Edward B. Burger David J. Chard Earlene J. Hall Paul A. Kennedy Steven J. Leinwand Freddie L. Renfro Dale G. Seymour Bert K. Waits


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1 Edward B. Burger David J. Chard Earlene J. Hall Paul A. Kennedy Steven J. Leinwand Freddie L. Renfro Dale G. Seymour Bert K. Waits
2 Geometry Contents in Brief CHAPTER 1 Foundations for Geometry CHAPTER 2 Geometric Reasoning CHAPTER 3 Parallel and Perpendicular Lines CHAPTER 4 Triangle Congruence CHAPTER 5 Properties and Attributes of Triangles CHAPTER 6 Polygons and Quadrilaterals CHAPTER 7 Similarity CHAPTER 8 Right Triangles and Trigonometry CHAPTER 9 Extending Perimeter, Circumference, and Area CHAPTER 10 Spatial Reasoning CHAPTER 11 Circles CHAPTER 12 Extending Transformational Geometry 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, N. MoPac Expressway, Building 3, Austin, Texas 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 Cover photo: The Stata Center at MIT, Boston, Massachusetts, USA. Scott Gilchrist/Masterfile
3 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 PreK12 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 cofounder of T3 (Teachers Teaching with Technology), a national professional development program.
4 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 RumsonFair Haven Regional HS Rumson, NJ Marieta W. Harris Mathematics Specialist Memphis, TN Debbie Hecky Geometry Teacher Scott High School Covington, KY Cynthia Hodges Department Chair Shoemaker High School Killeen, TX 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
5 Jamae Sellari Mathematics Instructor Forest Hill High School Jackson, MS Caren Sorrells Mathematics Coordinator Birdville ISD Haltom City, TX E. Robin Staudenmeier Middle/High School Math Coordinator Olympia Community USD 16 Stanford, IL Anna Valdez Geometry Teacher Nikki Rowe High School McAllen, TX Lauralea Wright Mathematics Teacher Mauldin High School Mauldin, SC 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 Jill Morris Navasota High School Navasota, TX Ruth Stutzman Jefferson Forest High School Forest, VA Carey Carter Alvarado High School Alvarado, TX Walter Babst Bonita High School La Verne, CA
6 By the Centroid Theorem, the centroid of a triangle is 2 3 of the distance from each vertex to the midpoint of the This method confirms the first answer. Use Logical Reasoning Use a Venn Diagram Make an Organized List 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 of x? Find the value of s A and B are the remote interior angles of BCD in ABC. Which of these equations must be true? Multiple Choice choose your answer. Gridded Response write your answer in a grid and fill in the corresponding bubbles. Short Response write openended responses that are scored with a 2point rubric. Extended Response write openended responses that are scored with a 4point rubric. m A = m B m BCD = m BCA  m A m A = 90  m B m B = m BCD  m A 44. Extended Response The measures of the angles in a triangle are in the ratio 2 : 3 : 4. Describe how to use algebra to find the measures of these angles. Then find the measure of each angle and classify the triangle. CHALLENGE AND EXTEND 45. An exterior angle of a triangle measures 117. Its remote interior angles measure ( 2y 2 + 7) and (61  ) y2. Find the value of y. 46. Two parallel lines are intersected by a transversal. What type of triangle is formed by the intersection of the angle bisectors of two sameside interior angles? Explain. (Hint: Use geometry software or construct a diagram of the angle bisectors of two sameside interior angles.) 47. Critical Thinking Explain why an exterior angle of a triangle cannot be congruent to a remote interior angle. 48. Probability The measure of each angle in a triangle is a multiple of 30. What is the probability that the triangle has at least two congruent angles? 49. In ABC, m B is 5 less than times m A. m C is 5 less than 2 1 times m A. 2 What is m A in degrees? SPIRAL REVIEW Make a table to show the value of each function when x is 2, 0, 1, and 4. (Previous course) 50. f(x) = 3x f(x) = x f(x) = (x  3) Find the length of NQ. Name the theorem or postulate that justifies your answer. (Lesson 27) Classify each triangle by its side lengths. (Lesson 41) 54. ACD 55. BCD 56. ABD 57. What if? If CA = 8, What is the effect on the classification of ACD? 230 Chapter 4 Triangle Congruence If you can t think of a different method to use to check your answer, circle the question and come back to it later. Read each test item and answer the questions that follow. Item A Multiple Choice Given that is the perpendicular bisector of AB, AC = 3n + 1, and BC = 6n  11, what is the value of n? 4 4_ 3_ How can you use the given answer choices to solve this problem? 2. Describe how to solve this problem directly. Item B Multiple Choice Which number forms a Which number forms a Pythagorean triple with 15 and 17? How can you use the given answer choices to find the answer? 4. Describe a different method you can use to check your answer. Item C Gridded Response Find the area of the square in square centimeters. 5. How can you use special right triangles to answer this question? 6. Explain how you can check your answer by using the Pythagorean Theorem. Item D Short Response Do the ordered pairs A(8, 4), B(0, 2), and C(8, 4) form a right triangle? Explain your answer. 7. Explain how to use slope to determine if ABC is a right triangle. 8. How can you use the Converse of the Pythagorean Theorem to check your answer? Item E Short Response Find the orthocenter of RST. Show your work. Any Question Type: Check with a Different Method It is important to check all of your answers on a test. An effective way to do this is to use a different method to answer the question a second time. If you get the same answer with two different methods, then your answer is probably correct. Short Response What are the coordinates of the centroid of ABC with A(2, 4), B(4, 6), and C(1, 1)? Show your work. Method 1: The centroid of a triangle is the point of concurrency of the medians. Write the equations of two medians and find their point of intersection. Let D be the midpoint of AB and let E be the midpoint of BC. D = ( , = (1,5) + 1 E = (4 2 ) = (2.5, 2.5) _ 2 _ 2 ) 9. Describe how you would solve this problem. 10. How can you use the third altitude of the triangle to confirm that your answer is correct? _ 2, 6 + (1) _ The median from C to D contains C (1, 1) and D (1, 5). It is vertical, so its equation is x = 1. The median from A to E contains A (2, 4) and E (2.5, 2.5). slope of AE = _ = 1.5 _ 4.5 =  1_ 3 y  y 1 = m (x  x 1) Pointslope form y  4 =  1_ 3 (x + 2) Substitute 4 for y 1,  1_ 3 and 2 for x 1. for m, Solve the system x = 1 y  4 =  1 to find the point of intersection. (x + 2) y  4 =  1_ (1 + 2) Substitute 1 for x. 3 y = 3 Simplify. The coordinates of the centroid are (1, 3). Method 2: To check this answer, use a different method. 3 Test Tackler 373 Problem Solving Strategies Draw a Diagram Make a Model Test Tackler Use the Test Tackler to become familiar with and practice testtaking strategies. The first page of this feature explains and shows an example of a testtaking strategy. Guess and Test Work Backward opposite side. CD is vertical with a length of 6 units. 2 (6) = 4, 3 and the coordinates of the point that is 4 units up from C is (1, 3). Make a Table Solve a Simpler Problem Find a Pattern The second page guides you through applications of the testtaking strategy. 372 Chapter 5 Properties and Attributes of Triangles C2 Preparing For Standardized Tests
7 1 and 5 1 and 51 and 6 2 and Chapter 4 Triangle Congruence that triangles are congruent? AAA SAS ASA HL Standardized Test Prep Use the Standardized Test Prep to apply testtaking strategies. 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. Use this diagram for Items 12 and What is the measure of ACD? What type of triangle is ABC? Isosceles acute Equilateral acute Isosceles obtuse Scalene acute Take some time to learn the directions for filling in a grid. Check and recheck to make sure you are filling in the grid properly. You will only get credit if the ovals below the boxes are filled in correctly. To check your answer, solve the problem using a different method from the one you originally used. If you made a mistake the first time, you are unlikely to make the same mistake when you solve a different way. Gridded Response 14. CDE JKL. m E = (3x + 4), and m L = (6x  5). What is the value of x? 15. Lucy, Eduardo, Carmen, and Frank live on the same street. Eduardo s house is halfway between Lucy s house and Frank s house. Lucy s house is halfway between Carmen s house and Frank s house. If the distance between Eduardo s house and Lucy s house is 150 ft, what is the distance in feet between Carmen s house and Eduardo s house? 16. JKL XYZ, and JK = 102n. XY = 2, and YZ = n 2. Find KL. 17. An angle is its own supplement. What is its measure? 18. The area of a circle is 154 square inches. What is its circumference to the nearest inch? 19. The measure of P is times the measure of Q. If P and Q are complementary, what is m P in degrees? Short Response 20. Given m with transversal n, explain why 2 and 3 are complementary. 21. G and H are supplementary angles. m G = (2x + 12), and m H = x. a. Write an equation that can be used to determine the value of x. Solve the equation and justify each step. b. Explain why H has a complement but G does not. 22. A manager conjectures that for every 1000 parts a factory produces, 60 are defective. a. If the factory produces 1500 parts in one day, how many of them can be expected to be defective based on the manager s conjecture? Explain how you found your answer. b. Use the data in the table below to show that 23. the manager s conjecture is false. Day Parts Defective Parts CUMULATIVE ASSESSMENT, CHAPTERS 1 4 Multiple Choice Use the diagram for Items 1 and BD is the perpendicular bisector of AC. a. What are the conclusions you can make from this statement? BD intersects AC at D. Explain why BD is the shortest path from B to AC. b. Suppose Extended Response BC EF, and AC DF. m C = 42.5, and m E = ABC and DEF are isosceles triangles. a. What is m D? Explain how you determined your answer. b. Show that ABC and DEF are congruent. c. Given that EF = 2x + 7 and AB = 3x + 2, find the value for x. Explain how you determined your answer. 1. Which of these congruence statements can be proved from the information given in the figure? AEB CED ABD BCA BAC DAC DEC DEA 2. What other information is needed to prove that CEB AED by the HL Congruence Theorem? AD AB BE AE Cumulative Assessment, Chapters CB AD DE CE 3. Which biconditional statement is true? Tomorrow is Monday if and only if today is not Saturday. Next month is January if and only if this month is December. Today is a weekend day if and only if yesterday was Friday. This month had 31 days if and only if last month had 30 days. 4. What must be true if than one point? P, Q, S, and T are collinear. PQ intersects ST at more P, Q, S, and T are noncoplanar. PQ and ST are opposite rays. PQ and ST are perpendicular. 5. ABC DEF, EF = x 27, and BC = 4x  2. Find the values of x. 6. Which conditional statement has the same truth value as its inverse? If n < 0, then n 2 > 0. If a triangle has three congruent sides, then it is an isosceles triangle. If an angle measures less than 90, then it is an acute angle. If n is a negative integer, then n < On a map, an island has coordinates (3, 5), and a reef has coordinates (6, 8). If each map unit represents 1 mile, what is the distance between the island and the reef to the nearest tenth of a mile? 4.2 miles 9.0 miles 6.0 miles 15.8 miles 8. A line has an xintercept of 8 and a yintercept 9. of 3. What is the equation of the line? y = 8x + 3 y = 8_ 3 x  8 y = 3_ 8 x + 3 y = 3x  8 JK passes through points J(1, 3) and K(3, 11). Which of these lines is perpendicular to JK? y =  1_ 2 x + 1_ 3 y = 2x  1_ 5 y = 1_ 2 x + 6 y = 2x If PQ = 2 (RS) + 4 and RS = TU + 1, which equation is true by the Substitution Property of Equality? PQ = TU + 5 PQ = TU + 6 PQ = 2 (TU) + 5 PQ = 2 (TU) Which of the following is NOT valid for proving KEYWORD: MG7 TestPrep AB is perpendicular to XY. If A(3, 5), B(9, 3), and X(2, 5), what are the coordinates of Y? (6, 8) (1, 4) (3, 1) (2, 5) AB CD and AC BD. Which of the following is true? 1 is congruent to 5. 2 is supplementary to is congruent to is supplementary to 13. ABC is formed by AB and BC. If A(4, 1) and B(4, 6), what coordinates for C will result in an obtuse angle? (1, 2) (2, 9) (0, 5) (2, 6) What is the length of the given segment to the nearest unit? WEEK 9 Which equation best represents the line in the graph? Use the to practice for your state test every day. 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 C12 _ x + 3 y = 1_ 2 x + 3 y = 3x + 1 y = 2x  3 _ x + 3 y = _ 1_ 2 x + 3 Each week s page has five practice test items, one for each day of the week. TestTaking 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 testtaking strategies that can be found throughout this book, such as working backward and eliminating answer choices. Preparing For Standardized Tests C3
8 COUNTDOWN TO TESTING WEEK 1 Which statement about a number line is true? Values increase toward the right. Values increase toward the left. Whole numbers are toward the right and decimal numbers are toward the left. Negative numbers are toward the right and positive numbers are toward the left. If a = b and b = c, which statement must be true? a > c a  c = 0 a + c = 0 a = c 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 Which shape is NOT included in the figure? Which statement best describes these two figures? Circle Square Triangle Trapezoid They cover the same area. They are the same size. They have the same number of sides. The distance around each figure is the same. Geometry 2 C4
9 WEEK 2 What is the length of FD? ABC is an obtuse angle. Which of these could be the measure of ABC? Which point is described by the coordinates (2, 3)? A B C D 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? 8 feet 16 feet 24 feet 32 feet OB bisects AOC, and m AOC = 60. What is m BOE? C5
10 WEEK 3 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 Which of these is a unit that can describe the perimeter of a figure? Meters Square centimeters Cubic inches Seconds Point X is the midpoint of HI. What is the coordinate of the point X? Which expression best represents the perimeter of the figure below? 27x 5x x x + 5 A line segment is drawn between the points (5, 8) and (1, 6). What are the coordinates of the midpoint of the segment? (3, 1) (4, 14) (2, 7) (  1 _ 2, 3 ) Geometry 2 C6
11 WEEK 4 Which equation below represents the second step of the solution process? Step 1: 6x  12 = 3 (5  x) Step 2:? Step 3: 9x  12 = 15 Step 4: 9x = 27 Step 5: x = 3 6x  12 = 15  x 6x  12 = 153x 6x  12 = 53x 6x = 3 (5  x)  12 Which conjecture best describes a rule for the pattern below? Rotate counterclockwise 90 Rotate clockwise 90 Rotate counterclockwise 180 Rotate clockwise 180 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 right triangle have different lengths. is false because a right triangle cannot have two congruent angles. 007 SE Which of the following best describes the value of 4n + 1 when n is an integer? The value is always negative. The value is always positive. The value is always even. The value is always odd. MN ÈÈ bisects LMO. Which statement must be true? m LMN = m OMN m LMO = m OMN m LMN = m OML m LMO = m ONM C7
12 WEEK 5 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. Which of the following statements is true, based on the figure? 2 and 4 are not adjacent but form a linear pair. 2 and 4 are adjacent angles that form a linear pair. 1 and 3 are adjacent angles and form a linear pair. 1 and 3 are not adjacent angles but form a linear pair. 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? a b b a a b b a 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? 4 square units 100 square units 364 square units 1536 square units How many pairs of vertical angles are in the diagram? Geometry 2 C8
13 WEEK 6 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. 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. 1 and 2 are congruent. Two angles are labeled in the figure below. Which of the following statements best describes this angle pair? They are complementary angles. They are congruent angles. They are supplementary angles. They are parallel angles. If line a is parallel to line b, and m 8 = 62º, what is m 1? SE 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 m AOB = m BOC m AOB + m AOC = m BOC m BOC + m AOC = m AOB C9
14 WEEK 7 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 What must be true if two nonvertical lines are perpendicular? Their slopes add to 0. The product of their slopes is 1. Their slopes are equal. Their yintercepts are equal. Which line is parallel to y = 2x + 3? y = 2x  8 y = 3x + 2 2y = 4x + 6 y = 2x + 3 Which expression best represents the perimeter of the rectangle? 4x + 1 6x + 4 8x x 2 + 3x 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? steep downward straight line steep upward curve horizontal line upward straight line Geometry 2 C10
15 WEEK 8 What is the slope of the given line segment? 21 _ Which two lines are perpendicular? y = 5x + 2 and 2y  10x = 4 Which of the following is the best classification for the given triangle? y = _ 1 x + 1 and y = 4x y = 3x + 1 and y  4x = 6 y = _ 1 x + 2 and y + 2x = 4 2 Equilateral Isosceles Scalene Right 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 4π y 2 8π y SE C11
16 WEEK 9 AB is perpendicular to XY. If A (3, 5), B (9, 3), and X (2, 5), what are the coordinates of Y? (6, 8) (1, 4) (3, 1) (2, 5) ABC is formed by BA and BC. If A (4, 1) and B (4, 6), what coordinates for C will result in an obtuse angle? (1, 2) (2, 9) (0, 5) (2, 6) AB CD and AC BD. Which of the following is true? 1 is congruent to 5. 2 is supplementary to is congruent to is supplementary to 13. What is the length of the given segment to the nearest unit? Which equation best represents the line in the graph? y = 1 _ 2 x + 3 y = 3x + 1 y = 2x  3 y =  1_ 2 x + 3 C12
17 WEEK 10 Which set of angle measures can be used to conclude that lines x and y are parallel? 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 Which postulate or theorem can be used to prove that these triangles are congruent? SAS ASA AAS SSS 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. 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? Determining the angle at which Earth tilts Calculating the mass of Earth Measuring the surface area of Earth Finding the distance around the equator C13
18 WEEK 11 What conclusion can you draw from the figure? Jan drew the figure below and claims that line l is parallel to line m. Which of the following proves her statement true? ABC is isosceles. The perimeter of ABC is 45 centimeters. DE = 10 centimeters DE = 1 _ 2 AB Angles on opposite sides of the transversal are equal. Corresponding angles on the same side of the transversal are congruent. More than two angles in the diagram have the same value. Two straight lines pass through the same transversal. 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. If X (5, 5) and Y (0, 0), what are the coordinates of Z so that m XYZ = 90? (5, 5) (5, 5) (5, 0) (0, 5) OZ is a bisector of XOY. Which of the following statements is NOT true? 2m ZOY = m XOY 2m XOZ = m XOY m ZOY = m XOY m XOZ = 1 _ 2 m XOY C14
19 WEEK 12 Which of the following correctly completes the congruence statement? AB? Based on the figure, which inequality is correct? FD AF EF ED 2x > x x < 10 x < 10 x > 8 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 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? 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. C15
20 WEEK 13 What are the coordinates of point P? Which postulate or theorem can be used to verify the congruence of these two triangles? (3, 2) (2, 3) (3, 2) (2, 3) SSS ASA AAS SAS 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. 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? 5 meters 8 meters 10 meters 25 meters  1 _ 24 1_ 2 2 C16
21 WEEK 14 A ceramic tile is in the shape of a triangle. The side across from the 30 angle is 6.25 centimeters long. How long is the hypotenuse of the tile? centimeters 6.25 Ç3 centimeters 12.5 centimeters 15 centimeters What is the slope of this line? Which equation should Aretha use to find the distance between two points across a river? 1 1_ _ 3 c = a 2 + b 2 c = a + b c 2 = ÇÇÇ a + b c = ÇÇÇ a 2 + b 2 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? l m n o C17
22 WEEK 15 Three coordinates of ABCD are A (4, 5), C (7, 3), and D (1, 3). Which coordinates could represent point B? (1, 5) (3, 7) (5, 1) (10, 5) Which two lines are parallel? y = 6x + 8 and y + _ 1 6 x = 3 y = _ 1 x  1 and y = 3x y  2x = 2 and y = 22x y = 1 _ 4 x and y  1 _ 4 x = 1 What is the midpoint of QR? (1, 2) (2, 1) (1, 2) (1, 2) Which of these statements is true? All quadrilaterals are parallelograms. Every rectangle is a parallelogram. Every parallelogram is also a rectangle. The diagonals of a rhombus are congruent. Which expression describes the total number of diagonals in a polygon with n sides? No. of sides No. of diagonals n (n  3) _ 2 2n 3n_ 2 _ 2n C18
23 WEEK 16 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? (8, 2), (3, 2), (3, 9) (4, 1), (6, 2), (8, 10) (2, 5), (2, 9), (8, 3) (0, 0), (1, 8), (5, 2) Natalia is using indirect measurement to find the distance across a pond. Which Pythagorean triple is represented by the triangle? 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, , 6, 6.5 What is the area of the composite figure? What is the measure of 3 in the regular hexagon? 8 square meters 21 square meters 25 square meters 45 square meters C19
24 WEEK 17 Which two lines are perpendicular? y = x + 6 and y = x  6 y + 2 _ 3 x = 1 and y = 3 _ 2 x  4 y = 1 _ 2 x  2 and y = 1 _ 2 x + 3 y  2x = 5 and y = 2x + 2 What is the perimeter of the composite figure to the nearest centimeter? What is the measure of 1 in the triangle below? centimeters 52 centimeters 60 centimeters 83 centimeters What is the sixth item in the pattern below? 64, 32, 16, 8, 0 1_ 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? It is a square. Its width is 2 units. Its perimeter is 6 units. Its area is 9 square units. C20
25 WEEK 18 Based on the pattern of similar triangles below, what is the value of x? Ç3 8 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 Ç 3 Ç 3 _ 3 Ç 2 1_ millimeters 78.8 millimeters millimeters 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. C21
26 WEEK 19 Which two line segments are congruent? AB and DF CE and GH GH and AB CD and DE 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 24foot tree casts an 18foot shadow. How long is the shadow cast by a 4foot mailbox at the same time of day? No. of sides No. of triangles formed n 2n feet n feet n + 2 _ feet 5 feet 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? 30 meters 60 meters 200 meters 225 meters What is x? C22
27 WEEK 20 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) n _ 2 Which coordinates represent a vertex of the hexagon? (0, 2) (4, 2) (3, 2) (2, 2) The two triangles in the figure are similar. What is the length of MN? Two regular pentagons have perimeters of 30 and 75 respectively. What scale factor relates the smaller figure to the larger one? 1 : : 6 1 : 15 1 : 21 Alissa is painting a diagonal line across a square tile. What is the length of the line? 2 Ç 8 centimeters 6 centimeters 8 centimeters 8 Ç 2 centimeters C23
28 WEEK 21 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 Exterior angle _ 360 n  2 _ n 2 + n 360n 360 _ n Carrie is building a skateboard ramp with the dimensions below. What is the approximate measure of x? What is the value of z? Ç 2 12 Ç 3 17 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? y = 1 _ 2 x + 3 y = 5 y = x + 3 y =  1 _ 2 x  3 (0, 4) (1, 4) (2, 4) (3, 3) C24
29 WEEK 22 If this pattern is continued, how many shaded triangles will there be in the fourth element of the pattern? 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_ miles miles 22.5 miles 31.5 miles What are the side lengths of the triangle? 3, 4, and 5 2, 3, and 5 3, 3, and 3 3, 3, and 3 Ç 2 An 18foot 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 14.4 feet 30.9 feet C25
30 WEEK 23 RST is a triangle. What is the ycoordinate 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? How does the slope of the hypotenuse of ABC compare with that of DBC? They have the same value. They have opposite signs. They have the same sign. They are reciprocals. 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? 16 feet 300 feet 1052 feet 5724 feet C26
31 WEEK 24 Quadrilaterals ABCD and WXYZ are similar. What is XY? 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? 3.6 units 4.1 units 8.5 units 12.2 units Which Pythagorean triple would be most helpful in finding the value of a? What is the perimeter of the square? C27
32 Foundations for Geometry KEYWORD: MG7 TOC ARE YOU READY? Euclidean and Construction Tools 11 Understanding Points, Lines, and Planes Explore Properties Associated with Points Measuring and Constructing Segments Measuring and Constructing Angles Pairs of Angles MULTISTEP TEST PREP READY TO GO ON? QUIZ Coordinate and Transformation Tools 15 Using Formulas in Geometry Connecting Geometry to Algebra: Graphing in the Coordinate Plane Midpoint and Distance in the Coordinate Plane Transformations in the Coordinate Plane Explore Transformations... MULTISTEP TEST PREP READY TO GO ON? QUIZ Study Guide: Preview Reading and Writing Math Study Guide: Review Chapter Test 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 KnowIt 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 MultiStep 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
33 Geometric Reasoning ARE YOU READY? Inductive and Deductive Reasoning 21 Using Inductive Reasoning to Make Conjectures Connecting Geometry to Number Theory: Venn Diagrams Conditional Statements Using Deductive Reasoning to Verify Conjectures Solve Logic Puzzles Biconditional Statements and Definitions MULTISTEP TEST PREP READY TO GO ON? QUIZ Mathematical Proof 25 Algebraic Proof Geometric Proof Design Plans for Proofs Flowchart and Paragraph Proofs MULTISTEP TEST PREP READY TO GO ON? QUIZ EXT Introduction to Symbolic Logic Study Guide: Preview Reading and Writing Math Study Guide: Review Chapter Test Problem Solving on Location: South Carolina KEYWORD: MG7 TOC Table of Contents Tools for Success 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 KnowIt 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 Test Prep Exercises 79, 86, 93, 101, 109, 116, 125 MultiStep Test Prep 78, 85, 92, 100, 102, 109, 115, 124, 126 College Entrance Exam Practice 135 Test Tackler 136 Standardized Test Prep 138
34 Parallel and Perpendicular Lines ARE YOU READY? Lines with Transversals KEYWORD: MG7 TOC 31 Lines and Angles Connecting Geometry to Algebra: Systems of Equations Explore Parallel Lines and Transversals Angles Formed by Parallel Lines and Transversals Proving Lines Parallel Construct Parallel Lines Perpendicular Lines Construct Perpendicular Lines MULTISTEP TEST PREP READY TO GO ON? QUIZ Coordinate Geometry 35 Slopes of Lines Explore Parallel and Perpendicular Lines Lines in the Coordinate Plane Connecting Geometry to Data Analysis: Scatter Plots and Lines of Best Fit MULTISTEP TEST PREP READY TO GO ON? QUIZ Study Guide: Preview Reading and Writing Math Study Guide: Review Chapter Test Tools for Success Writing Math 150, 160, 168, 177, 186, 196 Vocabulary 143, 144, 148, 175, 185, 194, 202 Study Strategy 145 KnowIt Notes 146, 147, 148, 155, 156, 157, 162, 163, 173, 174, 182, 184, 185, 190, 192, 193 Graphic Organizers 148, 157, 165, 174, 185, 193 Homework Help Online 148, 158, 166, 175, 185, 194 Test Prep Exercises , , , , 187, MultiStep Test Prep 150, 160, 168, 176, 180, 186, 196, 200 College Entrance Exam Practice 207 Test Tackler 208 Standardized Test Prep Artists Rights Society (ARS), New York/ADAGP, Paris
35 Triangle Congruence ARE YOU READY? Triangles and Congruence 41 Classifying Triangles Develop the Triangle Sum Theorem Angle Relationships in Triangles Congruent Triangles MULTISTEP TEST PREP READY TO GO ON? QUIZ KEYWORD: MG7 TOC Proving Triangle Congruence 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 MULTISTEP TEST PREP READY TO GO ON? QUIZ EXT Proving Constructions Valid Study Guide: Preview Reading and Writing Math Study Guide: Review Chapter Test Problem Solving on Location: Michigan 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 KnowIt 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, , , 272, 279 MultiStep Test Prep 220, 229, 236, 238, 247, 258, 264, 271, 278, 280 College Entrance Exam Practice 289 Test Tackler 290 Standardized Test Prep 292
36 Properties and Attributes of Triangles ARE YOU READY? Segments in Triangles KEYWORD: MG7 TOC 51 Perpendicular and Angle Bisectors Bisectors of Triangles Medians and Altitudes of Triangles Special Points in Triangles The Triangle Midsegment Theorem MULTISTEP TEST PREP READY TO GO ON? QUIZ Relationships in Triangles 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 Handson Proof of the Pythagorean Theorem The Pythagorean Theorem Applying Special Right Triangles Graph Irrational Numbers MULTISTEP TEST PREP READY TO GO ON? QUIZ Study Guide: Preview Reading and Writing Math Study Guide: Review Chapter Test Tools for Success Reading Math 299, 300 Writing Math 306, 313, 318, 325, 338, 344, 354, 361 Vocabulary 297, 298, 304, 311, 317, 324, 336, 352, 366 KnowIt 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 Test Prep Exercises 306, 313, 319, 326, 339, 345, 355, 362 MultiStep Test Prep 305, 312, 319, 326, 328, 338, 344, 354, 361, 364 College Entrance Exam Practice 371 Test Tackler 372 Standardized Test Prep 374
37 Polygons and Quadrilaterals ARE YOU READY? Polygons and Parallelograms Construct Regular Polygons Properties and Attributes of Polygons Connecting Geometry to Algebra: Relations and Functions Explore Properties of Parallelograms Properties of Parallelograms Conditions for Parallelograms MULTISTEP TEST PREP READY TO GO ON? QUIZ KEYWORD: MG7 TOC Other Special Quadrilaterals 64 Properties of Special Parallelograms Predict Conditions for Special Parallelograms Conditions for Special Parallelograms Explore Isosceles Trapezoids Properties of Kites and Trapezoids MULTISTEP TEST PREP READY TO GO ON? QUIZ Study Guide: Preview Reading and Writing Math Study Guide: Review Chapter Test Problem Solving on Location: Ohio Tools for Success Writing Math 379, 388, 397, 404, 405, 414, 424, 434 Vocabulary 377, 378, 386, 395, 412, 432, 438 KnowIt 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 Test Prep Exercises 388, 397, 405, , 425, MultiStep Test Prep 387, 396, 404, 406, 414, 424, 434, 436 College Entrance Exam Practice 443 Test Tackler 444 Standardized Test Prep 446
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