C(b, c). Use theDistance Formula to find AB, CD,AC, and BD.AB (a ) 0 2 (0 0) 2 a 2 or ayC(b, c) D(a + b, c)O A(0, 0) B(a, 0) xCD [(a b) ] b 2 (c c) 2 AC BD (b ) 2 (c ) 2[2(a b) ] ( ) 2c So, AB CD and AC BD.or ab 2 c2 REVIEW ITWhat is the DistanceFormula? (Lesson 6-7)Therefore, and ; oppositesides of a parallelogram are .Your TurnWrite a coordinateproof to prove that parallelogramWXYZ is a rectangle by proving thediagonals are congruent.Z(0, b)W(0, 0) (a, 0) XY (a, b)© Glencoe/McGraw-Hill306 Geometry: Concepts and Applications

15–6REVIEW ITWhat is the MidpointFormula? (Lesson 2-5)Write a coordinate proof to provethat the length of the segmentjoining the midpoints of two sidesof a triangle is one-half the lengthof the third side.Given: EFG with midpoints J and K, of EF and FGProve: JK 1 2 EGyJF(2a, 2b)KO E(0, 0) G(2c, 0)xLabel the vertices E , F , and G(2c, 0).Use the Midpoint Formula to find the coordinates of J and K,and the Distance Formula to find JK and EG.Coordinates of J: 0 2a, 0 2b2 2Coordinates of K: 2a 2c, 2b 2 2 0 (a c, b)JK [(a c) ] a 2 (b b) 2 c 2 or cEG (2c 0) (0 2 ) 0 2 (2c) 2 or 1 2 EG 1 2 Therefore, .© Glencoe/McGraw-HillHOMEWORKASSIGNMENTYour TurnWrite acoordinate proof to provethat the length of a mediansegment joining themidpoints of two legs ofa trapezoid is one-halfthe sum of the length ofthe bases.U(0, 2b)M(0, b)R(0, 0)T (2a, 2b)N (a + c, b)S(2c, o)Page(s):Exercises:Geometry: Concepts and Applications 307

C H A P T E R15BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 15 Foldable tohelp you study for your chaptertest.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary Chapter 15,go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 290–291) to help yousolve the puzzle.15-1Logic and Truth TablesIndicate whether the statement is true or false.1. A table that lists all truth values of a statement is a truthtable.2. p → q is an example of a disjunction.3. p → q is the inverse of a conditional statement.4. p q is an example of a conjunction.5. Complete the truth table.p q p q p q p q p → qTTFFTFTF© Glencoe/McGraw-Hill308 Geometry: Concepts and Applications

Chapter15BRINGING IT ALL TOGETHER15-2Deductive ReasoningDraw a conclusion from statements (1) and (2).6. (1) All functions are relations.(2) x y 2 is a relation.7. (1) Integers are rational numbers.(2) (6) is an integer.8. (1) If it is Saturday, I see my friends.(2) If I see my friends, we laugh.15-3Paragraph ProofsIndicate whether the statement is true or false.9. A proof is a logical argument where each statement is backedup by a reason accepted as true.Write a paragraph proof.10. Given: m1 m2; m3 m4Prove: m1 m4 90© Glencoe/McGraw-Hill2 31 4Geometry: Concepts and Applications 309

Chapter15BRINGING IT ALL TOGETHER15-4Preparing for Two-Column ProofsComplete the statement.11. A proof containing statements and reasons and is organized bysteps is aproof.Complete the proof.12. Given: AB and AR aretangent to circle K.BProve: BAK RAKKAProof:StatementsReasonsR1. AB and AR are tangent 1.to circle K2. 2. If 2 segments from the sameexterior point are tangent toa circle, then they are .3. BK RK 3.4. 4.5. RAK 5.6. RAK 6.15-5Two Column Proofs13. Write a two-column proof.Given: AB is tangent to circle X at B.AC is tangent to circle X at C.Prove: AB ACABX© Glencoe/McGraw-HillC310 Geometry: Concepts and Applications

Chapter15BRINGING IT ALL TOGETHERProof:StatementsReasons1. AB is tangent to circle X 1.at B. AC is tangent tocircle X at C.2. Draw BX, CX , and AX. 2. Through any 2there is 1 .3. ABX and ACX are 3. If a line is tangent to acircle, then it is to the. radius drawn to the point oftangency.4. BX CX 4.5. 5. Reflexive Property6. AXB 6. HL7. 7. CPCTC15-6Coordinate ProofsComplete the statement.14. The vertex or center of the figure should be© Glencoe/McGraw-Hillplaced on the .15. Position and label a rhombus on a coordinate plane with base r and height t.Geometry: Concepts and Applications 311

C H A P T E R15ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.com toaccess your textbook,more examples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 15.• You may want to take the Chapter 15 Practice Test onpage 671 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 15 Study Guide and Reviewon pages 668–670 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 15 Practice Test onpage 671.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 15 Foldable.• Then complete the Chapter 15 Study Guide and Review onpages 668–670 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 15 Practice Test onpage 671.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature312 Geometry: Concepts and Applications

C H A P T E R16More Coordinate Graphing and TransformationsUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.Begin with six sheets of graph paper and an8 1 " 11" poster board.2StapleStaple the six sheets ofgraph paper onto theposter board.LabelLabel the six pages withthe lesson titles.© Glencoe/McGraw-HillNOTE-TAKING TIP: When taking notes, markanything you do not understand with a questionmark. Be sure to ask your instructor to explain theconcepts or sections before your next quiz or exam.Chapter 16Geometry: Concepts and Applications 313

C H A P T E R16BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 16.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term´s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundon PageDefinitionDescription orExamplecenter of rotationcomposition oftransformationsdilation[dye-LAY-shun]elimination[ee-LIM-in-AY-shun]reflectionrotationsubstitution[SUB-sti-TOO-shun]system of equationstranslation© Glencoe/McGraw-Hillturn314 Geometry: Concepts and Applications

16–1Solving Systems of Equations by GraphingBUILD YOUR VOCABULARY (page 314)WHAT YOU’LL LEARN• Solve systems ofequations by graphing.A set of two or more equations is called a system ofequations.ORGANIZE ITOn the page labeledSolving Systems ofEquations by Graphing,sketch graphs of systemsof equations. Explainwhy each graph producesthe result that it does.Solve each system of equations by graphing.y x 1y x 3Find ordered pairs by choosing values for x and finding thecorresponding y-values.y x 1y x 3x x 1 y (x, y)3 2 2 (3, 2)2 1 1 (2, 1)1 0 0 (1, 0)x x 3 y (x, y)3 0 0 (3, 0)2 1 1 (2, 1)1 2 2 (1, 2)Graph the ordered pairs and draw thegraphs of the equations. The graphsintersect at the point whose coordinatesy(2, 1)are. Therefore, the solutionof the system of equations is .y x 1Oxy x 3© Glencoe/McGraw-Hilly 2xy 2x 3Use the slope and y-intercept to grapheach equation.Equation Slope y-intercepty 2x 2 0y 2xyy 2x 3Oxy 2x 3 2 3The slope of each line isso the graphs areand do not intersect. Therefore, there is .Geometry: Concepts and Applications 315

16–1REVIEW ITExplain how to graph6x 2y 8 using theslope-intercept method.(Lesson 4-6)Your TurnSolve each system of equations by graphing.a. x 2y 23x y 6yOxWRITE ITExplain how to solve asystem of equations bygraphing.b. 3x 2y 123x 2y 6yOxToshiro wants a wildflower garden. He wants the lengthto be 1.5 times the width and he has 100 meters offencing to put around the garden. If w represents thewidth of the garden and represents the length, solvethe system of equations below to find the dimensions ofthe wildflower garden. 1.5w2w 2 100Solve the second equation for .2w 2 100 The perimeter is meters.2w 2 2w 100 2w Subtract from eachside. 100 2w© Glencoe/McGraw-Hill 2 100 2w2 2Divide. 316 Geometry: Concepts and Applications

16–1Use a graphing calculator to graph the equationsandto find the coordinates of theintersection point. Note that these equations can be written asy 1.5x and y 50 x and then graphed.Enter: [y ] 1.5 ENTER 50—GRAPHNext, use the intersection tool onof the point of intersection.F5to find the coordinatesThe solution is . Since w and ,the width of the garden ismeters and the length ismeters.Check your answer by examining the original problem.Is the length of the garden 1.5 times the width? ✔Does the garden have a perimeter of 100 meters? ✔The solution checks.REMEMBER ITCheck the solutionto a system of equationsby substituting it intoeach equation.Your TurnRuth wants to enclose an area of her yard forher children to play. She has 72 meters of fence. The length ofthe play area is 4 meters greater than 3 times the width. Whatare the dimensions of the play area?© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 317

16–2Solving Systems of Equations by Using AlgebraBUILD YOUR VOCABULARY (page 314)WHAT YOU’LL LEARN• Solve systems ofequations by using thesubstitution orelimination method.One algebraic method for solving a system of equations iscalled substitution.Another algebraic method for solving systems of equationsis called elimination.ORGANIZE ITOn the page labeledSolving Systems ofEquations by UsingAlgebra, write a systemof equations and solveit using substitution andelimination. Explain theprocess you used witheach method.Use substitution to solve the system of equations.y x 42x y 1Substitute x 4 for y in the second equation.2x y 12x 13x 4 1Combine like terms.3x 4 1 Subtract from eachside.3x 3 3 x 3 Divide each side by .3 3x Division PropertySubstitute 1 for x in the first equation and solve for y.y (1) 4 The solution to this system of equations is .Your TurnUse substitution to solve 2x y 4 and x y 5.© Glencoe/McGraw-Hill318 Geometry: Concepts and Applications

16–2Use elimination to solve the system of equations.3x 2y 44x 2y 103x 2y 4() 4x 2y 107x 0 14Add the equations to eliminate the y terms. 7 x 1 4 Divide each side by 7.7 7x The value of x in the solution is .Now substitute in either equation to find the value of y.3x 2y 43 2y 4 2y 46 2y 6 4 6 Subtract from each side.Subtraction Property 2 y2 2 Divide each side by .2y The value of y in the solution is .© Glencoe/McGraw-HillThe solution to the system is .Your TurnUse elimination to solve x y 7 and2x y 1.Geometry: Concepts and Applications 319

16–2WRITE ITExplain the differencebetween solving asystem of equations bysubstitution or by theelimination method.Use elimination to solve the system of equations.3x y 6x 2y 9( 2)3x y 6 6x 2y 12x 2y 9 x 2y 97x 0 21 Combine like terms. 7 x 2 17 7Divide.x Substitute 3 into either equation to solve for y.3x y 63 y 6 Replace x with .9 y 69 y 9 6 9 Subtract from each side.y Subtraction PropertyThe solution of this system is .Your Turn4x y 3.Use elimination to solve 7x 3y 1 andHOMEWORKASSIGNMENTPage(s):Exercises:© Glencoe/McGraw-Hill320 Geometry: Concepts and Applications

16–3TranslationsBUILD YOUR VOCABULARY (page 314)WHAT YOU’LL LEARN• Investigate and drawtranslations on acoordinate plane.A translation is a slide of a figure from one position toanother.ORGANIZE ITOn the page labeledTranslations, sketchgraphs of severaldifferent translations.Explain why eachtranslation producesthe result it does.Graph LMN with vertices L(0, 3), M(4, 2), and N(3, 1).Then find the coordinates of its vertices if it is translatedby (5, 0). Graph the translation image.To find the coordinates of the vertices of LMN, add 5 toeach x-coordinate and add 0 to each y-coordinate of LMN:(x 5, y 0).L(0, 3) (5, 0)M(4, 2) (5, 0)L(0 5, 3 0) LM(4 5, 2 0) MN(3, 1) (5, 0)N(3 5, 1 0) NyLMNOx© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Your TurnGraph ABC with vertices A(1, 2), B(3, 1),and C(2, 1). Then find the coordinates of its vertices if it istranslated by (3, 2). Graph the translation image.OyxGeometry: Concepts and Applications 321

16–4ReflectionsBUILD YOUR VOCABULARY (page 314)WHAT YOU’LL LEARN• Investigate and drawreflections on acoordinate plane.A reflection is the flip of a figure over a line to produce amirror image.Graph ABC with vertices A(0, 0), B(4, 1), and C(1, 5).Then find the coordinates of its vertices if it is reflectedover the x-axis and graph its reflection image.ORGANIZE ITOn the pages labeledReflections, sketch graphsof several differentreflections. Explain whyeach reflection producesthe result it does.To find the coordinates of the vertices ofABC, use the definition of reflectionover the x-axis: (x, y) (x, y).A(0, 0)B(4, 1)ABy COBxC(1, 5)CThe vertices of ABC are , ,and .In the same ABC, find the coordinates of the verticesof ABC after a reflection over the y-axis. Graph thereflected image.To find the coordinates of A, B, and C,use the definition of reflection over they-axis: (x, y) (x, y).A(0, 0)B(4, 1)C(1, 5)ABCyOCBx© Glencoe/McGraw-HillThe vertices of ABC are , , and.322 Geometry: Concepts and Applications

16–4Your Turna. Graph quadrilateral QUAD with vertices Q(3, 3), U(3, 2),A(4, 4), and D(4, 1). Then find the coordinates of itsvertices if it is reflected over the y-axis. Graph its reflectionimage.WRITE ITyReflect a figure over thex-axis and then reflectits image over the y-axis.Is this double reflectionthe same as atranslation? Explain.Oxb. Graph STU with vertices S(1, 2), T(4, 4), and U(3, 3).Then find the coordinates of its vertices if it is reflectedover the y-axis and graph its reflection image.yOx© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 323

16–5RotationsWHAT YOU’LL LEARN• Investigate and drawrotations on acoordinate plane.BUILD YOUR VOCABULARY (page 314)A rotation, also called a turn, is a movement of a figurearound a point. The fixed point may be in theof the object or a pointtheobject and is called the center of rotation.ORGANIZE ITOn the page labeledRotations, sketch graphsof several differentrotations. Explain whyeach rotation producesthe result it does.Rotate ABC 270° clockwise aboutpoint A.• The center of rotation is A. Use a protractor to draw anAABCCBangle ofclockwise about point A, using AB as abaseline for your protractor.• Draw segment ABto AB.• Trace the figure on a piece of paper and rotate the top paperclockwise, until the figure is rotatedclockwise.• Draw ABC congruent to ABC.Your TurnRotate XYZ 60° counterclockwise about point Y.© Glencoe/McGraw-HillZYX324 Geometry: Concepts and Applications

16–5Graph XYZ with vertices X(2, 1), Y(2, 3), and Z(3, 5).Then find the coordinates of the vertices after thetriangle is rotated 180° clockwise about the origin. Graphthe rotation image.• Draw a segment from the origin topoint X.YyZ• Use a protractor to reproduceOX at a 180° angle so thatOX OX.XXx• Repeat this procedure with pointsY and Z.ZYThe rotation image XYZ has vertices X ,Y , and Z .Your TurnRotate ABC 90° counterclockwise around theorigin. The vertices are A(0, 4), B(3, 1), and C(4, 3).AyCOBx© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Geometry: Concepts and Applications 325

16–6DilationsWHAT YOU’LL LEARN• Investigate and drawdilations on acoordinate plane.BUILD YOUR VOCABULARY (page 314)A dilation is a transformation that alters the size of a figure,but not its shape. It enlarges or reduces a figure by ak.ORGANIZE ITOn the page labeledDilations, sketch graphsof several differentdilations. Explain whyeach dilation producesthe result it does.Graph AB with vertices A(0, 2) and B(2, 1). Then findthe coordinates of the dilation image of AB with a scalefactor of 3, and graph its dilation image.Since k 1, this is an enlargement. To find the dilation image,multiply each coordinate in the ordered pairs by 3.preimage( 3)A(0, 2)imageAAAyBB(2, 1)( 3)BOBxThe coordinates of the endpoints of the dilation image areA and B .Your TurnGraph JKL with vertices J(1, 2), K(4, 3),and L(6, 1). Then find the coordinates of the dilation imageof JKL with a scale factor of 2, and graph its dilation.O4y4 812x© Glencoe/McGraw-Hill812326 Geometry: Concepts and Applications

16–6WRITE ITHow can you determinewhether a dilation is areduction or anenlargement?Graph DEF with vertices D(3, 3), E(0, 3), and F(6, 3).Then find the coordinates of the dilation image with ascale factor of 1 3 and graph its dilation image.Since k 1, this is a reduction.preimageD(3, 3) 1 3 imageDFFEyDDxE(0, 3) 1 3 EEF(6, 3) 1 3 FThe coordinates of the vertices of the dilation image areD , E , and F .Your TurnGraph quadrilateral MNOP with verticesM(1, 2), N(3, 3), O(3, 5), and P(1, 4). Then find the coordinatesof the dilation image with a scale factor of 2 3 and graph itsdilation image.5y4© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:321O 1 2 3 xGeometry: Concepts and Applications 327

C H A P T E R16BRINGING IT ALL TOGETHERSTUDY GUIDEUse your Chapter 16 Foldableto help you study for yourchapter test.VOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 16, go to:www.glencoe.com/sec/math/t_resources/free/index.phpBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(page 314) to help you solvethe puzzle.16-1Solving Systems of Equations by GraphingSolve each system of equations by graphing.1. x y 6 2. x y 27 3. y 4x 2y 9 3x y 41 12x 3y 916-2Solving Systems of Equations by Using AlgebraComplete each statement.4. Substitution and elimination are methods for solving16-2.5. A linear system of equations can have at most solution.Solve the system of equations using substitution orelimination.6. 3x y 4 7. y 3x 82x 3y 9y 4 x© Glencoe/McGraw-Hill8. 2x 7y 3 9. 3x 5y 11x 1 4y x 3y 1328 Geometry: Concepts and Applications

Chapter16BRINGING IT ALL TOGETHER16-3TranslationsComplete the statement.10. When a figure is moved from one position to another withoutturning, it is called a .Find the coordinates of the vertices after the translation.Graph each preimage and image.11. rectangle WXZY with vertices W(2, 2), X(2, 10),Z(7, 10), and Y(7, 2) translated (6, 9)y848Y4OW44 8x8ZX12. ABC with vertices A(4, 0), B(2, 1), and C(0, 1) translated(0, 4)yCOBA x© Glencoe/McGraw-Hill13. JKL with vertices J(5, 2), K(2, 7), and L(1, 6)translated (6, 2)K 84y8 4 OJ48L4 8xGeometry: Concepts and Applications 329

Chapter16BRINGING IT ALL TOGETHER16-4ReflectionsComplete the statement.14. A is a flip of a figure over a line.Find the coordinates of the vertices after the reflection.Graph each preimage and image.15. quadrilateral ABCD with vertices A(1, 1), B(1, 4), C(6, 4), andD(6, 1) flipped over the x-axisyBCOADx16. quadrilateral JKLM with vertices J(3, 5), K(4, 0), L(0, 3),and M(1, 2) flipped over the y-axisyJMOKxL17. XYZ with vertices X(1, 1), Y(4, 1), and Z(1, 3) flipped overthe x-axisyZXOYx© Glencoe/McGraw-Hill330 Geometry: Concepts and Applications

Chapter16BRINGING IT ALL TOGETHER16-5RotationsFind the coordinates of the vertices after a rotation aboutthe origin. Graph the preimage and image.18. RST with vertices R(4, 1), S(1, 5), and T(6, 9) rotated90° counterclockwiseTy6SR6222O2 6x616-6DilationsUnderline the best term to complete the statement.19. A [dilation/rotation] alters the size of a figure but does notchange its shape.20. A figure is [reduced/enlarged] in a dilation if the scale factor isbetween 0 and 1.Find the coordinates of the dilation image for the givenscale factor. Graph the preimage and image.21. quadrilateral STUV with vertices S(2, 1), T(0, 2), U(2, 0),and V(0, 0) and scale factor 3y© Glencoe/McGraw-HillUTO VSxGeometry: Concepts and Applications 331

C H A P T E R16ChecklistARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.Visit geomconcepts.net toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 16.• You may want to take the Chapter 16 Practice Test onpage 713 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 16 Study Guide and Reviewon pages 710–712 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 16 Practice Test onpage 713 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 16 Foldable.• Then complete the Chapter 16 Study Guide and Review onpages 710–712 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 16 Practice Test onpage 713 of your textbook.Student SignatureParent/Guardian Signature© Glencoe/McGraw-HillTeacher Signature332 Geometry: Concepts and Applications

12–5Volumes of Pyramids and ConesWHAT YOU’LL LEARN• Find the volumes ofpyramids and cones.Theorem 12-11 Volume of a PyramidIf a pyramid has a volume of V cubic units and a height ofh units and the area of the base is B square units, thenV 1 3 Bh.Lessons cover the content ofthe lessons in your textbook.As your teacher discusses eachexample, follow along andcomplete the fill-in boxes.Take notes as appropriate.Find the volume of the rectangularpyramid.12 cmB w (10)(4) orV 1 Bh Theorem 12-113 1 3 (40)(12)Substitution cm 310 cm4 cmExamples parallel theexamples in your textbook.© Glencoe/McGraw-HillORGANIZE ITUse the boxes forVolume of Pyramids andCones. Sketch and labela pyramid and a cone.Then write the formulafor finding the volumesof a pyramid and acone.SurfaceArea Volume Ch. 12PrismsCylindersPyramidsConesSpheresFind the volume of the cone to the nearest hundredth.Find the height hh 2 21 2 35 235 in.h 2 441 1225h 2 784h 242 in. 784 h V 1 3 r2 h Theorem 12-11 1 3 (21)2 (28)Substitution in 311–3In circle W, find XV if UW XV , VW 35, and WY 21.XVYW is a angle. Definition of perpendicularVYW is a right triangle.Definition of right triangleY21WU35VFoldables featurereminds you to takenotes in your Foldable.Theorem 12-12 Volume of a ConeIf a cone has a volume of V cubic units, a radius of r units,and a height of h units, then V 1 3 r2 h.Geometry: Concepts and Applications 241(WY) 2 (YV) 2 2 Pythagorean Theorem21 2 (YV) 2 35 2 Replace WY and VW. (YV) 2 1225(YV) 2 Subtract.(YV) 2 Take the square root ofeach side.YV XY Theorem 11-5XV YV XYSegment additionC H A P T E R12BRINGING IT ALL TOGETHERXV 28 SubstitutionXV © Glencoe/McGraw-HillUse your Chapter 12 Foldable tohelp you study for your chaptertest.12-1Solid FiguresSTUDY GUIDEVOCABULARYPUZZLEMAKERTo make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 12, go to:www.glencoe.com/sec/math/t_resources/free/index.phpComplete each sentence.1. Two faces of a polyhedron intersect at a(n) .2. A triangular pyramid is called a .3. A is a figure that encloses a part of space.4. Three faces of a polyhedron intersect at a point called a(n).12-2Surface Areas of Prisms and CylindersBUILD YOURVOCABULARYYou can use your completedVocabulary Builder(pages 228–229) to help yousolve the puzzle.© Glencoe/McGraw-HillHOMEWORKASSIGNMENTPage(s):Exercises:Your TurnIn circle G, if CG AE, EG 20, CG 12, find AE.Geometry: Concepts and Applications 215exercises on your own.AC12G 20Your Turn Exercises allowyou to solve similarE© Glencoe/McGraw-HillFind the lateral area and surface area ofeach solid to the nearest hundredth.5. a regular pentagonal prism with apothema 4, side length s 6, and height h 12a. L b. S 66. a cylinder with radius r 42 and height h 10412Bringing It All TogetherStudy Guide reviews themain ideas and keyconcepts from each lesson.a. L b. S Geometry: Concepts and Applications 247Geometry: Concepts and Applicationsvii

NOTE-TAKING TIPSYour notes are a reminder of what you learned in class. Taking good notescan help you succeed in mathematics. The following tips will help you takebetter classroom notes.• Before class, ask what your teacher will be discussing in class. Reviewmentally what you already know about the concept.• Be an active listener. Focus on what your teacher is saying. Listen forimportant concepts. Pay attention to words, examples, and/or diagramsyour teacher emphasizes.• Write your notes as clear and concise as possible. The following symbolsand abbreviations may be helpful in your note-taking.Word or PhraseSymbol orAbbreviationWord or PhraseSymbol orAbbreviationfor example e.g. not equal such as i.e. approximately with w/ therefore without w/o versus vsand angle • Use a symbol such as a star (★) or an asterisk (*) to emphasis importantconcepts. Place a question mark (?) next to anything that you do notunderstand.• Ask questions and participate in class discussion.• Draw and label pictures or diagrams to help clarify a concept.• When working out an example, write what you are doing to solve theproblem next to each step. Be sure to use your own words.• Review your notes as soon as possible after class. During this time, organizeand summarize new concepts and clarify misunderstandings.Note-Taking Don’ts• Don’t write every word. Concentrate on the main ideas and concepts.• Don’t use someone else’s notes as they may not make sense.• Don’t doodle. It distracts you from listening actively.• Don’t lose focus or you will become lost in your note-taking.© Glencoe/McGraw-HillviiiGeometry: Concepts and Applications