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Edward B. Burger David J. Chard Earlene J. Hall Paul A. Kennedy Steven J. Leinwand Freddie L. Renfro Dale G. Seymour Bert K. Waits Geometry Contents in Brief CHAPTER 1 Foundations for Geometry CHAPTER 2 Geometric Reasoning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 3 Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . 142 CHAPTER 4 Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 CHAPTER 5 Properties and Attributes of Triangles . . . . . . . . . . 296 CHAPTER 6 Polygons and Quadrilaterals CHAPTER 7 Similarity CHAPTER 8 CHAPTER 9 Right Triangles and Trigonometry . . . . . . . . . . . . . . . 514 Extending Perimeter, Circumference, and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 C H A P T E R 10 Spatial Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 C H A P T E R 11 Circles C H A P T E R 12 Extending Transformational Geometry . . . . . . . . . 820 ......................... 2 70 . . . . . . . . . . . . . . . . . . . . . 376 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 Student Handbook Extra Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4 Problem Solving Handbook . . . . . . . . . . . . . . . . . . . . . . . . . . . S40 Skills Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S50 Postulates, Theorems, and Corollaries . . . . . . . . . . . . . . . . S82 Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S88 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S115 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S161 Symbols and Formulas . . . . . . . . . . . . . . . . . . . Inside Back Cover Copyright © 2007 by Holt, Rinehart and Winston All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Holt, Rinehart and Winston, 10801 N. MoPac Expressway, Building 3, Austin, Texas 78759. HOLT and the “Owl Design” are trademarks licensed to Holt, Rinehart and Winston, registered in the United States of America and/or other jurisdictions. Printed in the United States of America If you have received these materials as examination copies free of charge, Holt, Rinehart and Winston retains title to the materials and they may not be resold. Resale of examination copies is strictly prohibited. Possession of this publication in print format does not entitle users to convert this publication, or any portion of it, into electronic format. ISBN 0-03-035828-0 Cover photo: The Stata Center at MIT, Boston, Massachusetts, USA. © Scott Gilchrist/Masterfile 1 2 3 4 5 048 09 08 07 06 ge07se_FM_i_c3.indd ii 5/25/06 4:07:27 PM AUTHORS Edward B. Burger, Ph.D. is Professor of Mathematics and Chair at Williams College and is the author of numerous articles, books, and videos. He has won several of the most prestigious writing and teaching awards offered by the Mathematical Association of America. Dr. Burger has appeared on NBC TV, National Public Radio, and has given innumerable mathematical performances around the world. Steven J. Leinwand spent 22 years as the Mathematics Supervisor with the Connecticut Department of Education. He is currently a Principal Research Analyst at the American Institutes for Research. David J. Chard, Ph.D., is an Associate Dean of Curriculum and Academic Programs at the University of Oregon. He is the President of the Division for Research at the Council for Exceptional Children, is a member of the International Academy for Research on Learning Disabilities, and is the Principal Investigator on two major research projects for the U.S. Department of Education. Freddie L. Renfro, BA, MA, has 35 years of experience in Texas education as a classroom teacher and director/coordinator of Mathematics PreK-12 for school districts in the Houston area. She has served as TEA TAAS/ TAKS reviewer, team trainer for Texas Math Institutes, TEKS Algebra Institute writer, and presenter at math workshops. Earlene J. Hall, Ed.D., is the middle school mathematics supervisor for Detroit Public Schools, and an adjunct professor at Wayne State University in Detroit Michigan where she teaches graduate courses in the College of Education. Dale G. Seymour is a retired mathematics teacher, author, speaker and publisher. Dale founded Creative Publications in 1968, and went on to found two other mathematics publishing companies. Creating mathematical sculptures is one of his many hobbies. Paul A. Kennedy, Ph.D. is a professor in the Department of Mathematics at Colorado State University. Dr. Kennedy is a leader in mathematics education. His research focuses on developing algebraic thinking by using multiple representations and technology. He is the author of numerous publications. Bert K. Waits, Ph.D., is a Professor Emeritus of Mathematics at The Ohio State University and co-founder of T3 (Teachers Teaching with Technology), a national professional development program. CONTRIBUTING AUTHORS Linda Antinone Fort Worth, TX Ms. Antinone teaches mathematics at R. L. Paschal High School in Fort Worth, Texas. She has received the Presidential Award for Excellence in Teaching Mathematics and the National Radio Shack Teacher award. She has coauthored several books for Texas Instruments on the use of technology in mathematics. Carmen Whitman Pflugerville, TX Ms. Whitman travels nationally helping districts improve mathematics education. She has been a program coordinator on the mathematics team at the Charles A. Dana Center, and has served as a secondary math specialist for the Austin Independent School District. REVIEWERS Robert Brouhle Mathematics Department Chair, retired Marina High School Huntington Beach, CA Carey Carter Mathematics Teacher Everman Joe C. Bean High School Everman, TX Greg Davis Department Chair, retired Lodi High School Lodi, WI Roger Fuller Mathematics Department Chair Grand Prairie High School Grand Prairie, TX Anthony Gugliotta Supervisor of Math & Science Rumson-Fair Haven Regional HS Rumson, NJ Marieta W. Harris Mathematics Specialist Memphis, TN Debbie Hecky Geometry Teacher Scott High School Covington, KY Kathleen Kelly Mathematics Department Chair, retired Lawrence High School Fairfield, ME Mike Kingery Mathematics Teacher Mayfield High School Las Cruces, NM Joy Lindsay Mathematics Instructor Bonita High School LaVerne, CA Kim Loggins Geometry Teacher Los Alamitos High School Los Alamitos, CA Elaine Pappas Mathematics Department Chair Cedar Shoals High School Athens, GA Terri Salas Mathematics Consultant Corpus Christi, TX Jane Schneider Mathematics Department Chair Parkway West High School Ballwin, MO Cynthia Hodges Department Chair Shoemaker High School Killeen, TX ge07se_FM_i_c3.indd iv 5/25/06 4:08:09 PM Jamae Sellari Mathematics Instructor Forest Hill High School Jackson, MS Anna Valdez Geometry Teacher Nikki Rowe High School McAllen, TX Caren Sorrells Mathematics Coordinator Birdville ISD Haltom City, TX Lauralea Wright Mathematics Teacher Mauldin High School Mauldin, SC E. Robin Staudenmeier Middle/High School Math Coordinator Olympia Community USD 16 Stanford, IL Denise Young Mathematics Teacher Blue Valley West High School Overland Park, KS Maureen “Marnie” Stockman Geometry Specialist and Consultant Cordova, MD CONTRIBUTING WRITER Karen Droga Campe Instructor Yale University New Haven, CT FIELD TEST PARTICIPANTS ge07se_FM_i_c3.indd v Jill Morris Navasota High School Navasota, TX Carey Carter Alvarado High School Alvarado, TX Ruth Stutzman Jefferson Forest High School Forest, VA Walter Babst Bonita High School La Verne, CA 5/25/06 4:16:08 PM Preparing for Standardized Tests Holt Geometry provides many opportunities for you to prepare for standardized tests. Test Prep Exercises Use the Test Prep Exercises for daily practice of standardized test questions in various formats. 41. What is the value 19 52 42. Find the value of s. 23 28 of x? 57 71 �� ��� 34 56 ���� ��� ��� ��������� 43. ∠A and ∠B � are the remo te interior angl equations must es of ∠BCD in be true? �ABC. Which of these m∠A - 180° = m∠B m∠A = 90° m∠B CD = m∠BCA m∠B - m∠A m∠B = m∠B 44. Extende CD - m∠A d Response The measure 2 : 3 : 4. Describe s of the angles how to use alge in a triangle the measure bra to find the are in the ratio of each angle measures of and classify the these angles. Then find triangle. Multiple Choice—choose your answer. CHALLENGE Gridded Response—write your answer in a grid and fill in the corresponding bubbles. AND EXTEND 45. An exter ior angle of a triangle measure (2y 2 + 7)° and (61 s 117°. Its remo - y 2)°. Find the te interior angl value of y. es measure 46. Two para llel lines are inter sected by a trans by the intersecti versal. What on type of triangle (Hint: Use geom of the angle bisectors of two is formed etry software same-side inter or construct same-side inter ior angles? Expl a diagram of ior angles.) ain. the angle bisec tors of two 47. Critical Thinking Expl ain why an exter to a remote inter ior angle of a trian ior angle. gle cannot be 48. Probabil congruent ity The mea sure of each angl What is the prob e in a triangle ability that the is a multiple triangle has at of 30°. 49. In �ABC, least two cong m∠B is 5° less ruent angles? than 1 __1 time What is m∠A s m∠A. m∠C 2 in degrees? is 5° less than __1 2 2 times m∠A . Short Response—write open-ended responses that are scored with a 2-point rubric. SPIRAL REVIE W Make a table to show the valu e of each func (Previous cour tion when x is se) -2, 0, 1, and 50. f(x) = 3x 4. -4 51. f(x) = x 2 +1 53. Find the −−− length of NQ. 52. f(x) = (x Name the theo - 3) 2 + 5 or postulate that rem justifies your ����� answer. (Lesson 2-7) � Extended Response—write openended responses that are scored with a 4-point rubric. � Classify each triangle by its side � ����� � ��� 230 e area nd th se Fi C spon Item ers. ed Re imet Gridd re cent ua in sq ���� ��� � � e to to us ethod d come rent m n an a diffe e questio th ink of n’t th er, circle ca u If yo answ your check it later. to back ( ) ( ) Te Method 2: To check this answ er, use a diffe By the Centroid rent method. Theorem, the centroid of a of the distance triangle is __2 from each verte x to the midp opposite side. −− oint of the3 CD is vertical with a length and the coord of 6 units. __2 ( ) inates of the point that is 3 6 = 4, 4 units up from C is (1, 3). This meth od confirms the C2 � � � � �� � les to iang ght tr ial ri spec by u use on? swer n yo ti ur an k yo . ow ca this ques H ec ch 5. er rem can answ you ean Theo e how or � is th 3n + 1, plain e Pythag Ex at th −− AC = 6. th n? , iven using A ce G or of AB value of Item 4), Choi ct e Any Qu is th (-8, ion tiple r bise irs Aest Mul cula 11, what ? Type: Check d pa endi deisre t triangle perp C = 6n with a Differ orIt impo � B o the d ent Method a righ rtant to check all of your an D se D � diffe answers on a form rent method Item Respon ) 4 test. An effec to 8, answer the ques ( t C tive way to do tion a second ��� Shor 2), and er. two different methods, then this is to use sw time. If you get ����� if a an your answer the same answ ine B(0, n your is probably corre �� er with term ai ���� ct. pl de Ex to ope � use sl w to triangle. the swer? ho of � an erse ght plain � Conv eck your 7. Ex BC is a ri 4 theSho _ rtch Response Wha �A u use rem to t are the coor 3 n yo eo �ABC with A(-2, dinates of the ow ca ean Th 4), B(4, 6), and 4 8. H C(1, -1)? Show centroid of agor your work. -4 Pyth � Method 1: The . centroid of �aRST triangle is the � of 3 � of r _ the medians. te point of conc ces urrency ocen Write the equations 4 choi th � find their or er of two medians e point of inter sw and section. nd th n an Fi ve E se e gi . Item Respon Let D be the midpoint −− rectly use thm? of k. AB t di u and let E be the midp � � le Shor your wor lem n yo −− oint of BC. -2 + 4 4 + prob prob ow ca D= _ 6 this Show ,_ 1. H solve this = (1,5) 4+1 _ 6 + (-1) solve � 2 E= _ 2 �� � to , w to � � = (2.5, 2.5) 2 e ho 2 The median � escrib � � from C to D � 2. D contains C(1, It is vertical, � -1 ) and D(1, 5). so its equa � tion a is x = 1. � rms fo � r be �� The medi num d 17? an from ch �� A to E contains hi �� B A(-2, 4) and ce W ith 15 an −− _ E(2.5, 2.5). 8 Item w Choi slope of AE = 4 - 2.5 = _ 1.5 tiple an triple 1 . Mul -2 - 2.5 -_ gore -4.5 = ob lem 10 y - y1 = m x Pytha is pr 3 ( - x 1) 5 lve th -slope form soPoint ld of 1 y - 4 = - _(ux wou oices tude swer 7 + 2) Subs altititute 3 w yo er ch 1 4 for y , - _ an ird e ho 1 for m, answ your the th atand 3 ven escrib -2 for x . u use nfirm th 9. D 1 the gi n yo use to Solveca cox = 1 u use ? ow the syste le tom n yo u can er 10. H e triang d yo to 1 y ow ca the answ __ find 4 ho = the H point of inter ( et th 3. t? 3 x + 2) section. nd ent m rr4ec 1 er to fi co y _ ff = is - (1 + 2) e a di Substitute 1 er. 3 for x. escrib ur answ y=3 4. D k yo Problem Simplify. chec Solving The coordina tes of the cent 373 Strategies roid are (1, 3 ). st Tackler • Draw 372 � re Test Tackler s The second page guides you through applications of the test-taking strategy. � e squa of th estion e qu th swer Chapter 4 Trian gle Congruen ce d an em an test it each Read llow. fo that � ����� lengths. (Less on 4-1) 55. �BCD 56. �ABD 57. What if…? If CA = 8, Wha t is the effect classification on the of �ACD? 54. �ACD first answer. Chapter 5 Prop erties and Attri butes of Trian gles Preparing For Standardized Tests Use the Test Tackler to become familiar with and practice test-taking strategies. � � a Diagram • Make a Mode l • Guess and Test • Work Backw ard • Find a Patte rn • Make a Table • Solve a Simpl er Problem • Use Logical Reasoning • Use a Venn Diagr • Make an Organ am ized List The first page of this feature explains and shows an example of a test-taking strategy. Standardized Test Prep Short Response 12 and 13. am for Items Use this diagr � � ∠2 explain why transversal n, m with 20. Given � � lementary. and ∠3 are comp � ���� � � � Use the Standardized Test Prep to apply test-taking strategies. ? ure of ∠ACD 100° meas 12. What is the 40° � supplementary 140° 80° of 13. What type � � � � angles. are 21. ∠G and ∠H 12)°, and m∠H = x°. m∠G = (2x + be used to ion that can ion a. Write an equatvalue of x. Solve the equat determine the ? triangle is �ABC and justify each Isosceles acute Equilateral acute ∠H b. Explain why step. t but ∠G has a complemen KEYWORD: MG7 TestPr ep does not. parts CUMcture for every 1000 ULA s that TIV Etive. ASSESSMENT, conje 22. A manager CHAPTERS 1–4 ces, 60 are defec Multiple a factory produ Choice parts in one day, ry produces 1500 ons for filling ted to be If the facto directi a. expec the be Use can learn to thethem am are 6. Which condi cture? Take some time for how many of diagrthe make sure you ger’s1conje Items tiona mana and 2. and recheck to on value as its invers l statement has the same only get credit in a grid. Check defective based truth your answer. e? properly. You will in correctly. in how you found filling in the grid � filled Expla that are If n < 0, then 2 to show the boxes n > 0. if the ovals below r, solve the problem using in the table below b. Use the data conjecture is false. If a triangle has To check your answe you originally three congruent the manager’s d from the one � metho it nt you is sides, then an time, isosceles triang a differe � 5 4 le. a mistake the first 3� 2 used. If you made the same mistake when 1 If an angle meas Day make ures less than 1500 2500 are unlikely to an acute angle 90°, then it is 500 nt way. . 1000 2000 � you solve a differe Parts If n is a negat 1. Which of these 150 ive integer, then 90 30 n < 0. 150congruence Defective statements can proved60 from the inform be 7. On a map, onse ation given in Parts an island has the figure? Gridded Resp �AEB coord and °, ) −− 4 � a inate + reef �CED s (3, 5), and has coordinate AC.�ABD � . m∠E = (3x s (6, 8). If each �BCA lar bisector of represents 1 14. �CDE � �JKL5)°. What is the value of x? ndicu �BAC −− map unit perpe � �DAC mile, what is make from 23. BD is the the distance m∠L = (6x the island and usions you can �DEC � �DEA between the reef to the are the concl on mile? a. What nearest tenth 2. What −− and Frank live other of a ent? information statem do, Carmen, this�CEB why BD −− into is halfway Explad 15. Lucy, Eduar � �AED by AC at isD.neede −− prove that 4.2 miles . Eduardo’s house house. −− the HL Cong BD intersects ’s the same street Suppose−−− AC. ruenc Frank to b. 9.0 B e and −− miles Theor house em? AD est path from � AB 6.0 miles −− −−− Carmen’s house between Lucy’s is the short ay between CB −− � halfw 15.8 is AD −− miles BE � AE ce between Lucy’s house −− −− . If the distan ft, 8. DE A 150 line is � CE has and Frank’s house an x-intercept and Lucy’s house en’s of -8 and a of 3. What is 3. Which bicononse Eduardo’s house ce in feet between Carm y-intercept −− −− the equation ditional statem Extended Resp BC � EF, of the line? ent is les. what is the distan true? isosceles triang y = -8x + 3 do’s house? Tomo �DEF rroware and 8 is Mond m∠E = 95°. house and Eduar 24. �ABC y=_ if and −− not−− x-8 = 42.5°,ayand 3 only if today m∠C y=_ Satur and day. 3 x+3 and AC � DF. mined is - 2n. XY = 2, 8 y = 3x - 8 in how you deter , and JK = 10 Next montExpla h is January if a. What is m∠D? 16. �JKL �2 �XYZKL. 9. �JK and only if this �� passes throu her. answ is December. yourmont YZ = n . Find gh points J(1, are congruent. 3) and K(-3, Which of these 11). is Today �ABC and �DEF lines is perpe ement. What b. Show thatis a weekend day ifAB + 2, find = 3x ndicular to �JK is its own suppl yesterday only if + 7 and and 1 ��? = 2x 1 y = -_ EFwas 17. An angle Friday x+_ . you determined c. Given that 2 1 y = -2x - _ Explain how 3 its measure? This forh x. mont value the had 31 days 1 y=_ 5 if and x s. + mont 6 only er. inche e h if had 30 days. last your answ 2 154 squar y = 2x - 4 of a circle is nearest inch? the to 18. The area ce feren 10. If PQ = 2( 4. What must RS) + 4 What is its circum be true if PQ equation is true and RS = TU + 1, which ��� intersects than one point �ST measure of ∠Q. �� at more by the Subst ? 3_1_ times the itution Prope of Equality? ure of ∠P is 2 rty m∠P 19. The meas P, Q, S, and T tary, what is are collinear. are complemen PQ = TU + 5 If ∠P and ∠Q P, Q, S, and T are noncoplana in degrees? PQ = TU r. +6 ��� and ST PQ ��� are oppo site rays. PQ = 293 ( 2 TU) + 5 ��� and �ST PQ Chapters 1–4 �� are lative Assessment, PQ = 2(TU) + Cumu perpendicu lar. 6 5. �ABC � �DEF , EF = x 2 - 7, 11. Which of and BC = 4x Find the value the following - 2. s of x. is NOT valid that triangles for proving are congruent? -1 and 5 AAA 1 and 5 -1 and 6 SAS ASA 2 and 3 HL e Isosceles obtus Scalene acute The Hot Tip provides testtaking tips to help you suceed on your tests. These pages include practice with multiple choice, gridded response, short response, and extended response test items. 292 Countdown 9 WEEK to Testing DA Y 2 4, 1) ��� If A(-4, ��� and BC. ed by AB ∠ABC is form inates for C will , what coord and B(-4, 6) se angle? result in an obtu ) 2 (-1, DA Y 1 , ��� If A(3, 5) ular to XY. ��� is perpendic AB are the X(-2, -5), what B(9, 3), and of Y? coordinates (-2, 9) (0, 5) (2, 6) (6, 8) (1, 4) (3, 1) (2, 5) � � � � � �� �� � �� � � � � �� � �� �� � t to ∠5. ∠1 is congruen ntary to ∠14. ∠2 is suppleme t to ∠11. ∠8 is congruen ntary to ∠13. ∠6 is suppleme DA Y 5 DA Y 4 segment to h of the given What is the lengt the nearest unit? Which equation the graph? the line in best represents � � � � Countdown to Testing Use the Countdown to Testing to practice for your state test every day. DA Y 3 is true? the following ���. Which of ��� � BD ��� and AC ��� � CD AB � Chapter 4 Triang le Congruenc e There are 24 pages of practice for your state test. Each page is designed to be used in a week so that all practice will be completed before your state test is given. � �� � � � �� �� 1 8 10 Countdown � � y = -2x - 3 1 _ y = - 2x + 3 5 C12 � 1 _ y= x+3 2 y = 3x + 1 to Testing Each week’s page has five practice test items, one for each day of the week. Test-Taking Tips ✔ Get plenty of sleep the night before the ❑ test. A rested mind thinks more clearly and you won’t feel like falling asleep while taking the test. ✔ ❑ Draw a figure when one is not provided with the problem. If a figure is given, write any details from the problem on the figure. ✔ Read each problem carefully. As you ❑ finish each problem, read it again to make sure your answer is reasonable. ✔ Review the formula sheet that will be ❑ supplied with the test. Make sure you know when to use each formula. ✔ First answer problems that you know ❑ how to solve. If you do not know how to solve a problem, skip it and come back to it when you have finished the others. ✔ Use other test-taking strategies that can ❑ be found throughout this book, such as working backward and eliminating answer choices. Preparing For Standardized Tests C3 COUNTDOWN TO TESTING WEEK 1 DAY 1 DAY 2 Which statement about a number line is true? If a = b and b = c, which statement must be true? Values increase toward the right. a>c Values increase toward the left. -a - c = 0 Whole numbers are toward the right and decimal numbers are toward the left. a+c=0 a=c Negative numbers are toward the right and positive numbers are toward the left. DAY 3 If the width of each square in the grid is 1 centimeter, what is the diameter of the circle? 1 centimeter 3 centimeters 6 centimeters 12 centimeters DAY 4 DAY 5 Which shape is NOT included in the figure? Which statement best describes these two figures? Circle Square Triangle They cover the same area. Trapezoid They are the same size. They have the same number of sides. The distance around each figure is the same. C4 Countdown to Testing Geometry 2 Countdown to Testing DAY 1 WEEK 2 DAY 2 −− What is the length of FD? ∠ABC is an obtuse angle. Which of these could be the measure of ∠ABC? 0° { Î Ó £ ä £ Ó Î { 53° 0 90° 3 108° 6 9 DAY 3 Which point is described by the coordinates (-2, 3)? { Þ Ý ä { { { A B C D DAY 4 DAY 5 An architect is sketching a blueprint of a patio for a new home. On the blueprint, −−− C is the midpoint of AD, which represents one side of the patio. Point B is the −− midpoint of AC. If BC = 8 feet, what is the −−− length of AD? bisects ∠AOC, and m∠AOC = 60°. OB What is m∠BOE? Îä 8 feet 30° 16 feet 60° 24 feet 120° 32 feet 150° " Countdown to Testing ge07se_FM_C04_C27.indd C5 C5 5/25/06 4:23:47 PM Countdown to Testing 3 WEEK DAY 1 The figure below shows the first three elements in a pattern. The area of the white region in the first element is 8 cm 2, and the area of the white region in the second element is 16 cm 2. What will the area of the white region be when an element contains six circles? ���� ��������� ���� ���������� ���� ���������� 36 square centimeters 48 square centimeters 144 square centimeters 168 square centimeters DAY 2 DAY 3 Which of these is a unit that can describe the perimeter of a figure? ̶̶ Point X is the midpoint of HI. What is the coordinate of the point X? Meters � Square centimeters �� �� �� �� Cubic inches � � � � � � -4 Seconds 0 1 3 DAY 4 DAY 5 Which expression best represents the perimeter of the figure below? A line segment is drawn between the points (5, 8) and (-1, 6). What are the coordinates of the midpoint of the segment? �� ������ 27x 5x + 11 9x + 9 11x + 5 C6 Countdown to Testing � (3, 1) (4, 14) (2, 7) (-_12 , 3) Geometry 2 2007 SE Countdown to Testing 4 WEEK DAY 1 DAY 2 Which equation below represents the second step of the solution process? Which conjecture best describes a rule for the pattern below? Step 1: 6x - 12 = 3(5 - x) Step 2: ? � Step 3: 9x - 12 = 15 � � Step 4: 9x = 27 Rotate counterclockwise 90° Step 5: x = 3 Rotate clockwise 90° 6x - 12 = 15 - x Rotate counterclockwise 180° 6x - 12 = 15 - 3x Rotate clockwise 180° ��� 6x - 12 = 5 - 3x 6x = 3(5 - x) - 12 DAY 3 Given: A triangle is a right triangle. Conclusion: Two of the sides are congruent. This conclusion— is true because right triangles have exactly one angle that measures 90°. is true because all right triangles have two congruent angles. is false because, for example, the sides of a 30°-60°-90° right triangle have different lengths. is false because a right triangle cannot have two congruent angles. DAY 4 DAY 5 Which of the following best describes the value of 4n + 1 when n is an integer? bisects ∠LMO. Which statement must MN be true? The value is always negative. m∠LMN = m∠OMN The value is always positive. m∠LMO = m∠OMN The value is always even. m∠LMN = m∠OML The value is always odd. m∠LMO = m∠ONM Countdown to Testing C7 Countdown to Testing WEEK 5 DAY 1 Which statement is the converse of the conditional statement “If m∠A = 48°, then ∠A is acute?” If ∠A is not acute, then m∠A ≠ 48º. If ∠A is acute, then m∠A = 48º. If m∠A ≠ 48º, then ∠A is not acute. If ∠A is not acute, then it must be obtuse. DAY 2 DAY 3 Which of the following statements is true, based on the figure? Let a represent “Three points are not collinear,” and let b represent “The three points lie in exactly one plane.” Which symbolic sentence represents the statement “If three points lie in exactly one plane, then the three points are not collinear”? ∠2 and ∠4 are not adjacent but form a linear pair. � �� � ∠2 and ∠4 are adjacent angles that form a linear pair. a→b ∠1 and ∠3 are adjacent angles and form a linear pair. ∼b → ∼a b→a ∼a → ∼b ∠1 and ∠3 are not adjacent angles but form a linear pair. DAY 4 DAY 5 The figure below shows a pattern of right triangles and their areas, A. Based on the pattern, what will be the area of a right triangle with a height of 64 units? How many pairs of vertical angles are in the diagram? 2 3 6 �� � ������������ ����� � �� ������������� ����� ������������� ����� 4 square units 100 square units 364 square units 1536 square units C8 12 � � Countdown to Testing Geometry 2 2007 SE Countdown to Testing WEEK 6 DAY 1 DAY 2 A transversal crosses two parallel lines. If two angles are on opposite sides of the transversal and inside the two parallel lines, then they are alternate interior angles. If two angles are alternate interior angles, then they are congruent. ∠1 and ∠2 are alternate interior angles. Two angles are labeled in the figure below. Which of the following statements best describes this angle pair? � Which conclusion can be drawn from the given information? � ∠1 and ∠2 are parallel. ∠1 and ∠2 are alternate interior angles. ∠1 and ∠2 are complementary. They are complementary angles. ∠1 and ∠2 are congruent. They are congruent angles. They are supplementary angles. They are parallel angles. DAY 3 If line a is parallel to line b, and m∠8 = 62º, what is m∠1? � � � � � � � � � � 28° 62° 118° 180° DAY 4 DAY 5 The area of a circle is about 7 cm 2. By how many times will the area increase if the radius of the circle is tripled? B is in the interior of ∠AOC. Which of the following statements must be true? m∠AOB + m∠BOC = m∠AOC 1.5 m∠AOB = m∠BOC 3 m∠AOB + m∠AOC = m∠BOC 6 m∠BOC + m∠AOC = m∠AOB 9 Countdown to Testing C9 Countdown to Testing WEEK 7 DAY 1 Four rays are drawn from the origin to each of the following points: S(-2, 5), T(0, 4), U(-1, -3), and V(2, 6). Which point is on the ray that forms an acute angle with the ray in the figure? � � � �� � � �� S T U V DAY 2 DAY 3 What must be true if two nonvertical lines are perpendicular? Which line is parallel to y = 2x + 3? y = 2x - 8 Their slopes add to 0. y = 3x + 2 The product of their slopes is -1. 2y = -4x + 6 Their slopes are equal. y = -2x + 3 Their y-intercepts are equal. DAY 4 DAY 5 Which expression best represents the perimeter of the rectangle? Sheena is drawing a line graph to relate the side length of a square to the area of the square. Which of the following best describes the graph? ����� �� 4x + 1 6x + 4 8x + 2 3x 2 + 3x C10 Countdown to Testing steep downward straight line steep upward curve horizontal line upward straight line Geometry 2 2007 SE Countdown to Testing WEEK 8 DAY 1 What is the slope of the given line segment? � -2 1 -_ 2 � 1 � 2 �� � � �� DAY 2 DAY 3 Which two lines are perpendicular? Which of the following is the best classification for the given triangle? y = -5x + 2 and 2y - 10x = 4 1 x + 1 and y = 4x + 2 y=_ 4 y = 3x + 1 and y - 4x = 6 1 x + 2 and y + 2x = -4 y=_ 2 Equilateral Isosceles Scalene Right DAY 4 DAY 5 ̶̶ △SQT is an equilateral triangle. QR bisects ∠SQT. What are the measures of the angles of △SQR? What is the area of a circle with a radius of 2y? � 2πy 4πy 4πy 2 8πy 2 � � � 30°-30°-30° 30°-60°-90° 30°-60°-60° 60°-60°-60° Countdown to Testing C11 Countdown to Testing WEEK 9 DAY 1 DAY 2 is perpendicular to XY . If A(3, 5), AB B(9, 3), and X(-2, -5), what are the coordinates of Y? . If A(-4, 1) and BC ∠ABC is formed by BA and B(-4, 6), what coordinates for C will result in an obtuse angle? (6, 8) (1, 4) (3, 1) (2, 5) (-1, 2) (-2, 9) (0, 5) (2, 6) DAY 3 and AC BD CD . Which of the following is true? AB £ Ó Î { x È Ç £Ó £{ £x £Î ££ n £ä ∠1 is congruent to ∠5. ∠2 is supplementary to ∠14. ∠8 is congruent to ∠11. ∠6 is supplementary to ∠13. DAY 4 DAY 5 What is the length of the given segment to the nearest unit? Which equation best represents the line in the graph? { Þ Þ È Ý { ä { 1 5 8 10 C12 { { ä Î Ý 1x + 3 y=_ 2 y = 3x + 1 y = -2x - 3 1x + 3 y = -_ 2 Countdown to Testing ge07se_FM_C04_C27.indd C12 5/25/06 4:31:05 PM Countdown to Testing WEEK 10 DAY 1 Which set of angle measures can be used to conclude that lines x and y are parallel? Ý £ Ó Þ Î { m∠1 = 87° and m∠3 = 93° m∠1 = 82° and m∠4 = 98° m∠1 = 80° and m∠2 = 100° m∠3 = 88° and m∠4 = 92° DAY 2 DAY 3 Which postulate or theorem can be used to prove that these triangles are congruent? Which of the following conjectures is false? The product of an even number and an odd number is even. The difference of two negative numbers is a positive number. If x is negative, then -x is positive. If x is even, then x + 1 is odd. SAS ASA AAS SSS DAY 4 DAY 5 How many line segments can be determined by four points, no three of which are collinear? Timothy sketches a sphere with a circle around the middle. He labels the radius of the circle, which is the same as the radius of the sphere. Which problem might he be trying to solve? 4 6 8 Determining the angle at which Earth tilts 10 Calculating the mass of Earth Measuring the surface area of Earth Finding the distance around the equator Countdown to Testing ge07se_FM_C04_C27.indd C13 C13 5/25/06 4:32:03 PM Countdown to Testing WEEK 11 DAY 1 DAY 2 What conclusion can you draw from the figure? Jan drew the figure below and claims that line is parallel to line m. Which of the following proves her statement true? Ű £xÊV ££ä ££ä Çä Çä ££ä ABC is isosceles. The perimeter of ABC is 45 centimeters. Angles on opposite sides of the transversal are equal. DE = 10 centimeters Corresponding angles on the same side of the transversal are congruent. 1 AB DE = _ 2 More than two angles in the diagram have the same value. Two straight lines pass through the same transversal. DAY 3 Which of the following can you use to prove that two angles are complementary? The sum of their measures is 90°. The sum of their measures is 180°. The angles have the same measure. The measure of one angle is twice the other measure. DAY 4 DAY 5 If X(5, 5) and Y(0, 0), what are the coordinates of Z so that m∠XYZ = 90°? is a bisector of ∠XOY. Which of the OZ following statements is NOT true? (5, -5) (-5, -5) (5, 0) (0, 5) C14 2m∠ZOY = m∠XOY 2m∠XOZ = m∠XOY m∠ZOY = m∠XOY 1 m∠XOY m∠XOZ = _ 2 Countdown to Testing ge07se_FM_C04_C27.indd C14 5/25/06 4:32:51 PM Countdown to Testing WEEK 12 DAY 1 DAY 2 Which of the following correctly completes the congruence statement? ̶̶ AB ≅ ? ̶̶̶̶̶ Based on the figure, which inequality is correct? � � � � ������ � ̶̶ FD ̶̶ AF ̶̶ EF ̶̶ ED �� 2x > x + 10 2x < 10 x < 10 x>8 DAY 3 Roberta is attaching wooden trim around a stained glass window. The window is made up of eight congruent isosceles triangles. ����� ����� What length of trim does Roberta need in order to surround the entire window? 22 centimeters 78 centimeters 176 centimeters 624 centimeters DAY 4 DAY 5 How many different segments can be created from eight points on a given segment (including the segment’s endpoints)? Which of these conditional statements is true? 8 13 28 36 If two angles are vertical angles, then they are congruent. If two angles are congruent, then they are right angles. If four points are given, then they lie in exactly one plane. If one angle of a triangle measures 60º, then the triangle is a right triangle. Countdown to Testing C15 Countdown to Testing WEEK 13 DAY 1 DAY 2 What are the coordinates of point P? Which postulate or theorem can be used to verify the congruence of these two triangles? � � � � � �� � �� (3, -2) SSS (-2, 3) ASA (3, 2) AAS (2, -3) SAS DAY 3 Which conjecture is true? If a figure is a rectangle, its perimeter is equal to its area. If a figure is a triangle, all three sides are congruent. If a figure is a quadrilateral, then it has four sides. If a figure is a circle, its area is always greater than its circumference. DAY 4 DAY 5 The layout of a swimming pool is plotted on the coordinate grid below. If each unit on the grid represents 2 meters, what is the length of the pool? △LMN is shown on the grid. What is the ̶̶̶ slope of MN? � � � � � � �� � � 1 -_ 2 8 meters 10 meters 2 25 meters C16 � -4 1 _ 2 5 meters Countdown to Testing � � � � Countdown to Testing 14 WEEK DAY 1 A ceramic tile is in the shape of a 30°-60°-90° triangle. The side across from the 30° angle is 6.25 centimeters long. How long is the hypotenuse of the tile? 3.125 centimeters centimeters 6.25 √3 12.5 centimeters 15 centimeters DAY 2 DAY 3 What is the slope of this line? Which equation should Aretha use to find the distance between two points across a river? � � � � �� � � �� � � 1 c = a2 + b2 1 _ 3 3 c=a+b c 2 = √ a+b 1 -_ 3 c = √ a2 + b2 DAY 4 DAY 5 The sums of the angle measures of three polygons are given. Based on the pattern, what will be the sum of the measures of a hexagon? Which line in the graph is described by the equation y = x + 2? � � � � � � � ���� ���� � ���� � �� 240° 420° ℓ 600° m 720° n o Countdown to Testing C17 Countdown to Testing WEEK 15 DAY 1 DAY 2 Three coordinates of ABCD are A(4, 5), C(7, 3), and D(1, 3). Which coordinates could represent point B? Which two lines are parallel? 1x = 3 y = 6x + 8 and y + _ 6 1 x - 1 and y = 3x + 1 y=_ 3 y - 2x = 2 and y = 2 - 2x (1, 5) (3, 7) (5, 1) (10, 5) 1 x and y - _ 1x = 1 y=_ 4 4 DAY 3 ̶̶ What is the midpoint of QR? (1, -2) (-2, 1) (1, 2) (-1, -2) � � � � �� � � � �� DAY 4 DAY 5 Which of these statements is true? Which expression describes the total number of diagonals in a polygon with n sides? All quadrilaterals are parallelograms. Every rectangle is a parallelogram. Every parallelogram is also a rectangle. The diagonals of a rhombus are congruent. No. of sides 3 4 5 6 7 No. of diagonals 0 2 5 9 14 n(n - 3) _ 2 2n 3n _ 2 2n + 6 _ 3 C18 Countdown to Testing Countdown to Testing 16 WEEK DAY 1 DAY 2 The coordinates of the vertices of △ABC are (1, 1), (6, 1) and (1, 8). Which of the following could be the coordinates of the vertices of a triangle congruent to △ABC? Natalia is using indirect measurement to find the distance across a pond. Which Pythagorean triple is represented by the triangle? (-8, -2), (-3, -2), (-3, -9) (4, 1), (6, 2), (8, 10) (-2, 5), (-2, -9), (-8, 3) (0, 0), (-1, 8), (5, 2) ���� ���� 3-4-5 ���� 5-12-13 8-15-17 7-24-25 DAY 3 Which of the following sets of measurements could represent the side lengths of a right triangle? 3, 5, 9 4.5, 12, 8.5 6, 7, 10 2.5, 6, 6.5 DAY 4 DAY 5 What is the area of the composite figure? What is the measure of ∠3 in the regular hexagon? ��� ��� � ��� � ��� ��� � ��� 8 square meters 30° 21 square meters 60° 25 square meters 90° 45 square meters 120° Countdown to Testing C19 Countdown to Testing WEEK 17 DAY 1 Which two lines are perpendicular? y = x + 6 and y = x - 6 3x - 4 2 x = 1 and y = _ y+_ 3 2 1 x - 2 and y = -_ 1x + 3 y=_ 2 2 y - 2x = 5 and y = 2x + 2 DAY 2 DAY 3 What is the perimeter of the composite figure to the nearest centimeter? What is the measure of ∠1 in the triangle below? £ÓÊvÌ £ÓÊV £ £ÓÊvÌ 30° £äÊV xÊV xÊV Ó°xÊV 45° Ó°xÊV {ÊV 60° 90° xÊV 44 centimeters 52 centimeters 60 centimeters 83 centimeters DAY 4 DAY 5 What is the sixth item in the pattern below? The vertices of polygon ABCD are A(1, 5), B(8, 5), C(8, 3), and D(1, 3). Which of the following statements about this polygon is true? 64, 32, 16, 8, … 0 1 _ 2 2 4 C20 It is a square. Its width is 2 units. Its perimeter is 6 units. Its area is 9 square units. Countdown to Testing ge07se_FM_C04_C27.indd C20 5/25/06 4:33:30 PM Countdown to Testing 18 WEEK DAY 1 Based on the pattern of similar triangles below, what is the value of x? 2 4 4 √3 � 8 � � ��� � � ��� ��� ��� ������ ������ DAY 2 DAY 3 Which ratio is equivalent to sin B? What is the value of x to the nearest tenth of a millimeter? � ��� ��� � � � � �� �� � � � ��� 52.0 millimeters 2 √3 _ 3 61.6 millimeters √3 140.4 millimeters 78.8 millimeters √3 _ 2 1 _ 2 DAY 4 DAY 5 What is the value of x in the regular pentagon below? Which conjecture about polygons is NOT true? ���� The area of a parallelogram is the product of its base and height. A rhombus has four right angles. A square has four congruent sides. �� A trapezoid has exactly one pair of parallel sides. 54° 90° 108° 180° Countdown to Testing C21 Countdown to Testing WEEK 19 DAY 1 Which two line segments are congruent? ̶̶ ̶̶ AB and DF ̶̶ ̶̶̶ CE and GH ̶̶̶ ̶̶ GH and AB ̶̶ ̶̶ CD and DE � � � � � � � � �� � � � � � �� DAY 2 DAY 3 Based on the table, which algebraic expression best represents the number of triangles formed by drawing all of the diagonals from one vertex in a polygon with n sides? At a certain time of the day, a 24-foot tree casts an 18-foot shadow. How long is the shadow cast by a 4-foot mailbox at the same time of day? No. of sides 3 4 5 8 No. of triangles formed 1 2 3 6 ����� ���� ����� n 2n - 1 1.3 feet n-2 3 feet n+2 _ 2 4.5 feet 5 feet DAY 4 DAY 5 A school increases the width of its rectangular playground from 25 meters to 40 meters and the length from 45 meters to 60 meters. By how much does the perimeter of the playground increase? What is x? � 60 meters 2 200 meters 5 225 meters 10 30 Countdown to Testing � ��� ��� 30 meters C22 ��� Countdown to Testing WEEK 20 DAY 1 The figure shows the measure of each interior angle for several regular polygons. ��� ��� ���� ���� Which algebraic expression best represents the measure of an interior angle of a regular polygon with n sides? (n - 2)180 __ n 360n _ n+2 (n - 2)180 180n _ 2 DAY 2 DAY 3 Which coordinates represent a vertex of the hexagon? The two triangles in the figure are similar. ̶̶̶ What is the length of MN? � � � �� � � � �� �� (0, 2) (4, -2) (3, 2) (-2, 2) � � �� �� � � � � 3.5 7 6 17.5 DAY 4 DAY 5 Two regular pentagons have perimeters of 30 and 75 respectively. What scale factor relates the smaller figure to the larger one? Alissa is painting a diagonal line across a square tile. What is the length of the line? ���� 1 : 2.5 1:6 1 : 15 1 : 21 2 √ 8 centimeters 6 centimeters 8 centimeters 8 √ 2 centimeters Countdown to Testing C23 Countdown to Testing 21 WEEK DAY 1 The table lists the measure of an exterior angle for the given regular polygon. Which expression best represents the measure of an exterior angle of a regular polygon with n sides? Figure Quadrilateral Pentagon Decagon 90° 72° 36° Exterior angle 360 _ n-2 360 + n _ 2+n 360n 360 _ n DAY 2 DAY 3 Carrie is building a skateboard ramp with the dimensions below. What is the approximate measure of x? What is the value of z? �� �� ����� ���� � � 4° 12 8° 12 √2 12° 12 √3 15° 17 DAY 4 DAY 5 Which equation best describes the line containing the hypotenuse of this triangle? The center of circle C is the midpoint ̶̶ of AB. What are the coordinates of the midpoint? � � � � � �� � � � �� � 1x + 3 y=_ 2 y=5 y=x+3 1x - 3 y = -_ 2 C24 Countdown to Testing � � (0, 4) (1, 4) (2, 4) (3, 3) � Countdown to Testing WEEK 22 DAY 1 If this pattern is continued, how many shaded triangles will there be in the fourth element of the pattern? � 9 27 13 40 � � DAY 2 DAY 3 What is the slope of the line? A delivery truck travels 13.5 mi east and then 18 mi north. How far is the truck from its starting point? � � � � �� � � ����� �� 1 -_ 2 1 _ 3 1 _ 2 3 ������� 4.5 miles 20.25 miles 22.5 miles 31.5 miles DAY 4 DAY 5 What are the side lengths of the triangle? An 18-foot ladder reaches the top of a building when placed at an angle of 45° with the horizontal. What is the approximate height of the building? � � � 9.0 feet 12.7 feet 3, 4, and 5 14.4 feet 2, 3, and 5 30.9 feet 3, 3, and 3 3, 3, and 3 √2 Countdown to Testing C25 Countdown to Testing 23 WEEK DAY 1 DAY 2 RST is a 30°-60°-90° triangle. What is the y-coordinate of R if a = -5 and c = -2? What is x if y is 12.8 and z is 16 in the right triangle below? ,>]ÊÞ® â Þ Ý /V]Êä® ->]Êä® 3.2 4.0 9.6 3 12.8 3 √2 3 √3 6 DAY 3 How does the slope of the hypotenuse of ABC compare with that of DBC? They have the same value. Þ They have opposite signs. x They have the same sign. They are reciprocals. ä { Ý { Ó DAY 4 DAY 5 How many sides does a regular polygon have if each interior angle measures 120°? An electrician is standing at the top of a tower. He sees a truck at an angle of depression of 3°. If the tower is 300 feet tall, about how far away is the truck? 3 4 6 8 16 feet 300 feet 1052 feet 5724 feet C26 Countdown to Testing ge07se_FM_C04_C27.indd C26 5/25/06 4:34:52 PM Countdown to Testing WEEK 24 DAY 1 Quadrilaterals ABCD and WXYZ are similar. What is XY? � � � � �� � � � � � � � �� � 3.5 21 24.5 35 DAY 2 DAY 3 What is the second term in a proportion in which the first, third, and fourth terms are 3, 9, and 12, respectively? The endpoints of a segment are Q(-2, 6) and R(5, -4). What is the length of the segment to the nearest tenth? 3 4 3.6 units 6 4.1 units 8 8.5 units 12.2 units DAY 5 Which Pythagorean triple would be most helpful in finding the value of a? What is the perimeter of the square? ��� �� DAY 4 �� � �� 6 3-4-5 12 5-12-14 24 8-15-17 36 7-24-25 Countdown to Testing C27 Foundations for Geometry KEYWORD: MG7 TOC ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Euclidean and Construction Tools 1-1 Understanding Points, Lines, and Planes . . . . . . . . . . . . . . . . . . . 6 Explore Properties Associated with Points . . . . . . . 12 1-2 Measuring and Constructing Segments . . . . . . . . . . . . . . . . . . . 13 1-3 Measuring and Constructing Angles . . . . . . . . . . . . . . . . . . . . . . 20 1-4 Pairs of Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Coordinate and Transformation Tools 1-5 Using Formulas in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connecting Geometry to Algebra: Graphing in the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . 1-6 Midpoint and Distance in the Coordinate Plane . . . . . . . . . . . 1-7 Transformations in the Coordinate Plane. . . . . . . . . . . . . . . . . . Explore Transformations ......................... MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 42 43 50 56 58 59 Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Tools for Success Test Prep Exercises 11, 19, 26, 33, Reading Math 5 Writing Math 10, 18, 26, 33, 40, 48, 54 Vocabulary 3, 4, 9, 17, 24, 31, 38, 47, 53, 60 Know-It Notes 6, 7, 8, 13, 14, 16, 20, 21, 22, 24, 28, 29, 31, 36, 37, 43, 44, 45, 46, 50, 52 Graphic Organizers 8, 16, 24, 31, 37, 46, 52 Homework Help Online 9, 17, 24, 31, 38, 47, 53 ge07se_FM_vi.indd vi 40–41, 49, 55 Multi-Step Test Prep 10, 18, 26, 32, 34, 39, 48, 54, 58 College Entrance Exam Practice 65 Test Tackler 66 Standardized Test Prep 68 5/25/06 4:21:55 PM Geometric Reasoning ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Inductive and Deductive Reasoning 2-1 Using Inductive Reasoning to Make Conjectures . . . . . . . . . . . 74 Connecting Geometry to Number Theory: Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2-2 Conditional Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2-3 Using Deductive Reasoning to Verify Conjectures . . . . . . . . . . 88 Solve Logic Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2-4 Biconditional Statements and Definitions . . . . . . . . . . . . . . . . . 96 MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 KEYWORD: MG7 TOC Table of Contents Mathematical Proof 2-5 Algebraic Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6 Geometric Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Plans for Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7 Flowchart and Paragraph Proofs . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EXT Introduction to Symbolic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 104 110 117 118 126 127 128 Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Problem Solving on Location: South Carolina . . . . . . . . . . . . . . . 140 Tools for Success Test Prep Exercises 79, 86, 93, 101, Reading Math 73 Writing Math 78, 81, 86, 92, 96, 100, 109, 111, 115, 125 Vocabulary 71, 72, 77, 84, 91, 99, 107, 113, 122, 130 Know-It Notes 75, 76, 81, 83, 84, 89, 90, 98, 104, 106, 107, 110, 111, 112, 113, 118, 120, 122, 128 Graphic Organizers 76, 84, 90, 98, 107, 113, 122 Homework Help Online 77, 84, 91, 99, 107, 113, 122 109, 116, 125 Multi-Step Test Prep 78, 85, 92, 100, 102, 109, 115, 124, 126 College Entrance Exam Practice 135 Test Tackler 136 Standardized Test Prep 138 Parallel and Perpendicular Lines ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Lines with Transversals KEYWORD: MG7 TOC 3-1 Lines and Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connecting Geometry to Algebra: Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explore Parallel Lines and Transversals ......... 3-2 Angles Formed by Parallel Lines and Transversals . . . . . . . . . 3-3 Proving Lines Parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construct Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4 Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construct Perpendicular Lines. . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 152 154 155 162 170 172 179 180 181 Coordinate Geometry 3-5 Slopes of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explore Parallel and Perpendicular Lines ......... 3-6 Lines in the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . Connecting Geometry to Data Analysis: Scatter Plots and Lines of Best Fit . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 188 190 198 200 201 144 145 202 206 Tools for Success Writing Math 150, 160, 168, 177, 186, 196 Vocabulary 143, 144, 148, 175, 185, 194, 202 Study Strategy 145 Test Prep Exercises 150–151, Know-It Notes 146, 147, 148, 155, 156, 160–161, 168–169, 177–178, 187, 196–197 157, 162, 163, 173, 174, 182, 184, 185, 190, 192, 193 Multi-Step Test Prep 150, 160, 168, Graphic Organizers 148, 157, 165, 174, 185, 193 Homework Help Online 148, 158, 166, 175, 185, 194 176, 180, 186, 196, 200 College Entrance Exam Practice 207 Test Tackler 208 Standardized Test Prep 210 ©2007 Artists Rights Society (ARS), New York/ADAGP, Paris ge07se_FM_vii_xii.indd viii 5/25/06 4:37:16 PM Triangle Congruence ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Triangles and Congruence 4-1 Classifying Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Develop the Triangle Sum Theorem . . . . . . . . . . . . . . . . . . 4-2 Angle Relationships in Triangles . . . . . . . . . . . . . . . . . . . . . . . . 4-3 Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 KEYWORD: MG7 TOC 222 223 231 238 239 Proving Triangle Congruence 4-4 4-5 4-6 4-7 4-8 EXT Explore SSS and SAS Triangle Congruence . . . . . . . . . . . Triangle Congruence: SSS and SAS . . . . . . . . . . . . . . . . . . . . . . Predict Other Triangle Congruence Relationships . . . . Triangle Congruence: ASA, AAS, and HL . . . . . . . . . . . . . . . . . Triangle Congruence: CPCTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connecting Geometry to Algebra: Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Coordinate Proof . . . . . . . . . . . . . . . . . . . . . . . . Isosceles and Equilateral Triangles . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proving Constructions Valid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Solving on Location: Michigan . . . . . . . . . . . . . . . . . . . . 214 215 284 288 294 242 250 252 260 266 267 273 280 281 282 Tools for Success Reading Math 215, 273 Writing Math 220, 229, 236, 248, 258, 264, 271, 278 Vocabulary 213, 214, 219, 227, 234, 245, 256, 262, 270, 276, 284 Know-It Notes 216, 217, 218, 223, 224, 225, 226, 231, 233, 242, 243, 245, 252, 254, 255, 262, 267, 269, 273, 274, 275, 276 Graphic Organizers 218, 226, 233, 245, 255, 262, 269, 276 Homework Help Online 219, 227, 234, 245, 256, 262, 270, 276 Test Prep Exercises 221, 230, 236, 248, 258–259, 264–265, 272, 279 Multi-Step Test Prep 220, 229, 236, 238, 247, 258, 264, 271, 278, 280 College Entrance Exam Practice 289 Test Tackler 290 Standardized Test Prep 292 Properties and Attributes of Triangles ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Segments in Triangles KEYWORD: MG7 TOC 5-1 Perpendicular and Angle Bisectors . . . . . . . . . . . . . . . . . . . . . . 5-2 Bisectors of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3 Medians and Altitudes of Triangles . . . . . . . . . . . . . . . . . . . . . . Special Points in Triangles ...................... 5-4 The Triangle Midsegment Theorem . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 307 314 321 322 328 329 Relationships in Triangles 5-5 5-6 5-7 5-8 Connecting Geometry to Algebra: Solving Compound Inequalities . . . . . . . . . . . . . . . . . . . . . . . Explore Triangle Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . Indirect Proof and Inequalities in One Triangle . . . . . . . . . . . Inequalities in Two Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connecting Geometry to Algebra: Simplest Radical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hands-on Proof of the Pythagorean Theorem . . . . . . . . The Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applying Special Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . Graph Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 331 332 340 346 347 348 356 363 364 365 298 299 366 370 Tools for Success Test Prep Exercises 306, 313, 319, Reading Math 299, 300 Writing Math 306, 313, 318, 325, 338, 344, 354, 361 Vocabulary 297, 298, 304, 311, 317, 324, 336, 352, 366 Know-It Notes 300, 301, 303, 307, 309, 310, 314, 317, 323, 324, 333, 334, 335, 340, 342, 350, 351, 352, 356, 358, 359 Graphic Organizers 303, 310, 317, 324, 335, 342, 352, 359 Homework Help Online 304, 311, 317, 324, 336, 343, 352, 360 326, 339, 345, 355, 362 Multi-Step Test Prep 305, 312, 319, 326, 328, 338, 344, 354, 361, 364 College Entrance Exam Practice 371 Test Tackler 372 Standardized Test Prep 374 Polygons and Quadrilaterals ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Polygons and Parallelograms Construct Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1 Properties and Attributes of Polygons . . . . . . . . . . . . . . . . . . . Connecting Geometry to Algebra: Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explore Properties of Parallelograms . . . . . . . . . . . . . . . . 6-2 Properties of Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3 Conditions for Parallelograms. . . . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 KEYWORD: MG7 TOC 382 389 390 391 398 406 407 Other Special Quadrilaterals 6-4 Properties of Special Parallelograms . . . . . . . . . . . . . . . . . . . . . Predict Conditions for Special Parallelograms ... 6-5 Conditions for Special Parallelograms . . . . . . . . . . . . . . . . . . . Explore Isosceles Trapezoids .................... 6-6 Properties of Kites and Trapezoids . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Solving on Location: Ohio . . . . . . . . . . . . . . . . . . . . . . . . 378 379 438 442 448 408 416 418 427 436 437 Tools for Success Test Prep Exercises 388, 397, 405, Writing Math 379, 388, 397, 404, 405, 414, 424, 434 Vocabulary 377, 378, 386, 395, 412, 432, 438 Know-It Notes 383, 384, 385, 391, 392, 394, 398, 399, 401, 408, 409, 411, 418, 419, 421, 427, 429, 431 Graphic Organizers 385, 394, 401, 411, 421, 431 Homework Help Online 386, 395, 402, 412, 422, 432 ge07se_FM_vii_xii.indd xi 414–415, 425, 434–435 Multi-Step Test Prep 387, 396, 404, 406, 414, 424, 434, 436 College Entrance Exam Practice 443 Test Tackler 444 Standardized Test Prep 446 5/25/06 4:37:59 PM Similarity KEYWORD: MG7 TOC ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Similarity Relationships 7-1 Ratio and Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explore the Golden Ratio ....................... 7-2 Ratios in Similar Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predict Triangle Similarity Relationships ........ 7-3 Triangle Similarity: AA, SSS, and SAS . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 460 462 468 470 478 479 Applying Similarity Investigate Angle Bisectors of a Triangle ........ 7-4 Applying Properties of Similar Triangles . . . . . . . . . . . . . . . . . 7-5 Using Proportional Relationships . . . . . . . . . . . . . . . . . . . . . . . . 7-6 Dilations and Similarity in the Coordinate Plane . . . . . . . . . . Connecting Geometry to Algebra: Direct Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 453 504 508 481 488 495 501 502 503 Tools for Success Test Prep Exercises 459, 467, 477, Reading Math 453, 455, 456 Writing Math 459, 463, 466, 476, 486, 493, 499 Vocabulary 451, 452, 457, 465, 491, 498, 504 Know-It Notes 455, 457, 462, 464, 470, 471, 473, 481, 482, 483, 484, 490, 497 Graphic Organizers 457, 464, 473, 484, 490, 497 Homework Help Online 457, 465, 474, 484, 491, 498 487, 493, 500 Multi-Step Test Prep 458, 466, 476, 478, 486, 492, 499, 502 College Entrance Exam Practice 509 Test Tackler 510 Standardized Test Prep 512 Right Triangles and Trigonometry ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 Trigonometric Ratios 8-1 Similarity in Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explore Trigonometric Ratios ................... 8-2 Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connecting Geometry to Algebra: Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3 Solving Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 KEYWORD: MG7 TOC 524 525 533 534 542 543 Applying Trigonometric Ratios 8-4 Angles of Elevation and Depression . . . . . . . . . . . . . . . . . . . . . Indirect Measurement Using Trigonometry . . . . . . . . . . 8-5 Law of Sines and Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . 8-6 Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EXT Trigonometry and the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . 544 Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Solving on Location: Illinois . . . . . . . . . . . . . . . . . . . . . . 516 517 572 576 582 550 551 559 568 569 570 Tools for Success Test Prep Exercises 523, 532, 540, Reading Math 517, 534, 570 Writing Math 523, 525, 531, 540, 548, 557, 566, 571 Vocabulary 515, 516, 521, 529, 547, 563, 572 Know-It Notes 518, 519, 520, 525, 528, 537, 546, 552, 553, 554, 561, 563 Graphic Organizers 520, 528, 537, 546, 554, 563 Homework Help Online 521, 529, 537, 547, 555, 563 549, 558, 567 Multi-Step Test Prep 522, 530, 539, 542, 548, 557, 565, 568 College Entrance Exam Practice 577 Test Tackler 578 Standardized Test Prep 580 Extending Perimeter, Circumference, and Area ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 Developing Geometric Formulas KEYWORD: MG7 TOC Connecting Geometry to Algebra: Literal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1 Developing Formulas for Triangles and Quadrilaterals . . . . Develop π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2 Developing Formulas for Circles and Regular Polygons . . . . 9-3 Composite Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Develop Pick’s Theorem for Area of Lattice Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 589 598 600 606 613 614 615 Applying Geometric Formulas 9-4 Perimeter and Area in the Coordinate Plane . . . . . . . . . . . . . . 9-5 Effects of Changing Dimensions Proportionally . . . . . . . . . . Connecting Geometry to Probability: Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-6 Geometric Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use Geometric Probability to Estimate π . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 587 640 644 622 628 630 637 638 639 Tools for Success Test Prep Exercises 596–597, 605, Writing Math 596, 605, 611, 620, 626, 635 Vocabulary 585, 586, 603, 609, 633, 640 Study Strategy 587 Know-It Notes 589, 590, 591, 593, 600, 601, 602, 608, 619, 623, 624, 630, 633 Graphic Organizers 593, 602, 608, 619, 624, 633 Homework Help Online 593, 603, 609, 619, 625, 633 611–612, 621, 627, 636 Multi-Step Test Prep 595, 604, 610, 614, 620, 626, 635, 638 College Entrance Exam Practice 645 Test Tackler 646 Standardized Test Prep 648 Spatial Reasoning ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 Three-Dimensional Figures 10-1 Solid Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2 Representations of Three-Dimensional Figures . . . . . . . . . . . Use Nets to Create Polyhedrons . . . . . . . . . . . . . . . . . . . . . . 10-3 Formulas in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 KEYWORD: MG7 TOC 661 669 670 678 679 Surface Area and Volume 10-4 Surface Area of Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . Model Right and Oblique Cylinders . . . . . . . . . . . . . . . . . . 10-5 Surface Area of Pyramids and Cones . . . . . . . . . . . . . . . . . . . . 10-6 Volume of Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . 10-7 Volume of Pyramids and Cones . . . . . . . . . . . . . . . . . . . . . . . . . Connecting Geometry to Algebra: Functional Relationships in Formulas . . . . . . . . . . . . . . . . . . 10-8 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compare Surface Areas and Volumes ........... MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EXT Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Solving on Location: Pennsylvania . . . . . . . . . . . . . . . . . 652 653 730 734 740 688 689 697 705 713 714 722 724 725 726 Tools for Success Writing Math 653, 659, 667, 676, 686, 695, 703, 711, 720 Vocabulary 651, 652, 657, 665, 674, 684, 693, 701, 709, 718, 730 Know-It Notes 654, 656, 664, 670, 671, 672, 673, 680, 681, 683, 689, 690, 692, 697, 699, 700, 705, 707, 708, 714, 716, 717, 726, 727 Graphic Organizers 656, 664, 673, 683, 692, 700, 708, 717 Homework Help Online 657, 665, 674, 684, 693, 701, 709, 718 ge07se_FM_xiii_xvii.indd xv Test Prep Exercises 659, 667, 677, 687, 695, 703–704, 712, 721 Multi-Step Test Prep 658, 666, 675, 678, 686, 695, 703, 711, 720, 724 College Entrance Exam Practice 735 Test Tackler 736 Standardized Test Prep 738 5/25/06 4:41:26 PM Circles ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 Lines and Arcs in Circles KEYWORD: MG7 TOC 11-1 Lines That Intersect Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connecting Geometry to Data Analysis: Circle Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2 Arcs and Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-3 Sector Area and Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 755 756 764 770 771 Angles and Segments in Circles 11-4 Inscribed Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explore Angle Relationships in Circles .......... 11-5 Angle Relationships in Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . Explore Segment Relationships in Circles ....... 11-6 Segment Relationships in Circles . . . . . . . . . . . . . . . . . . . . . . . . 11-7 Circles in the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EXT Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772 Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 745 810 814 780 782 790 792 799 806 807 808 Tools for Success Reading Math 745, 748 Writing Math 754, 756, 762, 769, 778, 788, 797, 804 Vocabulary 743, 744, 751, 760, 767, 776, 810 Know-It Notes 746, 747, 748, 749, 750, 756, 757, 759, 764, 765, 766, 772, 773, 774, 775, 782, 783, 784, 785, 786, 792, 793, 794, 795, 799, 801 Graphic Organizers 750, 759, 766, 775, 786, 795, 801 Homework Help Online 751, 760, 767, 776, 786, 795, 802 Test Prep Exercises 754, 763, 769, 778, 789, 798, 804 Multi-Step Test Prep 753, 762, 768, 770, 777, 788, 797, 803, 806 College Entrance Exam Practice 815 Test Tackler 816 Standardized Test Prep 818 Extending Transformational Geometry ARE YOU READY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821 Congruence Transformations 12-1 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connecting Geometry to Algebra: Transformations of Functions . . . . . . . . . . . . . . . . . . . . . . . . . 12-3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explore Transformations with Matrices . . . . . . . . . . . . 12-4 Compositions of Transformations . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KEYWORD: MG7 TOC 824 831 838 839 846 848 854 855 Patterns 12-5 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-6 Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use Transformations to Extend Tessellations . . . . . . . . . 12-7 Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MULTI-STEP TEST PREP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . READY TO GO ON? QUIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EXT Using Patterns to Generate Fractals . . . . . . . . . . . . . . . . . . . . . 856 Study Guide: Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading and Writing Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study Guide: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Solving on Location: New Jersey . . . . . . . . . . . . . . . . . . 822 823 884 888 894 863 870 872 880 881 882 Tools for Success Writing Math 829, 836, 844, 852, 861, 868, 878, 883 Vocabulary 821, 822, 827, 851, 859, 866, 875, 884 Study Strategy 823 Know-It Notes 825, 826, 832, 833, 840, 841, 848, 849, 850, 856, 857, 858, 866, 873, 874 Graphic Organizers 826, 833, 841, 850, 858, 866, 874 Homework Help Online 827, 834, 842, 851, 859, 866, 875 Test Prep Exercises 829–830, 836–837, 845, 853, 862, 869, 878 Multi-Step Test Prep 829, 835, 843, 853, 854, 861, 868, 876, 880 College Entrance Exam Practice 889 Test Tackler 890 Standardized Test Prep 892 WHO USES MATHEMATICS? The Career Path features are a set of interviews with young adults who are either preparing for or just beginning in different career fields. These people share what math courses they studied in high school, how math is used in their field, and what options the future holds. Also, many exercises throughout the book highlight skills used in various career fields. KEYWORD: MG7 Career Career Applications Advertising 499 Agriculture 765 Animation 53, 835, 842 Anthropology 802 Archaeology 262, 787, 793 Architecture 47, 467, 667 Art 483, 657, 873 Aviation 277, 546, 564 Business 108, 194, 625 Carpentry 168, 418, 836 City Planning 305, 827 Communication 634, 802 Computer Graphics 495 Design 311, 317, 318 Electronics 692 Engineering 260, 554 Finance 108, 522 Forestry 548 Graphic Design 498, 752 Health 343 Industry 344 Interior Decorating 609, 867 Landscaping 607, 686, 702 Manufacturing 38, 754 Marine Biology 698, 720 Mechanics 434 Meteorology 801 Music 24 Navigation 228, 567, 729 Nutrition 107 Oceanography 174 Optometry 877 Photography 385, 459, 475 Political Science 79, 93 Real Estate 486 Surveying 25, 263, 556 xviii Who Uses Mathematics? ELECTRICIAN p. 320 Electricians install and maintain the systems that provide many of the modernday comforts we rely on, such as climate control, lighting, and technology. Look on page 320 to find out how Alex Peralta got started on this career path. TECHNICAL WRITER p. 612 Have you ever wondered who writes manuals for operating televisions or stereos? A technical writer not only writes manuals for operating electronics, but also documents maintenance procedures for airplanes. Look at the Career Path on page 612 to find out how to become a technical writer. FURNITURE MAKER p. 805 A furniture maker must take precise measurements and be aware of spatial relationships in order to build a quality finished product. The Career Path on page 805 describes the kind of experience needed to be successful as a furniture maker. WHY LEARN MATHEMATICS? Find a counterexample to show that the converse of each conditional is false. 38. If x = -5, then x 2 = 25. Links to interesting topics may accompany real-world applications in the examples or exercises. These links help you see how math is used in the real world. For a complete list of all applications in Holt Geometry, see page S162 in the Index. 39. If two angles are vertical angles, then they are congruent. 40. If two angles are adjacent, then they share a vertex. 41. If you use sunscreen, then you will not get sunburned. Geology Use the table and the statements below for Exercises 42–47. Write each conditional and find its truth value. p: calcite Animation q: not apatite r: a hardness of 3 Diamond is four times as hard as the next mineral on Mohs’ scale, corundum (ruby and sapphire). Real-World Animation 835 Geology Mohs’ scale is used to identify minerals. A mineral with a higher number is harder than a mineral with a lower number. s: a hardness less than 5 42. p → r 43. s → q 44. q → s 45. q → p 46. r → q 47. p → s 48. Critical Thinking Consider the conditional “If two angles are congruent, then they have the same measure.” Write the converse, inverse, and contrapositive and find the truth value of each. Use the related conditionals to draw a Venn diagram that represents the relationship between congruent angles and their measures. Mohs’ Scale Hardness Mineral 1 2 3 4 5 6 7 8 9 10 Talc Gypsum Calcite Fluorite Apatite Orthoclase Quartz Topaz Corundum Diamond 49. Write About It When is a conditional statement false? Explain why a true conditional statement can have a hypothesis that is false. Food 195 Navigation 278 50. What is the inverse of “If it is Saturday, then it is the weekend”? If it is the weekend, then it is Saturday. Geography 626 Recreation 92 If it is not Saturday, then it is the weekend. If it is not Saturday, then it is not the weekend. Geology 86, 804 Shuffleboard 305 If it is not the weekend, then it is not Saturday. History 48, 413, 531, 566, 595 Space Shuttle 548 51. Let a represent “Two lines are parallel to the same line,” and let b represent Marine Biology 720 “The two lines are parallel.” Which symbolic statement represents the conditional “If two lines are NOT parallel, then they are parallel to the same line”? b → ∼a ∼b → a b→a Math History 41, 78, 257, a → b 52. Which statement is a counterexample for the conditional statement 318, 493, 611, 703, 768 ��� “If f(x) = √25 - x , then f(x) is positive”? Shuttle Measurement 404 x=5 x=4 xSpace =3 x=0 53. Which statement has the same truth value as its converse? Mechanics 434 If a triangle has a right angle, its side lengths are 3 centimeters, 4 centimeters, Meteorology 476, 675, 797 and 5 centimeters. If an angle measures 104°, then the angle is obtuse. Monument 466 If a number is an integer, then it is a natural number. 2 If an angle measures 90°, then it is an acute angle. Each frame of a computer-animated 1 feature represents __ 24 of a second of film. 86 Chapter 2 Geometric Reasoning Monument Source: www.pixar.com Architecture 159, 220, 695 Art 876 Astronomy 752 Bicycles 337 Biology 100, 604 Chemistry 828 Conservation 271 Design 313 Ecology 248 Electronics 692 Engineering 115 Entertainment 149 Fitness 539 During its launch, a space shuttle accelerates to more than 27,359 km/h in just over 8 minutes. So the shuttle travels 3219 km/h faster each minute. The height of the Statue of Liberty from the foundation of the pedestal to the torch is 305 ft. Her index finger measures 8 ft, and the fingernail is 13 in. by 10 in. Sports 19, 635 Surveying 353, 556 Travel 458 Source: libertystatepark.org Why Learn Mathematics xix HOW TO STUDY GEOMETRY This book has many features designed to help you learn and study effectively. Becoming familiar with these features will prepare you for greater success on your exams. Learn Triangle Congruence: SSS and SAS 4-4 The vocabulary is listed at the beginning of every lesson. Who uses this? Engineers used the property of triangle rigidity to design the internal support for the Statue of Liberty and to build bridges, towers, and other structures. (See Example 2.) Objectives Apply SSS and SAS to construct triangles and to solve problems. Prove triangles congruent by using SSS and SAS. Vocabulary triangle rigidity included angle Study the examples to apply new concepts and skills. Examples include stepped out solutions. In Lesson 4-3, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent. The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape. For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate. Postulate 4-4-1 Look for the Know-It-Note icons to identify important information. Side-Side-Side (SSS) Congruence POSTULATE HYPOTHESIS � If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. 1 EXAMPLE � CONCLUSION � ���� ���� �ABC � �FDE � ���� � ���� ���� ���� � Using SSS to Prove Triangle Congruence Use SSS to explain why �PQR � �PSR. −− −− −− −− It is given that PQ � PS and that QR � SR. By −− −− the Reflexive Property of Congruence, PR � PR. Therefore �PQR � �PSR by SSS. Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts. � � � � 1. Use SSS to explain why �ABC � �CDA. � � � � An included angle is an angle formed by two adjacent sides of a polygon. ∠B is the included −− −− angle between sides AB and BC. Practice 242 � � � 1. Describe three ways you could prove that �ABC � �DEF. DEF DEF. 2. Explain why the SSS and SAS Postulates are shortcuts for proving triangles congruent. � � 3. GET ORGANIZED Copy and complete the graphic organizer. Use it to compare the SSS and SAS postulates. 4-4 � Chapter 4 Triangle Congruence THINK AND DISCUSS Use a graphic organizer to summarize each lesson. Test your understanding of examples by trying the Check It Out problems. Check your work in the Selected Answers. � � � ��� ��� ������������������� ����������������������� Exercises KEYWORD: MG7 4-4 KEYWORD: MG7 Parent GUIDED PRACTICE Refer to the examples from the lesson to solve the Guided Practice exercises. −− −− 1. Vocabulary In �RST which angle is the included angle of sides ST and TR? SEE EXAMPLE 1 p. 242 If you get stuck, use the internet for Homework Help Online. Use SSS to explain why the triangles in each pair are congruent. 2. �ABD � �CDB 3. �MNP � �MQP � � � � � � � SEE EXAMPLE 2 p. 243 � H J 4. Sailing Signal flags are used to communicate messages when radio silence is required. The Zulu signal flag means, “I require a tug.” GJ = GH = GL = GK = 20 in. Use SAS to explain why �JGK � �LGH. G L K SEE EXAMPLE p. 244 3 Show that the triangles are congruent for the given value of the variable. 5. �GHJ � �IHJ, x = 4 � � 6. �RST � �TUR, x = 18 � �� � ������ � ��� � � ��� ������� � ������ � � Review 4-4 Triangle Congruence: SSS and SAS 245 For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary Study and review vocabulary from the entire chapter. acute triangle . . . . . . . . . . . . . . 216 CPCTC . . . . . . . . . . . . . . . . . . . . . 260 isosceles triangle . . . . . . . . . . . 217 auxiliary line . . . . . . . . . . . . . . . 223 equiangular triangle . . . . . . . . 216 legs of an isosceles triangle . . 273 base . . . . . . . . . . . . . . . . . . . . . . . 273 equilateral triangle . . . . . . . . . 217 obtuse triangle . . . . . . . . . . . . . 216 base angle . . . . . . . . . . . . . . . . . . 273 exterior . . . . . . . . . . . . . . . . . . . . 225 remote interior angle . . . . . . . 225 congruent polygons . . . . . . . . . 231 exterior angle . . . . . . . . . . . . . . 225 right triangle . . . . . . . . . . . . . . . 216 coordinate proof . . . . . . . . . . . . 267 included angle. . . . . . . . . . . . . . 242 scalene triangle . . . . . . . . . . . . . 217 corollary . . . . . . . . . . . . . . . . . . . 224 included side . . . . . . . . . . . . . . . 252 triangle rigidity . . . . . . . . . . . . . 242 corresponding angles . . . . . . . 231 interior . . . . . . . . . . . . . . . . . . . . 225 vertex angle . . . . . . . . . . . . . . . . 273 corresponding sides. . . . . . . . . 231 interior angle . . . . . . . . . . . . . . . 225 Complete the sentences below with vocabulary words from the list above. Use the list on p. S82 to review the postulates and theorems found in the chapter. 1. A(n) ? is a triangle with at least two congruent sides. −−−− 2. A name given to matching angles of congruent triangles is 3. A(n) ? . −−−− ? is the common side of two consecutive angles in a polygon. −−−− 4-1 Classifying Triangles (pp. 216–221) EXERCISES EXAMPLE � Classify the triangle by its angle measures and side lengths. isosceles right triangle Classify each triangle by its angle measures and side lengths. 4. 5. ��� ��� ��� ���� 4-2 Angle Relationships in Triangles (pp. 223–230) EXERCISES EXAMPLE � Find m∠S. 12x = 3x + 42 + 6x 12x = 9x + 42 ���������� � ���� � ��� � Find m∠N. � 6. �� 3x = 42 x = 14 m∠S = 6 (14) = 84° � �� ���� � 7. In�LMN, m∠L = 8x °, m∠M = (2x + 1)°, and m∠N = (6x - 1)°. 284 xx How To Study Geometry Chapter 4 Triangle Congruence Test yourself with practice problems from every lesson in the chapter. TOOLS GEOMETRY OF In geometry, it is important to use tools correctly in order to measure accurately and produce accurate figures. One important tool is your pencil. Always use a sharp pencil with a good eraser. Ruler Protractor The ruler shown has a mark every __18 inch, so the accuracy is to the nearest __18 inch. � � � � � Line up one end with 0, not the edge. � � � � To use a protractor to measure an angle, you may need to extend the sides of the angle. � � � � � �� �� �� For acute angles, use the smaller measurement. For obtuse angles, use the larger measurement. �� � �� �� Straightedge A compass is used to draw arcs and circles. If you have trouble keeping the point in place, try keeping the compass still and turning the paper. A straightedge is used to draw a line through two points. If you use a ruler as a straightedge, do not use the marks on the ruler. Tilt the compass slightly. � �� � Place the center of your protractor on the vertex. Compass Keep your wrist flexible. Turn the compass with your index finger and thumb. �� �� Line up one � ray with 0.� � Choose the measurement that is the closest. �� �� First place your pencil on one points. Place the straightedge against your pencil and the other point. Draw the line. � � � � � � � � � � � � �� �� �� �� � � �� �� �� �� �� � � �� �� � � � Geometry Software Geometry software can be used to create figures and explore their properties. Use the toolbar to select, draw; and label figures. Drag points to explore properties of a figure. Use the menus to construct, transform, and measure figures. The parts of each figure are linked. To avoid deleting a whole figure, hide parts instead of deleting them. Tools of Geometry xxi Scavenger H Use this scavenger hunt to discover a few of the many tools in Holt Geometry that you can use to become an independent learner. On a separate sheet of paper, write the answers to each question below. Within each answer, one letter will be in a yellow box. After you have answered every question, identify the letters that would be in yellow boxes and rearrange them to reveal the answer to the question at the bottom of the page. 1. ■ ■ ■ ■ ■ What is the first Vocabulary term in the Study Guide: Preview for Chapter 1? 2. ■ ■ ■ ■ ■ ■ 3. ■ ■ ■ ■ ■ ■ ■ 4. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 5. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 6. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 7. ■ ■ ■ ■ ■ ■ 8. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ What keyword should you enter for Homework Help for Lesson 3-3? In Lesson 8-2, what is Example 4 teaching you to find? What theorem are you asked about in the Know-It Note on page 352? What mathematician is featured in the Math History link on page 318? Whose job is described in the Career Path on page 612? In the Study Guide: Review for Lesson 11-1, what do the lines intersect? What advice does Chapter 1’s Test Tackler give about how to answer a multiple choice test item you don’t know how to solve? What did the little acorn say when it grew up? ■■■■■■■■ xxii Scavenger Hunt The Problem Solving Plan Mathematical problems are a part of daily life. You need to use a good problem-solving plan to be a good problem solver. The plan used in this textbook is outlined below. UNDERSTAND the Problem First make sure you understand the problem you are asked to solve. What are you asked to find? What information is given? What information do you need? Do you have all the information needed? Do you have too much information? Restate the question in your own words. Identify the key facts given in the problem. Determine what information you need to solve the problem. Determine if you need more information. Determine if there is unnecessary information and eliminate it from your list of important facts. Make a PLAN Plan how to use the information you are given. Have you solved similar problems? What problem solving strategy or strategies could you use to solve this problem? Think about similar problems you have solved successfully. Choose an appropriate problem solving strategy and decide how you will use it. SOLVE Use your plan to solve the problem. Show the steps in the solution, and write a final statement that gives the solution to the problem. LOOK BACK Check your answer against the original problem. Have you answered the question? Is the answer reasonable? Are your calculations correct? Can you use another strategy or solve the problem in another way? Did you learn anyting that could help you solve similar problems in the future? Make sure you have answered the original question. The answer must make sense in relation to the question. Check to make sure your calculations are accurate. Using another strategy is a good way to check your answer. Try to remember the types of problems you have solved and the strategies you applied. Focus on Problem Solving xxiii