Chapter 15 Resource Masters New York, New York Columbus, Ohio Woodland Hills, California Peoria, Illinois CONSUMABLE WORKBOOKS Many of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks in both English and Spanish. Study Guide Workbook Skills Practice Workbook Practice Workbook Spanish Study Guide and Assessment 0-07-869623-2 0-07-869312-8 0-07-869622-4 0-07-869624-0 ANSWERS FOR WORKBOOKS The answers for Chapter 15 of these workbooks can be found in the back of this Chapter Resource Masters booklet. StudentWorksTM This CD-ROM includes the entire Student Edition along with the English workbooks listed above. TeacherWorksTM All of the materials found in this booklet are included for viewing and printing in the Geometry: Concepts and Applications TeacherWorks CD-ROM. Copyright © The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Geometry: Concepts and Applications. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 0-07-869281-4 1 2 3 4 5 6 7 8 9 10 Geometry: Concepts and Applications Chapter 15 Resource Masters 024 11 10 09 08 07 06 05 04 Contents Lesson 15-5 Study Guide and Intervention . . . . . . . . . . . . . . . 649 Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 Reading to Learn Mathematics . . . . . . . . . . . . . . 652 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 Vocabulary Builder . . . . . . . . . . . . . . . . . vii-viii Lesson 15-1 Study Guide and Intervention . . . . . . . . . . . . . . . 629 Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 Reading to Learn Mathematics . . . . . . . . . . . . . . 632 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 Lesson 15-6 Study Guide and Intervention . . . . . . . . . . . . . . . 654 Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 Reading to Learn Mathematics . . . . . . . . . . . . . . 657 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 Lesson 15-2 Study Guide and Intervention . . . . . . . . . . . . . . . 634 Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 Reading to Learn Mathematics . . . . . . . . . . . . . . 637 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 Chapter 15 Assessment Chapter 15 Test, Form 1A. . . . . . . . . . . . . . . 659-660 Chapter 15 Test, Form 1B . . . . . . . . . . . . . . . 661-662 Chapter 15 Test, Form 2A. . . . . . . . . . . . . . . 663-664 Chapter 15 Test, Form 2B . . . . . . . . . . . . . . . 665-666 Chapter 15 Extended Response Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 Chapter 15 Mid-Chapter Test . . . . . . . . . . . . . . . . 668 Chapter 15 Quizzes A & B. . . . . . . . . . . . . . . . . . 669 Chapter 15 Cumulative Review . . . . . . . . . . . . . . 670 Chapter 15 Standardized Test Practice . . . . . . . . . . . . . . . . . . . . . . . 671-672 Lesson 15-3 Study Guide and Intervention . . . . . . . . . . . . . . . 639 Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 Reading to Learn Mathematics . . . . . . . . . . . . . . 642 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 Lesson 15-4 Study Guide and Intervention . . . . . . . . . . . . . . . 644 Skills Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 Reading to Learn Mathematics . . . . . . . . . . . . . . 647 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 © Glencoe/McGraw-Hill Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . . . . . A1 ANSWERS . . . . . . . . . . . . . . . . . . . . . . . . . . A2-A23 iii Geometry: Concepts and Applications A Teacher’s Guide to Using the Chapter 15 Resource Masters The Fast File Chapter Resource system allows you to conveniently file the resources you use most often. The Chapter 15 Resource Masters include the core materials needed for Chapter 15. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet. All of the materials found in this booklet are included for viewing and printing in the Geometry: Concepts and Applications TeacherWorks CD-ROM. Vocabulary Builder Pages vii-viii include a student study tool that presents the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Practice There is one master for each lesson. These problems more closely follow the structure of the Practice section of the Student Edition exercises. These exercises are of average difficulty. When to Use Give these pages to students practice options or may be used as homework for second day teaching of the lesson. When to Use These provide additional before beginning Lesson 15-1. Encourage them to add these pages to their Geometry: Concepts and Applications Interactive Study Notebook. Remind them to add definitions and examples as they complete each lesson. Reading to Learn Mathematics One master is included for each lesson. The first section of each master presents key terms from the lesson. The second section contains questions that ask students to interpret the context of and relationships among terms in the lesson. Finally, students are asked to summarize what they have learned using various representation techniques. Study Guide There is one Study Guide master for each lesson. When to Use Use these masters as reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for those students who have been absent. When to Use This master can be used as a study tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learners) students. Skills Practice There is one master for each lesson. These provide computational practice at a basic level. When to Use These worksheets can be used with students who have weaker mathematics backgrounds or need additional reinforcement. © Glencoe/McGraw-Hill iv Geometry: Concepts and Applications Enrichment There is one master for each lesson. These activities may extend the concepts in the lesson, offer a historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. These are not written exclusively for honors students, but are accessible for use with all levels of students. Intermediate Assessment • A Mid-Chapter Test provides an option to assess the first half of the chapter. It is composed of free-response questions. • Two free-response quizzes are included to offer assessment at appropriate intervals in the chapter. When to Use These may be used as extra Continuing Assessment credit, short-term projects, or as activities for days when class periods are shortened. • The Cumulative Review provides students an opportunity to reinforce and retain skills as they proceed through their study of geometry. It can also be used as a test. The master includes free-response questions. Assessment Options The assessment section of the Chapter 15 Resources Masters offers a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use. • The Standardized Test Practice offers continuing review of geometry concepts in multiple choice format. Answers Chapter Assessments Chapter Tests • Page A1 is an answer sheet for the Standardized Test Practice questions that appear in the Student Edition on page 673. This improves students’ familiarity with the answer formats they may encounter in test taking. • Forms 1A and 1B contain multiple-choice questions and are intended for use with average-level and basic-level students, respectively. These tests are similar in format to offer comparable testing situations. • The answers for the lesson-by-lesson masters are provided as reduced pages with answers appearing in red. • Forms 2A and 2B are composed of freeresponse questions aimed at the averagelevel and basic-level student, respectively. These tests are similar in format to offer comparable testing situations. • Full-size answer keys are provided for the assessment options in this booklet. All of the above tests include a challenging Bonus question. • The Extended Response Assessment includes performance assessment tasks that are suitable for all students. A scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. © Glencoe/McGraw-Hill v Geometry: Concepts and Applications Chapter 15 Leveled Worksheets Glencoe’s leveled worksheets are helpful for meeting the needs of every student in a variety of ways. These worksheets, many of which are found in the FAST FILE Chapter Resource Masters, are shown in the chart below. • The Prerequisite Skills Workbook provides extra practice on the basic skills students need for success in geometry. • Study Guide and Intervention masters provide worked-out examples as well as practice problems. • Reading to Learn Mathematics masters help students improve reading skills by examining lesson concepts more closely. • Noteables™: Interactive Study Notebook with Foldables™ helps students improve note-taking and study skills. • Skills Practice masters allow students who are progressing at a slower pace to practice concepts using easier problems. Practice masters provide average-level problems for students who are moving at a regular pace. • Each chapter’s Vocabulary Builder master provides students the opportunity to write out key concepts and definitions in their own words. The Proof Builder master provides students the opportunity to write the chapter’s postulates and theorems in their own words. • Enrichment masters offer students the opportunity to extend their learning. Ten Different Options to Meet the Needs of Every Student in a Variety of Ways primarily skills primarily concepts primarily applications BASIC AVERAGE 1 Prerequisite Skills Workbook 2 Study Guide and Intervention 3 Reading to Learn Mathematics 4 NoteablesTM: Interactive Study Notebook with FoldablesTM 5 Skills Practice 6 Vocabulary Builder 7 Proof Builder 8 Parent and Student Study Guide (online) © Glencoe/McGraw-Hill 9 Practice 10 Enrichment vi ADVANCED Geometry: Concepts and Applications 15 NAME DATE PERIOD Reading to Learn Mathematics Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter 15. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Vocabulary Term Found on Page Definition/Description/Example compound statement conjunction contrapositive coordinate proof deductive reasoning dee•DUK•tiv disjunction indirect proof indirect reasoning inverse (continued on the next page) © Glencoe/McGraw-Hill vii Geometry: Concepts and Applications 15 NAME DATE PERIOD Reading to Learn Mathematics Vocabulary Builder (continued) Vocabulary Term Found on Page Definition/Description/Example Law of Detachment Law of Syllogism SIL•oh•jiz•um logically equivalent negation paragraph proof proof proof by contradiction statement truth table truth value two-column proof © Glencoe/McGraw-Hill viii Geometry: Concepts and Applications 15–1 NAME DATE PERIOD Study Guide Logic and Truth Tables A statement is any sentence that is either true or false, but not both. The table below lists different kinds of statements. Term negation conjunction disjunction conditional converse Symbol Definition ~p not p pq p and q pq p or q p→q if p, then q q→p if q, then p Every statement has a truth value. It is convenient to organize the truth values in a truth table like the one shown at the right. Conjunction p q pq T T F F T F T F T F F F Complete a truth table for each compound statement. 1. p q 2. ~p q p q pq p ~p q ~p q T T F F T F T F T T T F T T F F F F T T T F T F T F T T ~( p q) 3. p ~q 4. ~(p q) p q ~q p ~q p q pq T T F F T F T F F T F T F T F F T T F F T F T F T F F F © Glencoe/McGraw-Hill 629 F T T T Geometry: Concepts and Applications 15–1 NAME DATE PERIOD Skills Practice Logic and Truth Tables For Exercises 1–16, use conditionals a, b, c and d. a: A triangle has three sides. b: January is a day of the week. c: 5 5 20 d: Parallel lines do not intersect. Write the statements for each negation. 1. a A triangle does not have three sides. 2. b January is not a day of the week. 3. c 5 5 20 4. d Parallel lines intersect. Write a statement for each conjunction or disjunction. Then find the truth value. 5. a b A triangle has three sides or January is a day of the week; true. 6. a b A triangle has three sides and January is a day of the week; false. 7. a c A triangle has three sides or 5 5 20; true. 8. a c A triangle has three sides and 5 5 20; false. 9. a d A triangle has three sides or parallel lines do not intersect; true. 10. a d A triangle has three sides and parallel lines do not intersect; true. 11. b c January is a day of the week or 5 5 20; false. 12. b c January is a day of the week and 5 5 20; false. 13. b d January is a day of the week or parallel lines do not intersect; true. 14. b d January is a day of the week and parallel lines do not intersect; false. 15. c d 5 5 20 or parallel lines do not intersect; true. 16. c d 5 5 20 and parallel lines do not intersect; false. © Glencoe/McGraw-Hill 630 Geometry: Concepts and Applications 15–1 NAME DATE PERIOD Practice Logic and Truth Tables Use conditionals p, q, r, and s for Exercises 1–9. p: Labor Day is in April. q: A quadrilateral has 4 sides. r: There are 30 days in September. s: (5 3) 3 5 Write the statements for each negation. 1. p Labor Day is not in April. 2. q A quadrilateral does not have 4 sides. 3. r There are not 30 days in September. Write a statement for each conjunction or disjunction. Then find the truth value. 4. p q 5. p q 6. p r 7. q s 8. p s 9. q r Labor Day is in April or a quadrilateral has 4 sides; true. Labor Day is in April and a quadrilateral has 4 sides; false. Labor Day is not in April or there are 30 days in September; true. A quadrilateral does not have 4 sides and (5 3) 3 5; false. Labor Day is in April and (5 3) 3 5; false. A quadrilateral does not have 4 sides or there are not 30 days in September; false. Construct a truth table for each compound statement. 10. p q 11. p q p q ~p ~q ~(p ~q) T T F F T F T F F F T T F T F T F T T T © Glencoe/McGraw-Hill 631 p ~p q ~p q T T F F T F T F F F T F F F T T Geometry: Concepts and Applications 15–1 NAME DATE PERIOD Reading to Learn Mathematics Logic and Truth Tables Key Terms statement a statement that is either true or false, but not both truth value the true or false nature of a statement negation the negative of a statement truth table a convenient way to organize truth values compound statement two or more logic statements joined by and or or inverse a statement formed by negating both p and q in the conditional p → q Reading the Lesson 1. Supply one or two words to complete each sentence. a. Any two statements can be joined to form a compound statement. b. A statement that is formed by joining two statements with the word or is called a disjunction. truth value. d. A statement that is formed by joining two statements with the word and is called a conjunction. e. The statement represented by not p is the negation of p. f. If you negate both p and q in a statement p → q, the new statement is called the inverse. 2. Let p represent “you live in the United States,” q represent “July is a month in the c. The true or false nature of a statement is called its summer,” and r represent “red is a color.” For each exercise, explain what the symbols mean and then write the statement indicated by the symbols. a. p q The symbols mean the conjunction of p and q. You live in the United States and July is a month in the summer. b. p q The symbols mean the disjunction of p and q. You live in the United States or July is a month in the summer. c. q r The symbols mean the conjunction of the negation of q and r. July is not a month in the summer and red is a color. Helping You Remember 3. Prefixes can often help you to remember the meaning of words or to distinguish between similar words. Use the dictionary to find the meanings of the prefixes con and dis. Explain how these meanings can help you remember the difference between a conjunction and a disjunction. Sample answer: Con means together and dis means apart, so a conjunction is an and (or both together) statement and a disjunction is an or statement. © Glencoe/McGraw-Hill 632 Geometry: Concepts and Applications 15–1 NAME DATE PERIOD Enrichment Counterexamples When you make a conclusion after examining several specific cases, you have used inductive reasoning. However, you must be cautious when using this form of reasoning. By finding only one counterexample, you disprove the conclusion. Example: Is the statement 1 1 true when you replace x with 1, x 2, and 3? Is the statement true for all reals? If possible, find a counterexample. 1 1 1 1 1 1, 1, and 1. But when x , then 2. This 1 2 3 2 x counterexample shows that the statement is not always true. Answer each question. 1. The coldest day of the year in Chicago occurred in January for five straight years. Is it safe to conclude that the coldest day in Chicago is always in January? no 2. Suppose John misses the school bus four Tuesdays in a row. Can you safely conclude that John misses the school bus every Tuesday? no 3. Is the equation k 2 k true when you replace k with 1, 2, and 3? Is the equation true for all integers? If possible, find a counterexample. 4. Is the statement 2x x x true when 1 you replace x with 2, 4, and 0.7? Is the statement true for all real numbers? If possible, find a counterexample. It is true for 1, 2, and 3. It is not true for negative integers. Sample: 2 5. Suppose you draw four points A, B, C, and D and then draw A B , B C , C D , and D A . Does this procedure give a quadrilateral always or only sometimes? Explain your answers with figures. only sometimes Example: © Glencoe/McGraw-Hill It is true for all real numbers. 6. Suppose you draw a circle, mark three points on it, and connect them. Will the angles of the triangle be acute? Explain your answers with figures no, only sometimes Example: Counterexample: Counterexample: 633 Geometry: Concepts and Applications 15–2 NAME DATE PERIOD Study Guide Deductive Reasoning Two important laws used frequently in deductive reasoning are the Law of Detachment and the Law of Syllogism. In both cases you reach conclusions based on if-then statements. Law of Detachment Law of Syllogism If p → q is a true conditional and p is true, then q is true. If p → q and q → r are true conditionals, then p → r is also true. Example: Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. (1) If you break an item in a store, you must pay for it. (2) Jill broke a vase in Potter’s Gift Shop. (3) Jill must pay for the vase. Yes, statement (3) follows from statements (1) and (2) by the Law of Detachment. Determine if a valid conclusion can be reached from the two true statements using the Law of Detachment or the Law of Syllogism. If a valid conclusion is possible, state it and the law that is used. If a valid conclusion does not follow, write no valid conclusion. 1. (1) If a number is a whole number, then it is an integer. (2) If a number is an integer, then it is a rational number. If a number is a whole number, then it is a rational number; Syllogism. 2. (1) If a dog eats Dogfood Delights, the dog is happy. (2) Fido is a happy dog. no conclusion 3. (1) If people live in Manhattan, then they live in New York. (2) If people live in New York, then they live in the United States. If people live in Manhattan, then they live in the United States; Syllogism. 4. (1) Angles that are complementary have measures with a sum of 90. (2) A and B are complementary. m A m B 90; Detachment 5. (1) All fish can swim. (2) Fonzo can swim. no conclusion 6. Look for a Pattern Find the next number in the list 83, 77, 71, 65, 59 and make a conjecture about the pattern. 53; Each number is 6 less than the preceding one. © Glencoe/McGraw-Hill 634 Geometry: Concepts and Applications 15–2 NAME DATE PERIOD Skills Practice Deductive Reasoning Use the Law of Detachment to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, write no valid conclusion. 1. (1) If a figure is a triangle, then it is a polygon. (2) The figure is a triangle. The figure is a polygon. 2. (1) If I sell my skis, then I will not be able to go skiing. (2) I did not sell my skis. no valid conclusion 3. (1) If two angles are complementary, then the sum of their measures is 90. (2) Angle A and B are complementary. The sum of the measures of angles A and B is 90. 4. (1) If the measures of the lengths of two sides of a triangle are equal, then the triangle is isosceles. (2) Triangle ABC has two sides with lengths of equal measure. Triangle ABC is isosceles. 5. (1) If it rains, we will not go on a picnic. (2) We do not go on a picnic. no valid conclusion Use the Law of Syllogism to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, write no valid conclusion. 6. (1) If my dog does not bark all night, I will give him a treat. (2) If I give my dog a treat, then he will wag his tail. If my dog does not bark all night, then he will wag his tail. 7. (1) If a polygon has three sides, then the figure is a triangle. (2) If a figure is a triangle, then the sum of the measures of the interior angles is 180. If a polygon has three sides, then the sum of the measures of the interior angles is 180. 8. (1) If the concert is postponed, then I will be out of town. (2) If the concert is postponed, then it will be held in the gym. no valid conclusion 9. (1) All whole numbers are rational numbers. (2) All whole numbers are real numbers. no valid conclusion 10. (1) If the temperature reaches 70°, then the swimming pool will open. (2) If the swimming pool opens, then we will not go to the beach. If the temperature reaches 70°, then we will not go to the beach. © Glencoe/McGraw-Hill 635 Geometry: Concepts and Applications 15–2 NAME DATE PERIOD Practice Deductive Reasoning Determine if a valid conclusion can be reached from the two true statements using the Law of Detachment or the Law of Syllogism. If a valid conclusion is possible, state it and the law that is used. If a valid conclusion does not follow, write no valid conclusion. 1. If Jim is a Texan, then he is an American. Jim is a Texan. Jim is an American; Detachment. 2. If Spot is a dog, then he has four legs. Spot has four legs. no valid conclusion 3. If Rachel lives in Tampa, than Rachel lives in Florida. If Rachel lives in Florida, then Rachel lives in the United States. If Rachel lives in Tampa, then Rachel lives in the United States; Syllogism. 4. If October 12 is a Monday, then October 13 is a Tuesday. October 12 is a Monday. October 13 is a Tuesday; Detachment. 5. If Henry studies his algebra, then he passes the test. If Henry passes the test, then he will get a good grade. If Henry studies his algebra, then he will get a good grade; Syllogism. Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write no valid conclusion. 6. (1) If the measure of an angle is greater than 90, then it is obtuse. (2) m T is greater than 90. (3) T is obtuse. yes; Detachment 7. (1) If Pedro is taking history, then he will study about World War II. (2) Pedro will study about World War II. (3) Pedro is taking history. no valid conclusion 8. (1) If Julie works after school, then she works in a department store. (2) Julie works after school. (3) Julie works in a department store. yes; Detachment 9. (1) If William is reading, then he is reading a magazine. (2) If William is reading a magazine, then he is reading a magazine about computers. (3) If William is reading, then he is reading a magazine about computers. yes; Syllogism 10. Look for a Pattern Tanya likes to burn candles. She has found that, once a candle has burned, she can melt 3 candle stubs, add a new wick, and have one more candle to burn. How many total candles can she burn from a box of 15 candles? 22 © Glencoe/McGraw-Hill 636 Geometry: Concepts and Applications 15–2 NAME DATE PERIOD Reading to Learn Mathematics Deductive Reasoning Key Terms deductive reasoning (dee•DUK•tiv) the process of using facts, rules, definitions, or properties in logical order to reach a conclusion Law of Detachment a logic rule that states “if p → q is a true conditional and p is true, then q is true” Law of Syllogism (SIL•oh•jiz•um) a logic rule that states “if p → q and q → r are true conditionals, then p → r is also true” Reading the Lesson If s, t, and u are three statements, match each description from the list on the left with a symbolic statement from the list on the right. 1. negation of u e a. s u 2. conjunction of s and u g b. If s → t is true and s is true, then t is true. 3. negation of t h c. s → u 4. disjunction of s and u a d. s t 5. Law of Detachment b e. u 6. inverse of u → t j f. If s → t and t → u are true, then s → u is true. 7. inverse of s → u c g. s u 8. conjunction of s and t d h. t 9. Law of Syllogism f i. t 10. negation of t i j. u → t 11. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid. a. (1) Every square is a parallelogram. (2) Every parallelogram is a polygon. (3) Every square is a polygon. yes; Law of Syllogism b. (1) If two lines that lie in the same plane do not intersect, they are parallel. (2) Lines and m lie in plane A and do not intersect. (3) Lines and m are parallel. yes; Law of Detachment Helping You Remember 12. A good way to remember something is to explain it to someone else. Suppose that a classmate is having trouble remembering what the Law of Detachment means. Explain this rule in a way that will help him to understand. Sample answer: The word detach means to take something off of another thing. The Law of Detachment says that when a conditional and its hypothesis are both true, you can detach the conclusion and feel confident that it too is a true statement. © Glencoe/McGraw-Hill 637 Geometry: Concepts and Applications 15–2 NAME DATE PERIOD Enrichment Valid and Faulty Arguments Consider the statements at the right. What conclusions can you make? (1) Boots is a cat. (2) Boots is purring. (3) A cat purrs if it is happy. From statements 1 and 3, it is correct to conclude that Boots purrs if it is happy. However, it is faulty to conclude from only statements 2 and 3 that Boots is happy. The if-then form of statement 3 is If a cat is happy, then it purrs. Advertisers often use faulty logic in subtle ways to help sell their products. By studying the arguments, you can decide whether the argument is valid or faulty. Decide if each argument is valid or faulty. 1. (1) If you buy Tuff Cote luggage, it will survive airline travel. (2) Justin buys Tuff Cote luggage. Conclusion: Justin’s luggage will survive airline travel. valid 2. (1) If you buy Tuff Cote luggage, it will survive airline travel. (2) Justin’s luggage survived airline travel. Conclusion: Justin has Tuff Cote luggage. faulty 3. (1) If you use Clear Line long distance service, you will have clear reception. (2) Anna has clear long distance reception. Conclusion: Anna uses Clear Line long distance service. faulty 4. (1) If you read the book Beautiful Braids, you will be able to make beautiful braids easily. (2) Nancy read the book Beautiful Braids. Conclusion: Nancy can make beautiful braids easily. valid 5. (1) If you buy a word processor, you will be able to write letters faster. (2) Tania bought a word processor. Conclusion: Tania will be able to write letters faster. valid 6. (1) Great swimmers wear AquaLine swimwear. (2) Gina wears AquaLine swimwear. Conclusion: Gina is a great swimmer. faulty 7. Write an example of faulty logic that you have seen in an advertisement. Answers will vary. © Glencoe/McGraw-Hill 638 Geometry: Concepts and Applications 15–3 NAME DATE PERIOD Study Guide Paragraph Proofs A proof is a logical argument in which each statement you make is backed up by a reason that is accepted as true. In a paragraph proof, you write your statements and reasons in paragraph form. Example: Write a paragraph proof for the conjecture. Given: Prove: WXYZ W and X are supplementary. X and Y are supplementary. Y and Z are supplementary. Z and W are supplementary. Z Y and W Z X Y. By the definition of a parallelogram, WX X and Z Y , WZ and X Y are transversals; for For parallels W and X Y , WX and Z Y are transversals. Thus, the parallels WZ consecutive interior angles on the same side of a transversal are supplementary. Therefore, W and X, X and Y, Y and Z, Z and W are supplementary. Write a paragraph proof for each conjecture. 1. Given: Prove: PSU PTR TR SU 2. Given: S P TP DEF and RST are rt. triangles. E and S are ST and right angles. EF ED SR Prove: DEF RST 1. We know that PSU PTR and SU TR . By the Reflexive Property of Congruent Angles,P P. Then SUP TRP by AAS and TP by CPCTC. SP F ST , ED SR , and E and S are right angles. Since 2. We know that E all right angles are congruent, E S. Therefore, by SAS, DEF RST. © Glencoe/McGraw-Hill 639 Geometry: Concepts and Applications 15–3 NAME DATE PERIOD Skills Practice Paragraph Proofs Write a paragraph proof for each conjecture. A 1. If ABD is an isosceles triangle with base BD and C is the midpoint of B D , then ACD ACB. D B C If ABD is an isosceles triangle with base B D , then AD AB . If C is the , then CD CB . AC AC by the Reflexive Property, so midpoint of BD ACD ACB by SSS. 2. If lines a and b are parallel and W X X Y , then WXS YXZ. S W a X b Z Y If lines a and b are parallel, then SWX XYZ since they are alternate interior angles. WXS YXZ since they are vertical angles. Then it is XY , so WXS YXZ by ASA. given that WX 3. If ACDE is an isosceles trapezoid with bases and E D , then AED CDE. AC A C B E D If ACDE is an isosceles trapezoid with bases A C and E D , then the legs CD . Also, an isosceles trapezoid has congruent are congruent, so AE ED by the Reflexive Property, base angles, so AED CDE. Now, ED so AED CDE by SAS. 4. If RSTX is a rhombus, then RXT RST. S R X T If RSTX is a rhombus, then RS RX and XT ST. RT RT by the Reflexive Property, so RXT RST by SSS. © Glencoe/McGraw-Hill 640 Geometry: Concepts and Applications 15–3 NAME DATE PERIOD Practice Paragraph Proofs Write a paragraph proof for each conjecture. 1. If p q and p and q are cut by a transversal t, then 1 and 3 are supplementary. Since p q, we know that 1 2 since they are corresponding angles. We also know that 2 3 180 since they form a linear pair. Therefore, by substitution, 1 3 180. So, 1 and 3 are supplementary. 2. If E bisects B D and A C , then BA CD . Since E bisects B D and A C , we know that BE ED and CE EA. We also know that BEA CED since they are vertical angles. Therefore, BEA DEC by SAS. So, BAE DCE because corresponding parts of congruent triangles are congruent. So, line BA line CD since we have alternate interior angles that are congruent. 3. If 3 4, then ABC is isosceles. We know that 3 1 180 and 2 4 180 since they form linear pairs. Since 3 4, we can write 3 1 180 and 2 3 180. So, 1 180 3 and 2 180 3. Therefore, 1 2 by substitution. This implies that AC BC. So, ABC is isosceles by definition of isosceles. © Glencoe/McGraw-Hill 641 Geometry: Concepts and Applications 15–3 NAME DATE PERIOD Reading to Learn Mathematics Paragraph Proofs Key Terms proof a logical argument used to validate a conjecture in which each statement you make is backed up by a reason that is accepted as true paragraph proof a logical argument used to validate a conjecture in paragraph form Reading the Lesson 1. Complete each sentence with one or two words to make a true statement. a. In a proof, the given information comes from the hypothesis of the conditional. b. A proof is a logical argument in which each statement you make is backed up by a reason. c. A paragraph proof is written in paragraph form. d. One problem-solving strategy that you might use for writing a proof is work backward. e. In mathematics, proofs are used to validate a conjecture. 2. Use the diagram and the information. Write a plan for proving the conjecture. You do not need to write the proof. Given: a b; XY XZ Prove: 1 3 1 a X 3 Y 2 Z b Sample answer: First, use parallel lines and corresponding angles to show that 1 2. Then use the fact that XYZ is isosceles to show that 2 3. Then use the Transitive Property of Congruence to conclude that 1 3. 3. Write a paragraph proof for the conjecture. First, write a plan for the proof. Given: C; A T is tangent to C at T. Prove: CAT is a right triangle. T A C Sample answer: Plan: Since AT is a tangent to the circle, it is perpendicular to a radius at point T. Perpendicular segments form 90° angles. If a triangle has a 90° angle, then it is a right triangle. T is a tangent to C, then it is Proof: By Theorem 14-4, if A . By the definition of perpendicular perpendicular to the radius TC T C , then CTA is a right angle. By the definition of right lines, if AT triangle, if CTA is a right angle, then CAT is a right triangle. Helping You Remember 4. Some students like to use sayings like “My Dear Aunt Sally” to help them remember a mathematical idea. My Dear Aunt Sally stands for multiplication, division, addition, and subtraction for order of operations. Think of a saying to help you remember that definitions, postulates, and theorems can be used to justify statements when you write a proof. Sample answer: Down the Parallel Tracks for definition, postulate, theorem © Glencoe/McGraw-Hill 642 Geometry: Concepts and Applications NAME 15–3 DATE PERIOD Enrichment Logic Problems The following problems can be solved by eliminating possibilities. It may be helpful to use charts such as the one shown in the first problem. Mark an X in the chart to eliminate a possible answer. Solve each problem. 1. Nancy, Olivia, Mario, and Kenji each have one piece of fruit in their school lunch. They have a peach, an orange, a banana, and an apple. Mario does not have a peach or a banana. Olivia and Mario just came from class with the student who has an apple. Kenji and Nancy are sitting next to the student who has a banana. Nancy does not have a peach. Which student has each piece of fruit? Peach Orange Banana Nancy Olivia Mario X X X X X X Apple X X X 2. Victor, Leon, Kasha, and Sheri each play one instrument. They play the viola, clarinet, trumpet, and flute. Sheri does not play the flute. Kasha lives near the student who plays flute and the one who plays trumpet. Leon does not play a brass or wind instrument. Which student plays each instrument? Victor-flute, Leon-viola, Kasha-clarinet, Sheri-trumpet Kenji X X X Nancy-apple, Olivia-banana, Mario-orange, Kenji-peach 3. Mr. Guthrie, Mrs. Hakoi, Mr. Mirza, and Mrs. Riva have jobs of doctor, accountant, teacher, and office manager. Mr. Mirza lives near the doctor and the teacher. Mrs. Riva is not the doctor or the office manager. Mrs. Hakoi is not the accountant or the office manager. Mr. Guthrie went to lunch with the doctor. Mrs. Riva’s son is a high school student and is only seven years younger than his algebra teacher. Which person has each occupation? 4. Yvette, Lana, Boris, and Scott each have a dog. The breeds are collie, beagle, poodle, and terrier. Yvette and Boris walked to the library with the student who has a collie. Boris does not have a poodle or terrier. Scott does not have a collie. Yvette is in math class with the student who has a terrier. Which student has each breed of dog? Yvette, poodle; Lana, collie; Boris, beagle; Scott, terrier Mr. Guthrie-teacher, Mrs. Hakoi-doctor, Mr. Mirza-office manager, Mrs. Riva-accountant © Glencoe/McGraw-Hill 643 Geometry: Concepts and Applications 15–4 NAME DATE PERIOD Study Guide Preparing for Two-Column Proofs Many rules from algebra are used in geometry. Properties of Equality for Real Numbers Reflexive Property Symmetric Property Transitive Property Addition Property Subtraction Property Multiplication Property Division Property Substitution Property Distributive Property aa If a b, then b a. If a b and b c, then a c. If a b, then a c b c. If a b, then a c b c. If a b, then a c b c. a b If a b and c 0, then . c c If a b, then a may be replaced by b in any equation or expression. a(b c) ab ac Example: Prove that if 4x 8 8, then x 0. Given: 4x 8 8 Prove: x 0 Proof: Statements Reasons a. 4x 8 8 b. 4x 0 c. x 0 a. Given b. Addition Property () c. Division Property () Name the property that justifies each statement. 1. Prove that if 3x 9, then x 15. 5 Given: 3x 9 5 Prove: x 15 Proof: Statements Reasons a. 3x 9 a. Given 5 b. 3x 45 b. Multiplication Property () c. x 15 c. Division Property () 2. Prove that if 3x 2 x 8, then x 3. Given: 3x 2 x 8 Prove: x 3 Proof: Statements Reasons a. b. c. d. 3x 2 x 8 2x 2 8 2x 6 x 3 © Glencoe/McGraw-Hill a. b. c. d. Given Subtraction Property () Addition Property () Division Property () 644 Geometry: Concepts and Applications 15–4 NAME DATE PERIOD Skills Practice Preparing for Two-Column Proofs Complete each proof. 1. If m1 m2 and m3 m4, then mABC mROD. Given: m1 m2, m3 m4 Prove: mABC mROD Proof: Statements a. m1 m2, m3 m4 b. mABC m1 m3 mROD m2 m4 c. m1 m3 m2 m4 d. mABC mROD 2. If 7x 6 D R 2 4 O A 1 3 B Reasons C a. Given b. Angle Addition Postulate c. Addition Property of Equality d. Substitution Property of Equality 14, then x 12. Given: 7x 6 14 Prove: x 12 Proof: Statements Reasons 7x a. 14 a. Given b. 7x 84 c. x 12 b. Multiplication Property of Equality c. Division Property of Equality 6 3. If ABC is a right triangle with C a right angle and mA mB, then mA 45. Given: ABC is a right triangle with C a right angle and mA mB. Prove: mA 45 Proof: Statements Reasons a. ABC is a right triangle with C a right angle and mA mB b. mA mB mC 180 c. mC 90 d. mA mB 90 e. mA mA 90 f. 2mA 90 g. mA 45 © Glencoe/McGraw-Hill A C B a. Given b. c. d. e. f. g. Angle Sum Theorem Definition of Right Angle Subtraction Property of Equality Substitution Property of Equality Substitution Property of Equality Division Property of Equality 645 Geometry: Concepts and Applications NAME 15–4 DATE PERIOD Practice Preparing for Two-Column Proofs Name the property or equality that justifies each statement. 1. If mA mB, then mB mA. 2. If x 3 17, then x 14. 3. xy xy 4. If 7x 42, then x 6. 5. If XY YZ XM, then XM YZ XY. 6. 2(x 4) 2x 8 7. If mA mB 90, and mA 30, then 30 mB 90. 8. If x y 3 and y 3 10, then x 10. Symmetric Subtraction Reflective Division Addition Distributive Substitution Transitive Complete each proof by naming the property that justifies each statement. 9. Prove that if 2(x 3) 8, then x 7. Given: 2(x 3) 8 Prove: x 7 Proof: Statements Reasons a. 2(x 3) 8 a. Given b. 2x 6 8 b. Distributive Property c. 2x 14 c. Addition Property () d. x 7 d. Division Property () 10. Prove that if 3x 4 1x 6, then x 4. 2 Given: 3x 4 Prove: x 4 Proof: 1 x 2 6 Statements Reasons a. 3x 4 1x 6 a. Given b. 5x 4 6 b. Subtraction Property () 2 c. 2 5 x 2 c. Addition Property () 10 d. Multiplication Property () d. x 4 © Glencoe/McGraw-Hill 646 Geometry: Concepts and Applications 15–4 NAME DATE PERIOD Reading to Learn Mathematics Preparing for Two-Column Proofs Key Terms two-column proof a deductive argument that contains statements and reasons organized in two columns Reading the Lesson 1. State whether each statement is true or false. If the statement is false, explain why. a. Algebraic properties can be used as reasons in proofs. true b. When you solve an equation, you are using inductive reasoning. False; you are using deductive reasoning. c. In a two-column proof, you must give a reason for each statement. true d. The last statement in a two-column proof is the given information. False; the last statement is what you want to prove. 2. Fill in the missing statements and reasons in the two-column proof. Given: a b, c d Prove: m2 m7 m8 c 1 2 3 4 d 6 5 7 8 a b Proof: Statements Reasons a. a b, c d a. Given b. 2 4 b. Postulate 4-1 Corresponding Angles c. 4 5 c. Theorem 4-1 Alternate Interior Angles d. 2 5 d. Transitive Property of Congruence e. m2 m5 e. Definition of Congruent Angles f. m5 m7 m8 f. Exterior Angle Theorem g. m2 m7 m8 g. Substitution Property of Equality Helping You Remember 3. A good way to remember some terms is to compare them. Write several sentences comparing the similarities and differences between paragraph proofs and two-column proofs. Sample answer: Similarities: Both types of proofs contain a figure, the given information, a statement about what to prove, and a justification for each statement. Difference: A paragraph proof is written in paragraph form, while a two-column proof is written in two columns where one column has the statements and the second column has the reasons. © Glencoe/McGraw-Hill 647 Geometry: Concepts and Applications NAME 15–4 DATE PERIOD Enrichment More Counterexamples Some statements in mathematics can be proven false by counterexamples. Consider the following statement. For any numbers a and b, a b b a. You can prove that this statement is false in general if you can find one example for which the statement is false. Let a 7 and b 3. Substitute these values in the equation above. 7337 4 4 In general, for any numbers a and b, the statement a b b a is false. You can make the equivalent verbal statement: subtraction is not a commutative operation. In each of the following exercises a, b, and c are any numbers. Prove that the statement is false by counterexample. Sample answers are given. 1. a (b c) (a b) c 2. a (b c) (a b) c 6 (4 2) (6 4) 2 6222 40 6 (4 2) (6 4) 2 1.5 2 3 0.75 6 2 3. a b b a 4. a (b c) (a b) (a c) 6446 3 2 6 (4 2) (6 4) (6 2) 6 6 1.5 3 1 4.5 2 3 5. a (bc) (a b)(a c) 6. a2 a2 a4 6 (4 2) (6 4)(6 2) 6 8 (10)(8) 14 80 62 62 64 36 36 1296 72 1296 7. Write the verbal equivalents for Exercises 1, 2, and 3. 1. Subtraction is not an associative operation. 2. Division is not an associative operation. 3. Division is not a commutative operation. 8. For the Distributive Property a(b c) ab ac it is said that multiplication distributes over addition. Exercises 4 and 5 prove that some operations do not distribute. Write a statement for each exercise that indicates this. 4. Division does not distribute over addition. 5. Addition does not distribute over multiplication. © Glencoe/McGraw-Hill 648 Geometry: Concepts and Applications 15–5 NAME DATE PERIOD Study Guide Two-Column Proofs The reasons necessary to complete the following proof are scrambled up below. To complete the proof, number the reasons to match the corresponding statements. Given: Prove: CD ⊥ BE AB ⊥ BE AD CE BD DE AD CE Proof: Statements Reasons ⊥ BE 1. CD 1. Definition of Right Triangle 2. AB ⊥ BE 2. Given 3. 3 and 4 are right angles. 3. Given 2 4. ABD and CDE are right triangles. 4. Definition of Perpendicular Lines CE 5. AD 5. Given 6. BD DE 6. CPCTC 7. ABD CDE 7. In a plane, if two lines are cut by a transversal so that a pair of corresponding angles is congruent, then the lines are parallel. (Postulate 4-2) 9 8. 1 2 8. Given 6 CE 9. AD 9. HL 7 © Glencoe/McGraw-Hill 649 4 1 3 5 8 Geometry: Concepts and Applications 15–5 NAME DATE PERIOD Skills Practice Two-Column Proofs Write a two-column proof. 1. Given: ITC is an isosceles triangle with base IC , TS bisects ITC C S Prove: IS Proof: Statements Sample Answer: Reasons I S C T a. ITC is an isosceles triangle a. Given b. TS bisects ITC b. Given c. IT C T c. Definition of isosceles triangle d. ITS CTS d. Definition of angle bisector e. S T S T e. Reflexive Property f. ITS CTS f. SAS g. IS C S g. CPCTC with base IC . A B D C 2. Given: ABCD is a square. Prove: AC D B Proof: Statements Sample Answer: Reasons a. ABCD is a square. a. Given b. A D BC b. Definition of a square c. ADC and BCD are right c. Definition of a square d. ADC BCD d. Definition of congruent angles e. D C DC e. Reflexive Property f. ∆ADC ∆BCD f. SAS g. A C BD g. CPCTC angles. © Glencoe/McGraw-Hill 650 Geometry: Concepts and Applications 15–5 NAME DATE PERIOD Practice Two-Column Proofs Write a two-column proof. 1. Given: B is the midpoint of AC. Prove: AB CD BD Proof: Statements a. b. c. d. Reasons B is the midpoint of AC. AB BC BC CD BD AB CD BD a. b. c. d. Given Definition of midpoint Segment Addition Postulate Substitution Property () 2. Given: AEC DEB Prove: AEB DEC Proof: Statements Reasons a. AEC DEB a. Given b. mAEC mDEB b. Definition of congruent angles c. mAEC mAEB mBEC, mDEB mDEC mBEC c. Angle Addition Postulate d. mAEB mBEC mDEC mBEC d. Transitive Property () e. mAEB mDEC e. Subtraction Property () f. AEB DEC f. Definition of congruent angles © Glencoe/McGraw-Hill 651 Geometry: Concepts and Applications 15–5 NAME DATE PERIOD Reading to Learn Mathematics Two-Column Proofs Reading the Lesson 1. Determine whether each statement is true or false. If the statement is false, explain why. a. The given information for a proof can be found in the conclusion of the conjecture. False; the given information is found in the hypothesis. b. You cannot use definitions of geometric terms as a reason for a statement in a proof. False; definitions are one of the three things that can be used for a reason. c. If the figure you are given to work with for a proof has overlapping triangles, you can redraw the triangles as separate triangles. true False; the given information is always used in a two-column proof. d. The given information is never used in a two-column proof. 2. Write the statements and reasons for a two-column proof for each set of information. First, write a short plan for your proof. Given: UR , R S UT TS Prove: RUS TUS R Plan: Sample answer: Show that the two triangles are congruent and then use CPCTC. U S T Proof: Statements a. b. c. d. e. U R U T R S T S U SU S RUS TUS RUS TUS Reasons a. b. c. d. e. Given Given Reflexive Prop. of SSS Postulate CPCTC Helping You Remember 3. Sometimes it is helpful to summarize information that you need to remember. Summarize the steps you would take to write a two-column proof. Sample answer: First, write the Given and Prove and draw a diagram for the situation. Look at the given information and mark the diagram with that information. Look at what you are to prove and make a plan for using the given information to reach that conclusion. You can use the work backward strategy as well. Then write each statement and its reason in a logical order to arrive at the conclusion. © Glencoe/McGraw-Hill 652 Geometry: Concepts and Applications 15–5 NAME DATE PERIOD Enrichment Pythagorean Theorem Use the Pythagorean Theorem to find the area of the shaded region in the figure at the right. Think of the figure as four triangles and a square. b c c a a c c b a 2 a b 2 a b b b a area of large square b area of the center square 2 a b a b area of the 4 triangles a a 1 4 2 a b b a b Think of the figure as a large square. c2 a2 a b b2 a b c2 a2 b2 a b a b c2 c2 2 a b c2 a2 b2 2 a b a2 b2 a2 b2 The relationship c2 a2 b2 is true for all right triangles. Use the Pythagorean Theorem to find the area of A, B, and C in each of the following. Then, answer true or false for the statement A B C. 1. 2. C 3. C 1.5 3 C A 9 15 A 2.5 3 3 5 A 3 5 4 4 12 Squares B Equilateral Triangles B 2 3 Semicircles B A: 81, B: 144, C: 225; true © Glencoe/McGraw-Hill A: 2.253 , B: 43 , C: 6.253 ; true 653 A: 1.125, B: 2.000, C: 3.125; true Geometry: Concepts and Applications 15–6 NAME DATE PERIOD Study Guide Coordinate Proofs You can place figures in the coordinate plane and use algebra to prove theorems. The following guidelines for positioning figures can help keep the algebra simple. • • • • Use the origin as a vertex or center. Place at least one side of a polygon on an axis. Keep the figure within the first quadrant if possible. Use coordinates that make computations simple. The Distance Formula, Midpoint Formula, and Slope Formula are useful tools for coordinate proofs. Example: Use a coordinate proof to prove that the diagonals of a rectangle are congruent. Use (0, 0) as one vertex. Place another vertex on the x-axis at (a, 0) and another on the y-axis at (0, b). The fourth vertex must then be (a, b). Use the Distance Formula to find OB and AC. 2 2 ) 02 b ( ) 0 a b2 OB (a 2 2 AC (0 ) a2 b ( ) 0 a b2 Since OB AC, the diagonals are congruent. Name the missing coordinates in terms of the given variables. 1. ABCD is a rectangle. 2. HIJK is a parallelogram. D(a, c) A(a, 0) H (0, 0) K(0, c) J(a, b c) 3. Use a coordinate proof to show that the opposite sides of any parallelogram are congruent. Label the vertices A(0, 0), B(a, 0), C(b, c), and D(a b, c). Then use the Distance Formula to find AB, CD, AC, and BD. 2 ) 02 0 ( ) 0 a 2 a AB (a 2 CD (( a ) b ) b2 c ( ) c a 2 a 2 a ) b ) a2 c ( ) 0 b 2 c2 BD (( 2 AC (b ) 02 c ( ) 0 b 2 c2 So AB CD and AC BD. Therefore, the opposite sides of a parallelogram are congruent. © Glencoe/McGraw-Hill 654 Geometry: Concepts and Applications 15–6 NAME DATE PERIOD Skills Practice Coordinate Proofs Position and label each figure on a coordinate plane. 1–5. Sample answers given. 1. a square with sides m units long 2. a right triangle with legs p and r units long 3. a rhombus with sides c units long 4. an isosceles triangle with base d units long and heights s units long 5. a rectangle with length x units and width y units © Glencoe/McGraw-Hill 655 Geometry: Concepts and Applications 15–6 NAME DATE PERIOD Practice Coordinate Proofs Name the missing coordinates in terms of the given variables. 1. XYZ is a right isosceles triangle. 2. MART is a rhombus. M(0, 0), R(a b, a 2 b2) X(0, 0) Y(a, 0) 3. RECT is a rectangle. 4. DEFG is a parallelogram. R(0, 0) C(a, b) D(0, 0), F(a c, b) 5. Use a coordinate proof to prove that the diagonals of a rhombus are perpendicular. Draw the diagram at the right. Sample proof: a b2 0 b2 2 a 2 slope of AC ab0 ab a b2 0 b2 2 a 2 slope of BD ba ba 2 2 2 2 2 2 b a b a a b 1 ab ba b2 a2 © Glencoe/McGraw-Hill 656 Geometry: Concepts and Applications 15–6 NAME DATE PERIOD Reading to Learn Mathematics Coordinate Proofs Key Terms coordinate proof a proof that uses figures on a coordinate plane Reading the Lesson 1. Complete each sentence with one or two words to make a true statement. a. If you are writing a coordinate proof and need to show that two segments are congruent, Distance Formula a formula you may want to use is the ______________________. b. When drawing a diagram for a coordinate proof, try to place a vertex of the figure at the origin ________. c. If you are writing a coordinate proof and want to show that two segments are parallel, a Slope Formula formula you may want to use is the ___________________. d. When drawing a diagram for a coordinate proof, try to place at least on side of the axis polygon on a(n) ______. e. If you are writing a coordinate proof and want to show that a segment has been bisected, Midpoint Formula a formula you may want to use is the _______________________. first f. When drawing a diagram for a coordinate proof, try to keep the figure in the ______ quadrant. 2. Find the missing coordinates in each figure. Explain how you find the coordinates. a. isosceles triangle b. isosceles trapezoid y y R(?, b) S(?, c) T(b, ?) T(a, ?) S(?, ?) R(?, ?) O x R is (0, b) because the point is on the y–axis; S is (0, 0) because the point is the b origin; T is a, because it 2 is half way between R and S in vertical distance. U(a, ?) x R is (–a, 0) since it is a units in the negative direction horizontally and lies on the x–axis; S is (–b, c) since it is b units in the negative direction horizontally; T is (b, c) since it is c units in a positive vertical direction; U is (a, 0) since it is on the x–axis. Helping You Remember 3. What is an easy way to remember how best to draw a diagram that will help you devise a coordinate proof? Sample answer: A key point in the coordinate plane is the origin. The everyday meaning of origin is place where something begins. So look to see if there is a good way to begin by placing a vertex of the figure at the origin. © Glencoe/McGraw-Hill 657 Geometry: Concepts and Applications 15–6 NAME DATE PERIOD Enrichment Coordinate Proofs with Circles You can prove many theorems about circles by using coordinate geometry. Whenever possible locate the circle so that its center is at the origin. 1. Prove that an angle inscribed in a semicircle is a right angle. Use the figure at right. (Hint: Write an equation for the circle. Use your equation to help show that P) (slope of PB (slope of A ) 1). b0 slope of AP b slope of PB y P (a, b) B (r, O) a ( r) ar b0 b ar ar A(r, O) x O (slope of AP PB ) (slope of ) 2 b 2 b b 2 ar ar a r a 2 b 2 r 2, since (a, b) is on the graph of x 2 y 2 r 2. Therefore b r a 2 2 1. b 2 r 2 a 2, and 2 2 2 a r 2 2 a r 2. Suppose PQ B, Q is between A and B, A and PQ is the geometric mean between A Q and Q B . Prove that P is on the circle that has A B as a diameter. Use the figure at the right. (PQ)2 (AQ) (QB) (b)2 (a r) (r a) b 2 (r a) (r a) b2 r2 a2 Therefore a 2 b 2 r 2, which means that (a, b) is on the circle with the equation x 2 y 2 r 2. This is the circle that has A B as a diameter. © Glencoe/McGraw-Hill 658 y P (a, b) A (r, O) B (r, O) O Q (a, O) x Geometry: Concepts and Applications NAME 15 DATE PERIOD Chapter 15 Test, Form 1A Write the letter for the correct answer in the blank at the right of each problem. For Questions 1–3, refer to the following statements. p: Quadrilateral ABCD has four right angles. q: Opposite sides of quadrilateral ABCD are parallel. r: Quadrilateral ABCD is a rhombus. 1. Which statement is the negation of p? A. Quadrilateral ABCD has three right angles. B. Quadrilateral ABCD is not a square. C. Quadrilateral ABCD could be a trapezoid. D. Quadrilateral ABCD does not have four right angles. 1. 2. How would you write the statement below using symbols? Opposite sides of quadrilateral ABCD are parallel and quadrilateral ABCD is not a rhombus. A. q ∧ r B. q ∨ r C. q ∧ r D. q ∨ r 2. 3. If quadrilateral ABCD is a square, which compound statement is true? A. p ∧ q B. p ∨ r C. q ∨ p D. r ∧ q 3. For Questions 4–5, use the Law of Detachment or Law of Syllogism to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, choose no valid conclusion. 4. (1) All prime numbers except 2 are odd. (2) 37 is an odd number. A. 37 is a prime number. B. 35 is not a prime number. C. The next prime after 37 is 39. D. no valid conclusion 4. 5. (1) In two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar. (2) If two triangles are similar, then the measures of their corresponding sides are proportional. A. If the corresponding sides of two triangles are proportional, then two pairs of corresponding angles are congruent. B. In two triangles, if two pairs of corresponding angles are congruent, then the measures of their corresponding sides are proportional. C. In two triangles, if all three pairs of corresponding angles are congruent, then the two triangles are similar. D. no valid conclusion 5. 6. If ABC at the right is an equilateral triangle, what are the coordinates of point C? A. (2a, a3 ) a a3 C. , 2 2 B. (a, a3 ) a3 D. , a3 2 y C O A(0, 0) B(2a, 0) x 6. 7. Rectangle ABCD has been positioned on a coordinate plane so that its vertices have the coordinates A(0, 0), B(a, 0), C(a, b), and D(0, b). Which expression is the length of diagonal AC? A. a b © Glencoe/McGraw-Hill B. ab 2 b2 C. a 659 D. (a b )2 7. Geometry: Concepts and Applications NAME 15 DATE PERIOD Chapter 15 Test, Form 1A (continued) For Questions 8–11, complete the proof by selecting the B missing information for each corresponding location. E D Given: ABC is a right triangle, and quadrilateral ADEF is a rectangle. A C F Prove: BDE EFC Since quadrilateral ADEF is a rectangle, it is a (Question 8) by the definition of a rectangle. Then DE AF and (Question 9) by the definition of a parallelogram. This means DE AC FE and A B . Then B FEC and (Question 10) C since if two parallel lines are cut by a transversal, then corresponding angles are congruent. Therefore, DBE FEC by (Question 11) Similarity. B. rhombus EF B. A D C. polygon D. parallelogram 8. C. A E ⊥ DF D. D E AF 9. 10. A. EBD B. FEC C. BED D. B 10. 11. A. AA B. SAS C. SSS D. ASA 11. 8. A. square EF 9. A. AD For Questions 12–16, complete the proof below by choosing the statement or reason for each location. Theorem: If two segments from the same exterior point are tangent to a circle, then they are congruent. Given: X A is tangent to C at A; X B is tangent to C at B. Prove: X A XB C A B X Statements Reasons A is tangent to C at A. a. X X B is tangent to C at B. a. Given A and C B ; CAX and b. Draw C (Question 12) are right angles. b. (Question 13) X ; C X CX c. Draw C c. (Question 14) d. A C BC d. All radii of a circle are congruent. e. ACX (Question 15) e. (Question 16) A XB f. X f. CPCTC 12. A. ACB 13. A. B. C. D. B. BXC C. CBX D. AXB Pythagorean Theorem A tangent is perpendicular to a radius at the point of tangency. Definition of tangent Definition of perpendicular 12. 13. 14. A. Transitive B. Substitution C. Symmetric D. Reflexive 14. 15. A. CBX B. CXB C. BCX D. BXC 15. 16. A. HA B. LA C. LL D. HL 16. Bonus The coordinates of the vertices of ABC are A(0, 0), B(2a, 0), and B . C(0, b). Write an equation of the line containing the median to A b a A. y x b B. x 2a © Glencoe/McGraw-Hill C. y 2b 660 b a D. y x Bonus Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Test, Form 1B Write the letter for the correct answer in the blank at the right of each problem. For Questions 1–3, refer to the following statements. p: ABC is an isosceles right triangle. q: In ABC, B is a right angle. r: In ABC, AB BC. 1. Which statement is the negation of p? A. ABC is an equilateral triangle. B. ABC is a right triangle that is not isosceles. C. ABC is a scalene triangle. D. ABC is not an isosceles right triangle. 1. 2. How would you write the statement below using symbols? In ABC, B is not a right angle or AB BC. A. q ∨ r B. q ∧ r C. q ∨ r D. q ∧ r 2. 3. If ABC is equilateral, which compound statement is true? A. p ∨ q B. p ∧ r C. q ∨ r D. q ∧ r 3. For Questions 4–5, use the Law of Detachment or Law of Syllogism to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, choose no valid conclusion. 4. (1) (2) A. C. All right angles are congruent. A and B are right angles. mA 90 B. mA mB A B D. no valid conclusion 4. 5. (1) (2) A. B. C. If a parallelogram is a rhombus, then its diagonals bisect each other. All squares are rhombuses. All squares are parallelograms. All squares have diagonals that bisect each other. If the diagonals of a parallelogram bisect each other, then the parallelogram is a square. D. no valid conclusion 6. ABC at the right is an isosceles triangle. Which ordered pair could be the coordinates of point C? A. (b, a) B. (0, b) C. (a, b) D. (b, 0) y 5. C B(2b, 0) x O A(0, 0) 6. 7. Rectangle WXYZ has been positioned on a coordinate plane so that its vertices have the coordinates W(0, 0), X(2a, 0), Y(2a, 2b), and Z(0, 2b). What are the coordinates of the midpoint of diagonal WY ? A. (a, a) © Glencoe/McGraw-Hill B. (b, b) C. (a, b) 661 ab 2 ab 2 D. , 7. Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Test, Form 1B (continued) For Questions 8–11, complete the proof by selecting the missing information for each corresponding location. DE Given: AB Prove: ABC EDC A D C B E DE , then BAC (Question 8) because when two parallel lines Since AB are cut by a transversal the (Question 9) angles are congruent. Also, BCA (Question 10) because vertical angles are congruent. Therefore, ABC EDC by (Question 11) Similarity. 8. A. CED B. CDE 9. A. corresponding C. alternate interior C. ABC D. BCE B. alternate exterior D. consecutive interior 8. 9. 10. A. BCA B. DCE C. EDC D. CED 10. 11. A. SSS B. SAS C. ASA D. AA 11. For Questions 12–16, complete the proof below by choosing the statement or reason for each location. X Given: AB and C D are diameters of X. Prove: A C B D Reasons B and C D are diameters of X. a. A a. Given A XB ; X C XD b. X b. (Question 12) c. (Question 13) BXD c. (Question 14) d. ACX (Question 15) d. (Question 16) BD e. AC e. CPCTC Chords equidistant from the center of a circle are congruent. All radii of a circle are congruent. Distance Formula Definition of midpoint 13. A. AXC 14. A. B. C. D. B D Statements 12. A. B. C. D. C A B. BXC C. CBX D. AXB The angles opposite congruent sides of a triangle are congruent. Alternate interior angles are congruent. Vertical angles are congruent. The central angles of a circle are congruent. 12. 13. 14. 15. A. XBD B. CXA C. BDX D. XBD 15. 16. A. AA B. SSS C. ASA D. SAS 16. Bonus The coordinates of the vertices of ABC are A(0, 0), B(2a, 0), and C(0, 2b). Write an equation of the line containing the median to B C . A. x y 2b © Glencoe/McGraw-Hill B. x 2a C. y a 662 b a D. y x Bonus Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Test, Form 2A Let p represent “A and B are acute angles,” q represent “A and B are vertical angles,” and r represent “A and B are complementary angles.” 1. 1. Write the statement for the negation of r. 2. Use symbols to represent the disjunction below. A and B are not acute angles or they are vertical angles. 2. 3. Suppose mA 35 and mB 55. What is the truth value of the conjunction p ∧ r? 3. 4. If mA 35 and mB 55, what is the truth value of the conditional q → r? 4. Use the Law of Detachment or the Law of Syllogism to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, write no valid conclusion. 5. (1) If a rectangle is a square, then all four of its sides are congruent. (2) Quadrilateral ABCD has four congruent sides. 5. 6. (1) If two angles are complementary, then the sum of their measures is 90. (2) 1 and 2 are complementary. 6. 7. (1) If two solids are similar, then the ratio of their volumes is equal to the cube of the ratio of their heights. (2) All cubes are similar. 7. For Questions 8–13, complete the proof below by supplying the missing information for each corresponding location. Given: BAC BCA; BD BE Prove: BDC BEA 8. B 9. You know that BAC BCA. So BC (Question 8) because in a triangle (Question 9). Also, B B because (Question 10). It was also given that B D BE . So, BDC (Question 11) by (Question 12). Therefore, BDC BEA by (Question 13). D E 10. A C 11. 12. 13. © Glencoe/McGraw-Hill 663 Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Test, Form 2A (continued) Name the property of equality that justifies each statement. 14. If mABC mBCA 90 and mBCA 32, then mABC 32 90. 14. 15. If PQ BC AB BC, then PQ AB. 15. 16. If a b, then b a. 16. For Questions 17–22, complete the two-column proof below by supplying the statement or reason for each location. Given: Quadrilateral ABCD is a rectangle with YZ ⊥ DX . Prove: ADX ZYD X A 17. B Z D C Y 18. Statements Reasons a. ABCD is a rectangle. a. Given b. DAB is a right angle. b. (Question 17) Z ⊥ DX c. Y c. Given d. DZY is a right angle. d. Definition of perpendicular e. mZDY mZYD 90 e. (Question 18) f. mADX mZDY mADY f. (Question 19) g. mADY 90 g. Definition of rectangle h. mADX mZDY 90 h. (Question 20) i. mZDY mZYD mADX mZDY i. Substitution Property j. mZYD mADX j. (Question 21) k. ADX ZYD k. (Question 22) 19. 20. 21. 22. 23. Quadrilateral ABCD is an isosceles trapezoid. If the coordinates of three of its vertices are A(a, 0), B(b, c), and D(a, 0), find the coordinates of C. 23. 24. Parallelogram PQRS is placed on a coordinate plane to be used for a coordinate proof. What could be the coordinates of its vertices? 24. 25. Refer to Question 24. Show that opposite sides of the parallelogram are congruent. 25. Bonus You are asked to prove the theorem given below, using the figure shown at the right. What specifically do you need to prove in the figure? The altitude drawn to the hypotenuse of a right triangle divides it into two similar right triangles. © Glencoe/McGraw-Hill 664 B A D C Bonus Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Test, Form 2B Let p represent “The area of rectangle ABCD is 50 square inches,” q represent “The perimeter of rectangle ABCD is 30 inches,” and r represent “The length of rectangle ABCD is twice its width.” 1. Write the statement for the negation of q. 1. 2. Use symbols to represent the conjunction below. The area of rectangle ABCD is 50 square inches and its length is not twice its width. 2. 3. If rectangle ABCD is 9 inches long and 6 inches wide, what is the truth value of the disjunction q ∨ p? 3. 4. Suppose rectangle ABCD is 12.5 inches long and 4 inches wide. What is the truth value of the conditional p → r? 4. Use the Law of Detachment or the Law of Syllogism to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, write no valid conclusion. 5. (1) If a triangle is a right triangle, then it has two acute angles. (2) ABC is a right triangle. 5. 6. (1) If three points are noncollinear, then they determine a plane. (2) Points A, B, and C are coplanar. 6. 7. (1) All circles have diameters whose measure is twice the measure of their radii. (2) AB is a radius of B. 7. For Questions 8–13, complete the proof below by supplying the missing information for each corresponding location. Theorem: The angles opposite the congruent sides of an isosceles triangle are congruent. Given: isosceles ABC with AB BC B Prove: A C B BC . Let M be the You know that A midpoint of A C and draw B M . So A M (Question 8) by the (Question 9). Also, B M BM because (Question 10). So, ABM (Question 11) by (Question 12). Therefore, A C by (Question 13). 8. 9. 10. A M C 11. 12. 13. © Glencoe/McGraw-Hill 665 Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Test, Form 2B (continued) Name the property of equality that justifies each statement. , PQ PQ. 14. For any PQ 14. 15. If m1 m2 m1 m3, then m2 m3. 15. 16. If AB CD and CD EF, then AB EF. 16. For Questions 17–22, complete the two-column proof below by supplying the reason for each location. Given: Quadrilateral ABCD is a YB . rectangle with DX A Prove: AXD CYB X D 17. B Y C 18. Statements Reasons a. ABCD is a rectangle. a. Given b. DAX and BCY are right angles. b. (Question 17) c. DAX BCY c. (Question 18) d. ABCD is a parallelogram. d. Definition of rectangle D BC e. A CD B f. A e. (Question 19) g. BYC XBY YB X h. D g. (Question 20) h. Given i. XBY AXD i. (Question 21) j. BYC AXD j. Transitive Property k. AXD CYB k. (Question 22) 19. 20. f. Definition of parallelogram 21. 22. 23. ABCD is a square. If the coordinates of three of its vertices are A(a, 2a), B(a, 2a), and C(a, 0), find the coordinates of D. 23. 24. Isosceles triangle XYZ is placed on a coordinate plane to be used for a coordinate proof. What could be the coordinates of its vertices? 24. 25. Refer to Question 24. Show that the legs of the isosceles triangle are congruent. 25. Bonus You are asked to prove the theorem B D given below, using the figure shown at the right. What specifically do you need to A C prove in the figure? The median drawn to the hypotenuse of a right triangle divides it into two isosceles triangles. © Glencoe/McGraw-Hill 666 Bonus Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Extended Response Assessment Instructions: Demonstrate your knowledge by giving a clear, concise solution for each problem. Be sure to include all relevant drawings and to justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem. 1. Write a paragraph proof of the following theorem. In an isosceles right triangle, the angle bisector of the right angle is the perpendicular bisector of the hypotenuse. 2. Supply a reason for each statement to complete the proof below. Theorem: The measure of an inscribed angle is equal to one-half the measure of its intercepted arc. Given: P with inscribed angle ABC 1 Prove: mABC (mAC ) B C P 2 A Statements Reasons A , PB , and PC ; a. Draw P mAPC mBPC mAPB 360 a. b. AP BP; BP CP b. c. mPBC mPCB; mPAB mPBA c. d. mBPC mPBC mPCB 180; mAPB mPBA mPAB 180 d. e. mBPC 2(mPBC) 180; mAPB 2(mPBA) 180 e. f. mBPC 180 2(mPBC); mAPB 180 2(mPBA) f. g. mAPC [180 2(mPBC)] [180 2(mPBA)] 360 g. h. mAPC 2(mPBC) 2(mPBA) 0 h. i. mAPC 2(mPBC) 2(mPBA) i. j. mAPC 2(mPBC mPBA) j. k. mABC mPBC mPBA k. l. mAPC 2(mABC) m. mAPC mAC n. mAC 2(mABC) 1 o. mABC (mAC ) l. m. n. o. 2 3. Write a two-column proof of the following theorem. The segments joining consecutive midpoints of the sides of a rhombus form a parallelogram. Glencoe/McGraw-Hill 667 B Z D © W A X Y C Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Mid-Chapter Test (Lessons 15–1 through 15–3) Let p represent “x is divisible by 5,” q represent “100 x 120,” and r represent “x is divisible by 10.” 1. Write the statement for the negation of q. 1. 2. For what whole number value(s) of x is the conjunction p ∧ r true? 2. 3. If x 115, what is the truth value of q ∨ p? 3. 4. If x 113, which conditional statements below are true? p→q q→r r→p 4. Use the Law of Detachment or the Law of Syllogism to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, write no valid conclusion. 5. (1) If both pairs of opposite sides in a quadrilateral are congruent, then it is a parallelogram. (2) If a quadrilateral is a parallelogram, then both pairs of its opposite sides are parallel. 5. 6. (1) All regular hexagons are similar. (2) Hexagons ABCDEF and UVWXYZ are regular. 6. 7. (1) If two angles are vertical, then they are congruent. (2) Two right angles are congruent. 7. Determine whether each situation is an example of inductive or deductive reasoning. 8. The numbers 3, 13, and 23 are prime, so Raina conjectures that all numbers with 3 as their ones digit are prime. 8. 9. Mike knows that the product of two odd numbers is an odd number, so he concludes that the product 193 207 is odd. 9. For Questions 10–14, complete the proof by supplying the missing information for each corresponding location. Given: AE DE ; EBC ECB Prove: AB DC A D 10. E You know that EBC ECB, so B C 11. (Question 10) because in a triangle the EB 12. sides opposite congruent angles are E . And DE congruent. You also know that A AEB DEC because (Question 11). So 13. AEB (Question 12) by (Question 13). DC by (Question 14). Therefore, AB © Glencoe/McGraw-Hill 14. 668 Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Quiz A (Lessons 15–1 and 15–2) Let p represent “A hexagon has six sides,” q represent “Skew lines are coplanar,” and r represent “72 12 5.” 1. What is the truth value of the conjunction r ∧ q? 1. 2. What is the truth value of the disjunction p ∨ r? 2. 3. Determine the truth value of the conditional p → q. 3. Use the Law of Detachment or the Law of Syllogism to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, write no valid conclusion. 4. (1) If a quadrilateral is a square, then it is a rectangle. (2) All rectangles have four right angles. 4. 5. (1) If a triangle has one right angle, then it has two acute angles. (2) ABC has two acute angles. 5. 15 NAME DATE PERIOD Chapter 15 Quiz B (Lessons 15–3 through 15–6) Complete the proof of the theorem by supplying the missing information for each corresponding location. Theorem: The diagonal of a rhombus bisects a pair of opposite angles. Given: rhombus ABCD with diagonal BD Prove: BD bisects ABC and ADC. A D B C . Since BC CD DA By the definition of rhombus, AB congruence of segments is reflexive, B D BD . So, ABD CBD by (Question 1). Then, ABD CBD and ADB (Question 2) by CPCTC. Thus, by the (Question 3), BD bisects ABC and ADC. 1. 2. 3. 4. Draw and label a figure for the conjecture below. The medians of an equilateral triangle are congruent. 4. 5. Name the property of equality that justifies the following statement. If mA mB 180 and mA 110, then 110 mB 180. 6. Parallelogram ABCD is placed on a coordinate plane as part of a coordinate proof. If three of the vertices are labeled A(0, 0), B(a, b), and D(c, 0), what are the coordinates of vertex C? © Glencoe/McGraw-Hill 669 5. 6. Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Cumulative Review 1. Write this statement in if-then form. (Lesson 1–4) The sum of two odd integers x and y is even. bisects CXE. 2. In the figure, XD If mDXE 55, find mAXE. (Lessons 3–3 and 3–6) B A X C 1. E D 2. 3. Find the slope of the line passing through points at (2, 7) and (4, 3). (Lesson 4–5) 3. 4. In the figure, ACB DCE and BC CE . What other congruence relation is needed to prove ABC DEC by SAS? (Lesson 5–5) 4. A E C B D 5. If AX is the perpendicular bisector of BC in ABC, then ABC must be a(n) ? triangle with A its ? angle and BC its ? . (Lesson 6–4) 6. Refer to the figure at the right. Find the value of x. (Lesson 7–2) 5. (3x 5) x 115 6. 7. In parallelogram ABCD, mA 2x 30 and mD x. Find mB. (Lesson 8–2) 8. Determine if the pair of triangles shown at the right are similar. If so, state the reason and find the value of x. (Lesson 9–3) 7. x 3 5 8. 10 9. What is the measure of one interior angle of a regular nonagon? (Lesson 10–2) For Questions 10–11, refer to the figure. 10. In the figure, BF is a ? of C. (Lesson 11–1) 11. If mAB 72 and mEF 58, find mAGB. (Lesson 15–3) 9. B A C D G F 10. E 11. 12. Find the lateral area, to the nearest tenth, of a cone with height 9 feet and radius 12 feet. (Lesson 12–4) 12. For Questions 13–14, refer to the theorem and figure below. (Lesson 15–2) Theorem: The diagonals of a rectangle are congruent. 13. What is the hypothesis of the theorem? Glencoe/McGraw-Hill D F E 13. 14. 14. What is to be proved? © C 670 Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Standardized Test Practice (Chapters 1–15) Write the letter for the correct answer in the blank at the right of each problem. 1. What is the intersection of BF and plane EGH in the figure at the right? A. B B. E C. F D. G E A H B D F G C 1. 2. Identify the hypothesis of the converse of the statement below. All right triangles have two acute angles. A. A triangle has two acute angles. B. A triangle has one right angle. C. An isosceles triangle has two acute angles. D. A right triangle can be isosceles. 2. 3. Points R, Q, and T are collinear with point R between points Q and T. If QR 6.8 and RT 9.2, find QT. A. 2.4 B. 3.6 C. 11.5 D. 16 3. 4. If CAT and DAT form a linear pair and CAT is acute, then DAT is what kind of angle? A. acute B. obtuse C. right D. straight 4. 5. Find the value of x in the figure at the right so that m . A. 2.5 B. 17 C. 61 D. 119 7x m (3x 10) n 5. 6. Find the equation of the line passing through the point at (6, 4) and perpendicular to the line y 2x 1. A. y 2x 8 1 2 B. y x 2 1 2 1 2 C. y x 7 D. y x 1 7. Which congruence test can be used to prove ABC CDA? A. SSS B. ASA C. AAS D. SAS 8. In ABC, AB AC and mB 38. Find mA. A. 38 B. 76 C. 104 A B 6. D C 7. 8. D. 142 9. In XYZ, mX 58 and mY 49. List the sides of the triangle in order from least to greatest measure. A. XZ , YZ , XY B. X Y , Y Z , X Z C. YZ , X Y , X Z D. X Z , XY , YZ 9. 10. Quadrilateral WXYZ is a parallelogram whose diagonals intersect at point A. If YA 2t, WA 3t 4, and XZ 5t, find XA. A. 4 B. 9 C. 10 D. 20 10. © Glencoe/McGraw-Hill 671 Geometry: Concepts and Applications 15 NAME DATE PERIOD Chapter 15 Standardized Test Practice (Chapters 1–15) (continued) BC 11. If XY in the figure, find AC. A. 15 B. 25 C. 35 D. 45 A 12 X 9 B 20 Y 11. C 12. If RST XYZ, RS 18, XY 30, and the perimeter of RST is 57, find the perimeter of XYZ. A. 83 B. 95 C. 102 D. 110 12. 13. What is the area of a trapezoid with altitude 12 centimeters and bases 10 centimeters and 18 centimeters long? A. 168 cm2 B. 336 cm2 C. 968 cm2 D. 2160 cm2 13. 14. Which of the following statements is false? A. B. C. D. A is a chord of X. AD mAB 180 mBD mEBD mEA 180 EB is a radius of X. E B X C D 14. 15. To the nearest tenth, what is the volume of a cone with diameter 3 feet and height 5 feet? A. 11.8 ft3 B. 28.3 ft3 C. 35.3 ft3 D. 47.1 ft3 15. 16. The measure of the longer leg of a 30°-60°-90° triangle is 12 inches. What is the measure of the hypotenuse? A. 63 in. B. 83 in. C. 123 in. 17. Find the value of a to the nearest tenth. A. 10.7 B. 11.9 C. 14.4 D. 17.8 18. Find the value of x. A. 6 B. 9 C. 10 D. 12 D. 24 in. 16. B a C 42 16 A 17. C x 12 3 4 18. 19. Use the Law of Detachment to determine a conclusion that follows from statements (1) and (2). (1) Acute angles have measures less than 90. (2) 1 and 2 are acute angles. A. 1 2 B. m1 m2 C. m1 m2 90 D. 1 and 2 both have measure less than 90. 19. 20. Which of the following could be the coordinates of the vertices of parallelogram ABCD? A. A(0, 0), B(2a, 0), C(a, b), D(2a, b) B. A(0, 0), B(a, 0), C(a b, c), D(b, c) C. A(0, 0), B(a, 0), C(a b, c), D(b, c) D. A(0, 0), B(a, 0), C(a, b), D(a, b) 20. © Glencoe/McGraw-Hill 672 Geometry: Concepts and Applications Preparing for Standardized Tests Answer Sheet 1. A B C D E 2. A B C D E 3. A B C D E 4. A B C D E 5. A B C D E 6. A B C D E 7. A B C D E 8. A B C D E 9. / / • • • • 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10. Show your work. © Glencoe/McGraw-Hill A1 Geometry: Concepts and Applications Study Guide NAME DATE © Glencoe/McGraw-Hill p→q q→p if p, then q if q, then p Symbol Definition ~p not p pq p and q pq p or q A2 T F T F T T F F © Glencoe/McGraw-Hill q T F T F p T T F F 3. p ~q q p 1. p q F T F T ~q T T T F pq F T F F p ~q 629 T T F F p 4. ~(p q) T T F F p 2. ~p q Complete a truth table for each compound statement. Every statement has a truth value. It is convenient to organize the truth values in a truth table like the one shown at the right. conditional converse Term negation conjunction disjunction T F F F pq T F T F F T T T ~( p q) T F T T ~p q T F F F Geometry: Concepts and Applications T F T F q F F T T q T F T F T T F F ~p q pq PERIOD Conjunction p A statement is any sentence that is either true or false, but not both. The table below lists different kinds of statements. Logic and Truth Tables 15–1 Skills Practice NAME DATE PERIOD d: Parallel lines do not intersect. © Glencoe/McGraw-Hill 630 Geometry: Concepts and Applications 16. c d 5 5 20 and parallel lines do not intersect; false. 15. c d 5 5 20 or parallel lines do not intersect; true. 14. b d January is a day of the week and parallel lines do not intersect; false 13. b d January is a day of the week or parallel lines do not intersect; true. 12. b c January is a day of the week and 5 5 20; false. 11. b c January is a day of the week or 5 5 20; false. 10. a d A triangle has three sides and parallel lines do not intersect; true. 9. a d A triangle has three sides or parallel lines do not intersect; true. 8. a c A triangle has three sides and 5 5 20; false. 7. a c A triangle has three sides or 5 5 20; true. 6. a b A triangle has three sides and January is a day of the week; false. Write a statement for each conjunction or disjunction. Then find the truth value. 5. a b A triangle has three sides or January is a day of the week; true. Parallel lines intersect. 4. d 5 5 20 3. c January is not a day of the week. 2. b A triangle does not have three sides. Write the statements for each negation. 1. a c: 5 5 20 For Exercises 1–16, use conditionals a, b, c and d. a: A triangle has three sides. b: January is a day of the week. Logic and Truth Tables 15–1 Answers (Lesson 15-1) Geometry: Concepts and Applications Practice NAME DATE PERIOD © Glencoe/McGraw-Hill A quadrilateral does not have 4 sides. 2. q A3 T F T F T T F F F F T T ~p © Glencoe/McGraw-Hill q p 10. p q F T F T ~q F T T T ~(p ~q) T T F F F F T T p ~p F F T F ~p q Geometry: Concepts and Applications T F T F q A quadrilateral does not have 4 sides or there are not 30 days in September; false. 9. q r A quadrilateral does not have 4 sides and (5 3) 3 5; false. 7. q s 11. p q 631 There are not 30 days in September. 3. r Labor Day is in April and a quadrilateral has 4 sides; false. 5. p q Construct a truth table for each compound statement. Labor Day is in April and (5 3) 3 5; false. 8. p s Labor Day is not in April or there are 30 days in September; true. 6. p r Labor Day is in April or a quadrilateral has 4 sides; true. 4. p q Write a statement for each conjunction or disjunction. Then find the truth value. Labor Day is not in April. 1. p Write the statements for each negation. Use conditionals p, q, r, and s for Exercises 1–9. p: Labor Day is in April. q: A quadrilateral has 4 sides. r: There are 30 days in September. s: (5 3) 3 5 Logic and Truth Tables 15–1 NAME statement. negation of p. © Glencoe/McGraw-Hill 632 Geometry: Concepts and Applications conjunction is an and (or both together) statement and a disjunction is an or statement. 3. Prefixes can often help you to remember the meaning of words or to distinguish between similar words. Use the dictionary to find the meanings of the prefixes con and dis. Explain how these meanings can help you remember the difference between a conjunction and a disjunction. Sample answer: Con means together and dis means apart, so a Helping You Remember July is not a month in the summer and red is a color. c. q r The symbols mean the conjunction of the negation of q and r. United States or July is a month in the summer. b. p q The symbols mean the disjunction of p and q. You live in the United States and July is a month in the summer. a. p q The symbols mean the conjunction of p and q. You live in the 2. Let p represent “you live in the United States,” q represent “July is a month in the summer,” and r represent “red is a color.” For each exercise, explain what the symbols mean and then write the statement indicated by the symbols. f. If you negate both p and q in a statement p → q, the new statement is called the inverse. e. The statement represented by not p is the truth value. d. A statement that is formed by joining two statements with the word and is called a conjunction. c. The true or false nature of a statement is called its compound PERIOD b. A statement that is formed by joining two statements with the word or is called a disjunction. a. Any two statements can be joined to form a 1. Supply one or two words to complete each sentence. Reading the Lesson statement a statement that is either true or false, but not both truth value the true or false nature of a statement negation the negative of a statement truth table a convenient way to organize truth values compound statement two or more logic statements joined by and or or inverse a statement formed by negating both p and q in the conditional p → q Key Terms DATE Reading to Learn Mathematics Logic and Truth Tables 15–1 Answers (Lesson 15-1) Geometry: Concepts and Applications 15–2 Study Guide NAME DATE © Glencoe/McGraw-Hill A4 C B © Glencoe/McGraw-Hill D A Example: D A B C Counterexample: 5. Suppose you draw four points A, B, C, and D and then draw A B , B C , C D , and D A . Does this procedure give a quadrilateral always or only sometimes? Explain your answers with figures. only sometimes It is true for 1, 2, and 3. It is not true for negative integers. Sample: 2 3. Is the equation k 2 k true when you replace k with 1, 2, and 3? Is the equation true for all integers? If possible, find a counterexample. 1. The coldest day of the year in Chicago occurred in January for five straight years. Is it safe to conclude that the coldest day in Chicago is always in January? no Answer each question. 633 Example: Geometry: Concepts and Applications Counterexample: 6. Suppose you draw a circle, mark three points on it, and connect them. Will the angles of the triangle be acute? Explain your answers with figures no, only sometimes It is true for all real numbers. 4. Is the statement 2x x x true when 1 you replace x with 2, 4, and 0.7? Is the statement true for all real numbers? If possible, find a counterexample. 2. Suppose John misses the school bus four Tuesdays in a row. Can you safely conclude that John misses the school bus every Tuesday? no Example: Is the statement 1 1 true when you replace x with 1, x 2, and 3? Is the statement true for all reals? If possible, find a counterexample. 1 1 1 1 1 1, 1, and 1. But when x , then 2. This 1 2 3 2 x counterexample shows that the statement is not always true. If p → q is a true conditional and p is true, then q is true. © Glencoe/McGraw-Hill 634 PERIOD Geometry: Concepts and Applications 53; Each number is 6 less than the preceding one. 6. Look for a Pattern Find the next number in the list 83, 77, 71, 65, 59 and make a conjecture about the pattern. 5. (1) All fish can swim. (2) Fonzo can swim. no conclusion m A m B 90; Detachment (2) A and B are complementary. 4. (1) Angles that are complementary have measures with a sum of 90. the United States; Syllogism. 3. (1) If people live in Manhattan, then they live in New York. (2) If people live in New York, then they live in the United States. If people live in Manhattan, then they live in 2. (1) If a dog eats Dogfood Delights, the dog is happy. (2) Fido is a happy dog. no conclusion is a whole number, then it is a rational number; Syllogism. 1. (1) If a number is a whole number, then it is an integer. (2) If a number is an integer, then it is a rational number. If a number Determine if a valid conclusion can be reached from the two true statements using the Law of Detachment or the Law of Syllogism. If a valid conclusion is possible, state it and the law that is used. If a valid conclusion does not follow, write no valid conclusion. Yes, statement (3) follows from statements (1) and (2) by the Law of Detachment. (1) If you break an item in a store, you must pay for it. (2) Jill broke a vase in Potter’s Gift Shop. (3) Jill must pay for the vase. Example: Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. Law of Syllogism If p → q and q → r are true conditionals, then p → r is also true. Law of Detachment Two important laws used frequently in deductive reasoning are the Law of Detachment and the Law of Syllogism. In both cases you reach conclusions based on if-then statements. PERIOD Deductive Reasoning DATE When you make a conclusion after examining several specific cases, you have used inductive reasoning. However, you must be cautious when using this form of reasoning. By finding only one counterexample, you disprove the conclusion. Enrichment NAME Counterexamples 15–1 Answers (Lessons 15-1 and 15-2) Geometry: Concepts and Applications 15–2 Practice NAME DATE © Glencoe/McGraw-Hill A5 © Glencoe/McGraw-Hill 635 Geometry: Concepts and Applications If the temperature reaches 70°, then we will not go to the beach. (2) If the swimming pool opens, then we will not go to the beach. 10. (1) If the temperature reaches 70°, then the swimming pool will open. no valid conclusion (2) All whole numbers are real numbers. 9. (1) All whole numbers are rational numbers. no valid conclusion (2) If the concert is postponed, then it will be held in the gym. 8. (1) If the concert is postponed, then I will be out of town. If a polygon has three sides, then the sum of the measures of the interior angles is 180. (2) If a figure is a triangle, then the sum of the measures of the interior angles is 180. 7. (1) If a polygon has three sides, then the figure is a triangle. If my dog does not bark all night, then he will wag his tail. (2) If I give my dog a treat, then he will wag his tail. 6. (1) If my dog does not bark all night, I will give him a treat. Use the Law of Syllogism to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, write no valid conclusion. no valid conclusion (2) We do not go on a picnic. 5. (1) If it rains, we will not go on a picnic. Triangle ABC is isosceles. is isosceles. (2) Triangle ABC has two sides with lengths of equal measure. 4. (1) If the measures of the lengths of two sides of a triangle are equal, then the triangle The sum of the measures of angles A and B is 90. (2) Angle A and B are complementary. 3. (1) If two angles are complementary, then the sum of their measures is 90. no valid conclusion (2) I did not sell my skis. 2. (1) If I sell my skis, then I will not be able to go skiing. The figure is a polygon. (2) The figure is a triangle. 1. (1) If a figure is a triangle, then it is a polygon. PERIOD © Glencoe/McGraw-Hill 636 Geometry: Concepts and Applications 10. Look for a Pattern Tanya likes to burn candles. She has found that, once a candle has burned, she can melt 3 candle stubs, add a new wick, and have one more candle to burn. How many total candles can she burn from a box of 15 candles? 22 yes; Syllogism 9. (1) If William is reading, then he is reading a magazine. (2) If William is reading a magazine, then he is reading a magazine about computers. (3) If William is reading, then he is reading a magazine about computers. 8. (1) If Julie works after school, then she works in a department store. (2) Julie works after school. (3) Julie works in a department store. yes; Detachment 7. (1) If Pedro is taking history, then he will study about World War II. (2) Pedro will study about World War II. (3) Pedro is taking history. no valid conclusion 6. (1) If the measure of an angle is greater than 90, then it is obtuse. (2) m T is greater than 90. (3) T is obtuse. yes; Detachment Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write no valid conclusion. If Henry studies his algebra, then he will get a good grade; Syllogism. 5. If Henry studies his algebra, then he passes the test. If Henry passes the test, then he will get a good grade. October 13 is a Tuesday; Detachment. 4. If October 12 is a Monday, then October 13 is a Tuesday. October 12 is a Monday. If Rachel lives in Tampa, then Rachel lives in the United States; Syllogism. 3. If Rachel lives in Tampa, than Rachel lives in Florida. If Rachel lives in Florida, then Rachel lives in the United States. no valid conclusion 2. If Spot is a dog, then he has four legs. Spot has four legs. Jim is an American; Detachment. 1. If Jim is a Texan, then he is an American. Jim is a Texan. Determine if a valid conclusion can be reached from the two true statements using the Law of Detachment or the Law of Syllogism. If a valid conclusion is possible, state it and the law that is used. If a valid conclusion does not follow, write no valid conclusion. PERIOD Use the Law of Detachment to determine a conclusion that follows from statements (1) and (2). If a valid conclusion does not follow, write no valid conclusion. DATE Deductive Reasoning Skills Practice NAME Deductive Reasoning 15–2 Answers (Lesson 15-2) Geometry: Concepts and Applications © Glencoe/McGraw-Hill PERIOD A6 © Glencoe/McGraw-Hill 637 Geometry: Concepts and Applications detach means to take something off of another thing. The Law of Detachment says that when a conditional and its hypothesis are both true, you can detach the conclusion and feel confident that it too is a true statement. 12. A good way to remember something is to explain it to someone else. Suppose that a classmate is having trouble remembering what the Law of Detachment means. Explain this rule in a way that will help him to understand. Sample answer: The word Helping You Remember b. (1) If two lines that lie in the same plane do not intersect, they are parallel. (2) Lines ᐉ and m lie in plane A and do not intersect. (3) Lines ᐉ and m are parallel. yes; Law of Detachment 11. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid. a. (1) Every square is a parallelogram. (2) Every parallelogram is a polygon. (3) Every square is a polygon. yes; Law of Syllogism If s, t, and u are three statements, match each description from the list on the left with a symbolic statement from the list on the right. 1. negation of u e a. s u 2. conjunction of s and u g b. If s → t is true and s is true, then t is true. 3. negation of t h c. s → u 4. disjunction of s and u a d. s t 5. Law of Detachment b e. u 6. inverse of u → t j f. If s → t and t → u are true, then s → u is true. 7. inverse of s → u c g. s u 8. conjunction of s and t d h. t 9. Law of Syllogism f i. t 10. negation of t i j. u → t Reading the Lesson deductive reasoning (dee•DUK•tiv) the process of using facts, rules, definitions, or properties in logical order to reach a conclusion Law of Detachment a logic rule that states “if p → q is a true conditional and p is true, then q is true” Law of Syllogism (SIL•oh•jiz•um) a logic rule that states “if p → q and q → r are true conditionals, then p → r is also true” Key Terms DATE Reading to Learn Mathematics NAME Deductive Reasoning 15–2 Enrichment NAME (1) Boots is a cat. (2) Boots is purring. (3) A cat purrs if it is happy. DATE © Glencoe/McGraw-Hill Answers will vary. 7. Write an example of faulty logic that you have seen in an advertisement. 5. (1) If you buy a word processor, you will be able to write letters faster. (2) Tania bought a word processor. Conclusion: Tania will be able to write letters faster. valid 3. (1) If you use Clear Line long distance service, you will have clear reception. (2) Anna has clear long distance reception. Conclusion: Anna uses Clear Line long distance service. faulty 1. (1) If you buy Tuff Cote luggage, it will survive airline travel. (2) Justin buys Tuff Cote luggage. Conclusion: Justin’s luggage will survive airline travel. valid Decide if each argument is valid or faulty. PERIOD 638 faulty Geometry: Concepts and Applications 6. (1) Great swimmers wear AquaLine swimwear. (2) Gina wears AquaLine swimwear. Conclusion: Gina is a great swimmer. 4. (1) If you read the book Beautiful Braids, you will be able to make beautiful braids easily. (2) Nancy read the book Beautiful Braids. Conclusion: Nancy can make beautiful braids easily. valid 2. (1) If you buy Tuff Cote luggage, it will survive airline travel. (2) Justin’s luggage survived airline travel. Conclusion: Justin has Tuff Cote luggage. faulty Advertisers often use faulty logic in subtle ways to help sell their products. By studying the arguments, you can decide whether the argument is valid or faulty. From statements 1 and 3, it is correct to conclude that Boots purrs if it is happy. However, it is faulty to conclude from only statements 2 and 3 that Boots is happy. The if-then form of statement 3 is If a cat is happy, then it purrs. Consider the statements at the right. What conclusions can you make? Valid and Faulty Arguments 15–2 Answers (Lesson 15-2) Geometry: Concepts and Applications 15–3 Skills Practice NAME © Glencoe/McGraw-Hill WXYZ W and X are supplementary. X and Y are supplementary. Y and Z are supplementary. Z and W are supplementary. A7 SP TP PSU PTR TR SU DEF and RST are rt. triangles. E and S are ST and right angles. EF ED SR Prove: DEF RST 2. Given: © Glencoe/McGraw-Hill 639 Geometry: Concepts and Applications 1. We know that PSU PTR and SU TR . By the Reflexive Property of Congruent Angles,P P. Then SUP TRP by AAS and TP by CPCTC. SP F ST , ED S R , and E and S are right angles. Since 2. We know that E all right angles are congruent, E S. Therefore, by SAS, DEF RST. Prove: 1. Given: Write a paragraph proof for each conjecture. Z Y and W Z X Y. By the definition of a parallelogram, WX X and Z Y , WZ and X Y are transversals; for For parallels W and X Y , WX and Z Y are transversals. Thus, the parallels WZ consecutive interior angles on the same side of a transversal are supplementary. Therefore, W and X, X and Y, Y and Z, Z and W are supplementary. Given: Prove: Example: Write a paragraph proof for the conjecture. D DATE C A B PERIOD Y S X Z W b a E A B C D X R T S © Glencoe/McGraw-Hill 640 Geometry: Concepts and Applications If RSTX is a rhombus, then RS RX and XT ST. RT RT by the Reflexive Property, so RXT RST by SSS. 4. If RSTX is a rhombus, then RXT RST. If ACDE is an isosceles trapezoid with bases A C and E D , then the legs CD . Also, an isosceles trapezoid has congruent are congruent, so AE ED by the Reflexive Property, base angles, so AED CDE. Now, ED so AED CDE by SAS. 3. If ACDE is an isosceles trapezoid with bases and E D , then AED CDE. AC If lines a and b are parallel, then SWX XYZ since they are alternate interior angles. WXS YXZ since they are vertical angles. Then it is X Y , so WXS YXZ by ASA. given that WX 2. If lines a and b are parallel and WX X Y , then WXS YXZ. If ABD is an isosceles triangle with base B D , then AD A B . If C is the D , then CD C B . AC AC by the Reflexive Property, so midpoint of B ACD ACB by SSS. 1. If ABD is an isosceles triangle with base BD and C is the midpoint of B D , then ACD ACB. Write a paragraph proof for each conjecture. PERIOD Paragraph Proofs DATE A proof is a logical argument in which each statement you make is backed up by a reason that is accepted as true. In a paragraph proof, you write your statements and reasons in paragraph form. Study Guide NAME Paragraph Proofs 15–3 Answers (Lesson 15-3) Geometry: Concepts and Applications Practice DATE Reading to Learn Mathematics NAME © Glencoe/McGraw-Hill A8 © Glencoe/McGraw-Hill 3. If 3 4, then ABC is isosceles. 2. If E bisects B D and A C , then BA CD . 641 Geometry: Concepts and Applications We know that 3 1 180 and 2 4 180 since they form linear pairs. Since 3 4, we can write 3 1 180 and 2 3 180. So, 1 180 3 and 2 180 3. Therefore, 1 2 by substitution. This implies that AC BC. So, ABC is isosceles by definition of isosceles. Since E bisects B D and A C , we know that BE ED and CE EA. We also know that BEA CED since they are vertical angles. Therefore, BEA DEC by SAS. So, BAE DCE because corresponding parts of congruent triangles are congruent. So, line BA line CD since we have alternate interior angles that are congruent. Since p q, we know that 1 2 since they are corresponding angles. We also know that 2 3 180 since they form a linear pair. Therefore, by substitution, 1 3 180. So, 1 and 3 are supplementary. PERIOD 1 Y 2 Z X 3 b a C T A © Glencoe/McGraw-Hill postulate, theorem 642 Geometry: Concepts and Applications 4. Some students like to use sayings like “My Dear Aunt Sally” to help them remember a mathematical idea. My Dear Aunt Sally stands for multiplication, division, addition, and subtraction for order of operations. Think of a saying to help you remember that definitions, postulates, and theorems can be used to justify statements when you write a proof. Sample answer: Down the Parallel Tracks for definition, Sample answer: Plan: Since AT is a tangent to the circle, it is perpendicular to a radius at point T. Perpendicular segments form 90° angles. If a triangle has a 90° angle, then it is a right triangle. T is a tangent to C, then it is Proof: By Theorem 14-4, if A . By the definition of perpendicular perpendicular to the radius TC T C , then CTA is a right angle. By the definition of right lines, if AT triangle, if CTA is a right angle, then CAT is a right triangle. Helping You Remember 3. Write a paragraph proof for the conjecture. First, write a plan for the proof. Given: C; A T is tangent to C at T. Prove: CAT is a right triangle. Sample answer: First, use parallel lines and corresponding angles to show that 1 2. Then use the fact that XYZ is isosceles to show that 2 3. Then use the Transitive Property of Congruence to conclude that 1 3. 2. Use the diagram and the information. Write a plan for proving the conjecture. You do not need to write the proof. Given: a b; XY XZ Prove: 1 3 1. Complete each sentence with one or two words to make a true statement. a. In a proof, the given information comes from the hypothesis of the conditional. b. A proof is a logical argument in which each statement you make is backed up by a reason. c. A paragraph proof is written in paragraph form. d. One problem-solving strategy that you might use for writing a proof is work backward. e. In mathematics, proofs are used to validate a conjecture. Reading the Lesson proof a logical argument used to validate a conjecture in which each statement you make is backed up by a reason that is accepted as true paragraph proof a logical argument used to validate a conjecture in paragraph form Paragraph Proofs 15–3 Key Terms PERIOD 1. If p q and p and q are cut by a transversal t, then 1 and 3 are supplementary. DATE Write a paragraph proof for each conjecture. Paragraph Proofs 15–3 NAME Answers (Lesson 15-3) Geometry: Concepts and Applications 15–4 Study Guide NAME © Glencoe/McGraw-Hill A9 X X X X X X X X Mario X X X Kenji © Glencoe/McGraw-Hill Mr. Guthrie-teacher, Mrs. Hakoi-doctor, Mr. Mirza-office manager, Mrs. Riva-accountant 3. Mr. Guthrie, Mrs. Hakoi, Mr. Mirza, and Mrs. Riva have jobs of doctor, accountant, teacher, and office manager. Mr. Mirza lives near the doctor and the teacher. Mrs. Riva is not the doctor or the office manager. Mrs. Hakoi is not the accountant or the office manager. Mr. Guthrie went to lunch with the doctor. Mrs. Riva’s son is a high school student and is only seven years younger than his algebra teacher. Which person has each occupation? X Olivia Nancy Nancy-apple, Olivia-banana, Mario-orange, Kenji-peach Apple Banana Orange Peach 1. Nancy, Olivia, Mario, and Kenji each have one piece of fruit in their school lunch. They have a peach, an orange, a banana, and an apple. Mario does not have a peach or a banana. Olivia and Mario just came from class with the student who has an apple. Kenji and Nancy are sitting next to the student who has a banana. Nancy does not have a peach. Which student has each piece of fruit? Solve each problem. 643 Geometry: Concepts and Applications Yvette, poodle; Lana, collie; Boris, beagle; Scott, terrier 4. Yvette, Lana, Boris, and Scott each have a dog. The breeds are collie, beagle, poodle, and terrier. Yvette and Boris walked to the library with the student who has a collie. Boris does not have a poodle or terrier. Scott does not have a collie. Yvette is in math class with the student who has a terrier. Which student has each breed of dog? Victor-flute, Leon-viola, Kasha-clarinet, Sheri-trumpet 2. Victor, Leon, Kasha, and Sheri each play one instrument. They play the viola, clarinet, trumpet, and flute. Sheri does not play the flute. Kasha lives near the student who plays flute and the one who plays trumpet. Leon does not play a brass or wind instrument. Which student plays each instrument? aa If a b, then b a. If a b and b c, then a c. If a b, then a c b c. If a b, then a c b c. If a b, then a c b c. a b If a b and c 0, then . c c If a b, then a may be replaced by b in any equation or expression. a(b c) ab ac 9 3x 2 x 8 2x 2 8 2x 6 x 3 © Glencoe/McGraw-Hill a. b. c. d. PERIOD a. Given b. Multiplication Property () c. Division Property () Reasons Geometry: Concepts and Applications Given Subtraction Property () Addition Property () Division Property () 644 a. b. c. d. 2. Prove that if 3x 2 x 8, then x 3. Given: 3x 2 x 8 Prove: x 3 Proof: Statements Reasons a. b. 3x 45 c. x 15 3 x 5 1. Prove that if 3x 9, then x 15. 5 Given: 3x 9 5 Prove: x 15 Proof: Statements DATE a. Given b. Addition Property () c. Division Property () Name the property that justifies each statement. a. 4x 8 8 b. 4x 0 c. x 0 Example: Prove that if 4x 8 8, then x 0. Given: 4x 8 8 Prove: x 0 Proof: Statements Reasons Distributive Property Reflexive Property Symmetric Property Transitive Property Addition Property Subtraction Property Multiplication Property Division Property Substitution Property Properties of Equality for Real Numbers Many rules from algebra are used in geometry. PERIOD Preparing for Two-Column Proofs DATE The following problems can be solved by eliminating possibilities. It may be helpful to use charts such as the one shown in the first problem. Mark an X in the chart to eliminate a possible answer. Enrichment NAME Logic Problems 15–3 Answers (Lessons 15-3 and 15-4) Geometry: Concepts and Applications 1 3 2 4 D 15–4 Practice NAME DATE © Glencoe/McGraw-Hill A10 14 B C O d. Subtraction Property of Equality e. Substitution Property of Equality f. Substitution Property of Equality g. Division Property of Equality d. mA mB 90 e. mA mA 90 f. 2mA 90 g. mA 45 Geometry: Concepts and Applications c. Definition of Right Angle 645 b. Angle Sum Theorem B c. mC 90 a. Given C A 10 6 © Glencoe/McGraw-Hill d. x 4 c. b. 5x 4 6 2 5 x 2 a. 3x 4 1 x 2 Statements Given: 3x 4 1x 6 2 Prove: x 4 Proof: 646 Geometry: Concepts and Applications d. Multiplication Property () c. Addition Property () b. Subtraction Property () a. Given Reasons d. Division Property () d. x 7 10. Prove that if 3x 4 1x 6, then x 4. c. Addition Property () c. 2x 14 b. Distributive Property b. 2x 6 8 a. Given b. Multiplication Property of Equality c. Division Property of Equality a. Given a. 2(x 3) 8 Reasons 2 9. Prove that if 2(x 3) 8, then x 7. Given: 2(x 3) 8 Prove: x 7 Proof: Reasons a. ABC is a right triangle with C a right angle and mA mB b. mA mB mC 180 © Glencoe/McGraw-Hill Transitive 8. If x y 3 and y 3 10, then x 10. Distributive 6. 2(x 4) 2x 8 Division 4. If 7x 42, then x 6. Subtraction PERIOD 2. If x 3 17, then x 14. Complete each proof by naming the property that justifies each statement. Substitution 7. If mA mB 90, and mA 30, then 30 mB 90. Addition 5. If XY YZ XM, then XM YZ XY. Reflective 3. xy xy Symmetric 1. If mA mB, then mB mA. Statements c. Addition Property of Equality d. Substitution Property of Equality a. Given b. Angle Addition Postulate Reasons 3. If ABC is a right triangle with C a right angle and mA mB, then mA 45. Given: ABC is a right triangle with C a right angle and mA mB. Prove: mA 45 Proof: Statements Reasons b. 7x 84 c. x 12 a. 7x 6 Prove: x 12 Proof: Statements 6 7x Given: 14 6 7x 2. If 14, then x 12. d. mABC mROD a. m1 m2, m3 m4 b. mABC m1 m3 mROD m2 m4 c. m1 m3 m2 m4 Given: m1 m2, m3 m4 Prove: mABC mROD Proof: Statements 1. If m1 m2 and m3 m4, then mABC mROD. Name the property or equality that justifies each statement. R PERIOD Complete each proof. A DATE Preparing for Two-Column Proofs Skills Practice NAME Preparing for Two-Column Proofs 15–4 Answers (Lesson 14-4) Geometry: Concepts and Applications PERIOD © Glencoe/McGraw-Hill true A11 c. Theorem 4-1 Alternate Interior Angles d. Transitive Property of Congruence e. Definition of Congruent Angles f. Exterior Angle Theorem g. Substitution Property of Equality c. 4 5 d. 2 5 e. m2 m5 f. m5 m7 m8 g. m2 m7 m8 © Glencoe/McGraw-Hill 647 Geometry: Concepts and Applications figure, the given information, a statement about what to prove, and a justification for each statement. Difference: A paragraph proof is written in paragraph form, while a two-column proof is written in two columns where one column has the statements and the second column has the reasons. 3. A good way to remember some terms is to compare them. Write several sentences comparing the similarities and differences between paragraph proofs and two-column proofs. Sample answer: Similarities: Both types of proofs contain a Helping You Remember b. Postulate 4-1 Corresponding Angles b. 2 4 b a a. Given 8 7 Reasons 5 6 a. a b, c d 3 4 1 2 d Proof: Statements 2. Fill in the missing statements and reasons in the two-column proof. Given: a b, c d Prove: m2 m7 m8 c False; the last d. The last statement in a two-column proof is the given information. statement is what you want to prove. true False; you are c. In a two-column proof, you must give a reason for each statement. using deductive reasoning. b. When you solve an equation, you are using inductive reasoning. a. Algebraic properties can be used as reasons in proofs. 1. State whether each statement is true or false. If the statement is false, explain why. Reading the Lesson two-column proof a deductive argument that contains statements and reasons organized in two columns Key Terms DATE Reading to Learn Mathematics NAME Preparing for Two-Column Proofs 15–4 Enrichment DATE PERIOD 2 62 62 64 36 36 1296 72 1296 6. a2 a2 a4 6 (4 2) (6 4) (6 2) 6 6 1.5 3 1 4.5 4. a (b c) (a b) (a c) 6 2 1.5 2 3 0.75 6 (4 2) (6 4) 2 2. a (b c) (a b) c © Glencoe/McGraw-Hill 648 Geometry: Concepts and Applications 4. Division does not distribute over addition. 5. Addition does not distribute over multiplication. 8. For the Distributive Property a(b c) ab ac it is said that multiplication distributes over addition. Exercises 4 and 5 prove that some operations do not distribute. Write a statement for each exercise that indicates this. 1. Subtraction is not an associative operation. 2. Division is not an associative operation. 3. Division is not a commutative operation. 7. Write the verbal equivalents for Exercises 1, 2, and 3. 6 (4 2) (6 4)(6 2) 6 8 (10)(8) 14 80 3 5. a (bc) (a b)(a c) 3 2 6446 3. a b b a 6 (4 2) (6 4) 2 6222 40 1. a (b c) (a b) c In each of the following exercises a, b, and c are any numbers. Prove that the statement is false by counterexample. Sample answers are given. In general, for any numbers a and b, the statement a b b a is false. You can make the equivalent verbal statement: subtraction is not a commutative operation. 7337 4 4 Let a 7 and b 3. Substitute these values in the equation above. You can prove that this statement is false in general if you can find one example for which the statement is false. For any numbers a and b, a b b a. Some statements in mathematics can be proven false by counterexamples. Consider the following statement. More Counterexamples 15–4 NAME Answers (Lesson 14-4) Geometry: Concepts and Applications Study Guide NAME DATE © Glencoe/McGraw-Hill Statements AD CE A12 8. Given 7 9. HL 649 8. 1 2 CE 9. AD © Glencoe/McGraw-Hill 6 3 Geometry: Concepts and Applications 7. In a plane, if two lines are cut by a transversal so that a pair of corresponding angles is congruent, then the lines are parallel. (Postulate 4-2) 9 7. ABD CDE 8 5 6. CPCTC 5. Given 6. BD DE CE 5. AD 4. Definition of Perpendicular Lines 4. ABD and CDE are right triangles. 2 3. Given 3. 3 and 4 are right angles. 4 2. Given 1 1. Definition of Right Triangle Reasons PERIOD 2. AB ⊥ BE ⊥ BE 1. CD Proof: Prove: Given: CD ⊥ BE AB ⊥ BE AD CE BD DE The reasons necessary to complete the following proof are scrambled up below. To complete the proof, number the reasons to match the corresponding statements. Two-Column Proofs 15–5 Skills Practice NAME Geometry: Concepts and Applications g. CPCTC g. A C BD 650 f. SAS f. ∆ADC ∆BCD © Glencoe/McGraw-Hill e. Reflexive Property e. D C DC d. ADC BCD C d. Definition of congruent angles c. Definition of a square c. ADC and BCD are right angles. b. Definition of a square b. A D BC D a. Given Reasons a. ABCD is a square. Statements Sample Answer: 2. Given: ABCD is a square. Prove: AC B D Proof: B f. SAS f. ITS CTS A e. Reflexive Property e. S T S T g. CPCTC d. Definition of angle bisector d. ITS CTS g. IS C S c. Definition of isosceles triangle C c. IT C T T S PERIOD b. Given a. Given I DATE b. TS bisects ITC . with base IC a. ITC is an isosceles triangle 1. Given: ITC is an isosceles triangle with base IC , TS bisects ITC C S Prove: IS Proof: Statements Sample Answer: Reasons Write a two-column proof. Two-Column Proofs 15–5 Answers (Lesson 15-5) Geometry: Concepts and Applications Practice NAME © Glencoe/McGraw-Hill B is the midpoint of AC. AB BC BC CD BD AB CD BD A13 Geometry: Concepts and Applications f. Definition of congruent angles f. AEB DEC © Glencoe/McGraw-Hill e. Subtraction Property () c. Angle Addition Postulate c. mAEC mAEB mBEC, mDEB mDEC mBEC e. mAEB mDEC b. Definition of congruent angles b. mAEC mDEB d. Transitive Property () a. Given a. AEC DEB d. mAEB mBEC mDEC mBEC Reasons 651 PERIOD Given Definition of midpoint Segment Addition Postulate Substitution Property () Reasons a. b. c. d. DATE Statements 2. Given: AEC DEB Prove: AEB DEC Proof: a. b. c. d. Statements 1. Given: B is the midpoint of AC. Prove: AB CD BD Proof: Write a two-column proof. Two-Column Proofs 15–5 DATE Reading to Learn Mathematics NAME PERIOD U R U T R S T S U SU S RUS TUS RUS TUS a. e. d. c. b. Given Given Reflexive Prop. of SSS Postulate CPCTC Reasons U S T R © Glencoe/McGraw-Hill 652 Geometry: Concepts and Applications the Given and Prove and draw a diagram for the situation. Look at the given information and mark the diagram with that information. Look at what you are to prove and make a plan for using the given information to reach that conclusion. You can use the work backward strategy as well. Then write each statement and its reason in a logical order to arrive at the conclusion. 3. Sometimes it is helpful to summarize information that you need to remember. Summarize the steps you would take to write a two-column proof. Sample answer: First, write Helping You Remember e. d. c. b. a. Statements Proof: Plan: Sample answer: Show that the two triangles are congruent and then use CPCTC. Given: UR UT , R S TS Prove: RUS TUS 2. Write the statements and reasons for a two-column proof for each set of information. First, write a short plan for your proof. d. The given information is never used in a two-column proof. False; the given information is always used in a two-column proof. c. If the figure you are given to work with for a proof has overlapping triangles, you can redraw the triangles as separate triangles. true False; definitions are one of the three things that can be used for a reason. b. You cannot use definitions of geometric terms as a reason for a statement in a proof. False; the given information is found in the hypothesis. a. The given information for a proof can be found in the conclusion of the conjecture. 1. Determine whether each statement is true or false. If the statement is false, explain why. Reading the Lesson Two-Column Proofs 15–5 Answers (Lesson 15-5) Geometry: Concepts and Applications b c c c c a b area of the 4 triangles a b a A14 a b 2 a b 2 a b 2 a b 1 2 b b a a a area of the center square c2 c2 2 a b c2 c2 c2 DATE b a area of large square PERIOD a2 b2 2 a b a2 b2 a2 b2 9 B 12 15 C © Glencoe/McGraw-Hill A: 81, B: 144, C: 225; true Squares A 1. 3 4 B 2 3 5 2.5 3 653 A: 2.253 , B: 43 , C: 6.253 ; true Equilateral Triangles A 1.5 3 2. C Semicircles B 4 5 Geometry: Concepts and Applications A: 1.125, B: 2.000, C: 3.125; true A 3 3. C a2 b2 a b a b Use the Pythagorean Theorem to find the area of A, B, and C in each of the following. Then, answer true or false for the statement A B C. a2 a b b2 a b b b Think of the figure as a large square. The relationship c2 a2 b2 is true for all right triangles. 4 a Think of the figure as four triangles and a square. Use the Pythagorean Theorem to find the area of the shaded region in the figure at the right. Pythagorean Theorem Enrichment NAME © Glencoe/McGraw-Hill 15–5 Study Guide NAME DATE Use the origin as a vertex or center. Place at least one side of a polygon on an axis. Keep the figure within the first quadrant if possible. Use coordinates that make computations simple. H (0, 0) K(0, c) J(a, b c) © Glencoe/McGraw-Hill 654 Geometry: Concepts and Applications So AB CD and AC BD. Therefore, the opposite sides of a parallelogram are congruent. 2 ) 02 0 ( ) 0 a 2 a AB (a 2 2 CD (( a ) b ) b c ( ) c a 2 a 2 a ) b ) a2 c ( ) 0 b 2 c2 BD (( 2 AC (b ) 02 c ( ) 0 b 2 c2 D(a b, c). Then use the Distance Formula to find AB, CD, AC, and BD. D(a, c) A(a, 0) 2. HIJK is a parallelogram. PERIOD 3. Use a coordinate proof to show that the opposite sides of any parallelogram are congruent. Label the vertices A(0, 0), B(a, 0), C(b, c), and 1. ABCD is a rectangle. Name the missing coordinates in terms of the given variables. Since OB AC, the diagonals are congruent. 2 2 ) 02 b ( ) 0 a b2 OB (a 2 a 2 AC (0 ) a2 b ( ) 0 b2 Use the Distance Formula to find OB and AC. Use (0, 0) as one vertex. Place another vertex on the x-axis at (a, 0) and another on the y-axis at (0, b). The fourth vertex must then be (a, b). Example: Use a coordinate proof to prove that the diagonals of a rectangle are congruent. The Distance Formula, Midpoint Formula, and Slope Formula are useful tools for coordinate proofs. • • • • You can place figures in the coordinate plane and use algebra to prove theorems. The following guidelines for positioning figures can help keep the algebra simple. Coordinate Proofs 15–6 Answers (Lessons 15-5 and 15-6) Geometry: Concepts and Applications 15–6 Practice NAME DATE © Glencoe/McGraw-Hill B (m, 0) x A (0, 0) O x Y (p, 0) A15 R (c a, d) O x B (12 d, 0) © Glencoe/McGraw-Hill x B ( x2, 0) A ( x2, 0) O C ( x2 , y) D ( x2, y) y 655 5. a rectangle with length x units and width y units A ( 12d, 0) C (0, s) y Geometry: Concepts and Applications 4. an isosceles triangle with base d units long and heights s units long O P (0, 0) Q (c, 0) x S (a, d) y 3. a rhombus with sides c units long O X (0, 0) Z (0, r ) y 2. a right triangle with legs p and r units long C (m, m) D (0, m) y D(0, 0), F(a c, b) 4. DEFG is a parallelogram. 656 Sample proof: a b2 0 b2 2 a 2 slope of AC ab0 ab 2 2 a b b2 0 a2 slope of BD ba ba b2 a b2 a2 b2 1 a2 2 ab ba b2 a2 © Glencoe/McGraw-Hill PERIOD M(0, 0), R(a b, a 2 b2) 2. MART is a rhombus. Geometry: Concepts and Applications 5. Use a coordinate proof to prove that the diagonals of a rhombus are perpendicular. Draw the diagram at the right. R(0, 0) C(a, b) 3. RECT is a rectangle. X(0, 0) Y(a, 0) 1. XYZ is a right isosceles triangle. Name the missing coordinates in terms of the given variables. PERIOD Coordinate Proofs DATE Position and label each figure on a coordinate plane. 1–5. Sample answers given. 1. a square with sides m units long Skills Practice NAME Coordinate Proofs 15–6 Answers (Lesson 15-6) Geometry: Concepts and Applications © Glencoe/McGraw-Hill PERIOD A16 x T(a, ?) U(a, ?) x T(b, ?) R is (–a, 0) since it is a units in the negative direction horizontally and lies on the x–axis; S is (–b, c) since it is b units in the negative direction horizontally; T is (b, c) since it is c units in a positive vertical direction; U is (a, 0) since it is on the x–axis. R(?, ?) O S(?, c) y b. isosceles trapezoid © Glencoe/McGraw-Hill 657 Geometry: Concepts and Applications the origin. The everyday meaning of origin is place where something begins. So look to see if there is a good way to begin by placing a vertex of the figure at the origin. 3. What is an easy way to remember how best to draw a diagram that will help you devise a coordinate proof? Sample answer: A key point in the coordinate plane is Helping You Remember R is (0, b) because the point is on the y–axis; S is (0, 0) because the point is the b origin; T is a, because it 2 is half way between R and S in vertical distance. S(?, ?) R(?, b) y a. isosceles triangle 2. Find the missing coordinates in each figure. Explain how you find the coordinates. 1. Complete each sentence with one or two words to make a true statement. a. If you are writing a coordinate proof and need to show that two segments are congruent, Distance Formula a formula you may want to use is the ______________________. b. When drawing a diagram for a coordinate proof, try to place a vertex of the figure at the origin ________. c. If you are writing a coordinate proof and want to show that two segments are parallel, a Slope Formula formula you may want to use is the ___________________. d. When drawing a diagram for a coordinate proof, try to place at least on side of the axis polygon on a(n) ______. e. If you are writing a coordinate proof and want to show that a segment has been bisected, Midpoint Formula a formula you may want to use is the _______________________. first f. When drawing a diagram for a coordinate proof, try to keep the figure in the ______ quadrant. Reading the Lesson coordinate proof a proof that uses figures on a coordinate plane Key Terms DATE Reading to Learn Mathematics NAME Coordinate Proofs 15–6 Enrichment NAME DATE ar a r 2 2 © Glencoe/McGraw-Hill 658 (PQ)2 (AQ) (QB) (b)2 (a r) (r a) b 2 (r a) (r a) b2 r2 a2 Therefore a 2 b 2 r 2, which means that (a, b) is on the circle with the equation x 2 y 2 r 2. This is the circle that has A B as a diameter. a r 2. Suppose PQ B, Q is between A and B, A and PQ is the geometric mean between A Q and Q B . Prove that P is on the circle that has A B as a diameter. Use the figure at the right. a r 2 b r a 2 2 1. b 2 r 2 a 2, and 2 2 a 2 b 2 r 2, since (a, b) is on the graph of x 2 y 2 r 2. Therefore ar b 2 b b 2 2 (slope of AP PB ) (slope of ) slope of PB a ( r) ar b0 b ar ar b0 slope of AP b 1. Prove that an angle inscribed in a semicircle is a right angle. Use the figure at right. (Hint: Write an equation for the circle. Use your equation to help show that P) (slope of PB (slope of A ) 1). A (r, O) A(r, O) O O y y Q (a, O) P (a, b) P (a, b) x B (r, O) x B (r, O) PERIOD Geometry: Concepts and Applications You can prove many theorems about circles by using coordinate geometry. Whenever possible locate the circle so that its center is at the origin. Coordinate Proofs with Circles 15–6 Answers (Lesson 15-6) Geometry: Concepts and Applications Chapter 15 Answer Key Form 1A Page 659 Page 660 1. Form 1B Page 661 Page 662 D 8. 9. 2. A 3. C 4. D 10. 11. 1. D B C A 8. A 9. C B D D 10. 2. A 3. B 4. C 11. 12. 5. 5. 12. C 13. 6. 16. B D C D Bonus A 7. B 13. B A B 14. 6. B 7. C © Glencoe/McGraw-Hill 15. 16. C C D Bonus D 14. A17 15. A C Geometry: Concepts and Applications Chapter 15 Answer Key Form 2A Page 663 Page 664 A and B are not complementary angles. 1. 2. p ∨ q 3. false 14. Substitution 15. Subtraction 16. 17. true 4. 18. 5. no valid conclusion 6. The sum of the measures of 1 and 2 is 90. The ratio of the volumes of two cubes is equal to the cube of the ratio of their heights. 7. 9. 10. 20. Transitive or Substitution, 21. Subtraction, 12. SAS 13. CPCTC © Glencoe/McGraw-Hill C(b, c) Sample answer: PQ RS a, QR 25. BEA AA Similarity Sample answer: P(0, 0), Q(a, 0), R(a b, c), S(b, c) 24. The congruence of angles is reflexive. 11. The acute angles in a right triangle are complementary. 19. 23. The sides opposite congruent angles are congruent. Definition of rectangle Angle Addition Postulate 22. B A 8. Symmetric Bonus A18 b2 c2 PS ABD CAD Geometry: Concepts and Applications Chapter 15 Answer Key Form 2B Page 665 Page 666 The perimeter of rectangle ABCD is not 30 inches. 1. 14. Reflexive 15. 16. Subtraction Transitive or Substitution Definition of rectangle p ∧ r 2. 3. true 17. 4. false All right angles are congruent. 18. Opposite sides of a parallelogram are congruent. 19. 5. ABC has two acute angles. 20. 6. no valid conclusion 7. The measure of a diameter of B is 2 AB. 21. When two parallel lines are cut by a transversal, corresponding angles are congruent. 22. AAS 23. D(a, 0) 8. C M 9. Definition of midpoint 24. 10. The congruence of segments is reflexive. 25. 11. CBM 12. SSS 13. CPCTC © Glencoe/McGraw-Hill When two parallel lines are cut by a transversal, alternate interior angles are congruent. Sample answer: X(a, 0), Y(0, b), Z(a, 0) Sample answer: XY YZ a2 b2 ABD and ACD are isosceles triangles. Bonus A19 Geometry: Concepts and Applications Chapter 15 Assessment Answer Key Page 667, Extended Response Assessment Scoring Rubric Score General Description Specific Criteria 4 Superior A correct solution that is supported by welldeveloped, accurate explanations • Satisfactory A generally correct solution, but may contain minor flaws in reasoning or computation • Nearly Satisfactory A partially correct interpretation and/or solution to the problem • Nearly Unsatisfactory A correct solution with no supporting evidence or explanation • • Unsatisfactory An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given • 3 2 1 0 © Glencoe/McGraw-Hill • • • • • • • • • • • • • • • • • • • • • • • Shows thorough understanding of the concepts of paragraph proofs, and two-column proofs. Uses appropriate strategies to solve problems. Computations are correct. Written explanations are exemplary. Graphs are accurate and appropriate. Goes beyond requirements of some or all problems. Shows an understanding of the concepts of paragraph proofs, and two-column proofs. Uses appropriate strategies to solve problems. Computations are mostly correct. Written explanations are effective. Graphs are mostly accurate and appropriate. Satisfies all requirements of problems. Shows an understanding of most of the concepts of paragraph proofs, and two-column proofs. May not use appropriate strategies to solve problems. Computations are mostly correct. Written explanations are satisfactory. Graphs are mostly accurate. Satisfies the requirements of most of the problems. Final computation is correct. No written explanations or work is shown to substantiate the final computation. Graphs may be accurate but lack detail or explanation. Satisfies minimal requirements of some of the problems. Shows little or no understanding of the concepts of paragraph proofs, and two-column proofs. Does not use appropriate strategies to solve problems. Computations are incorrect. Written explanations are unsatisfactory. Graphs are inaccurate or inappropriate. Does not satisfy requirements of problems. No answer may be given. A20 Geometry: Concepts and Applications Chapter 15 Answer Key Extended Response Assessment Sample Answers Page 667 ⊥ B the definition of perpendicular, AX C . is the perpendicular Therefore, AX bisector of B C . 1. Given: Triangle ABC is isosceles with bisects BAC. right angle at A; AX is the perpendicular bisector Prove: AX of B C . 2. a. The sum of the measures of the central angles of a circle is 360. b. All radii of a circle have equal measures. c. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. d. Angle Sum Theorem e. Substitution Property of Equality f. Subtraction Property of Equality g. Substitution Property of Equality h. Subtraction Property of Equality i. Addition Property of Equality j. Distributive Property k. Angle Addition Postulate l. Substitution Property of Equality m. Definition of arc measure n. Substitution Property of Equality o. Division Property of Equality B X A C bisects BAC, so You know that AX CAX BAX by the definition of angle bisector. Since ABC is isosceles, AC and ACB ABC. Then, AB ACX ABX by ASA. By CPCTC, is a bisector of the C X XB . So AX hypotenuse, B C , by the definition of bisector. Also by CPCTC, CXA BXA. CXA and BXA are supplementary, because they form a linear pair and linear pairs of angles are supplementary. Since CXA and BXA are congruent and supplementary, their measures are 90. By definition, they are right angles. So, by 3. Given: rhombus ABCD; W, X, Y, and Z are the midpoints of A B , B C , C D, and A D, respectively. Prove: Quadrilateral WXYZ is a parallelogram. Statements Reasons 1. Quadrilateral ABCD is a rhombus. 2. AB BC CD AD 3. W, X, Y, and Z are the midpoints of A B , B C , C D, and A D, respectively. 1. Given 2. Definition of rhombus 3. Given 1 1 4. AW BW AB; BX CX BC; 4. Definition of midpoint CY DY 2 1 CD; 2 DZ AZ 2 1 AD 2 5. AW CY and AZ CX; BW DY and BX DZ 6. A C; B D 7. AWZ CYX; BWX DYZ 8. WZ Y X; ZY X W 9. WXYZ is a parallelogram. © Glencoe/McGraw-Hill 5. Substitution Property of Equality 6. In a parallelogram, opposite angles are congruent. 7. SAS 8. CPCTC 9. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. A21 Geometry: Concepts and Applications Chapter 15 Answer Key Mid-Chapter Test Page 668 Quiz A Page 669 1. x 100 or x 120 1. false any whole number whose ones digit is 5 2. 2. false 3. true 3. true 4. p → q, r → p 5. If both pairs of opposite sides in a quadrilateral are congruent, then they are parallel. 6. ABCDEF UVWXYZ 7. no valid conclusion 8. inductive 9. If a quadrilateral is a square, then it has 4. four right angles. 5. no valid conclusion Quiz B Page 669 1. SSS 2. CDB deductive Definition of angle bisector 3. EC 10. C Vertical angles are congruent. 11. 12. DEC 13. SAS 14. CPCTC © Glencoe/McGraw-Hill 4. A22 Sample answer: A Y X Z 5. Substitution 6. C(a c, b) B Geometry: Concepts and Applications Chapter 15 Answer Key Cumulative Review Page 670 1. Standardized Test Practice Page 671 Page 672 If x and y are both odd numbers, then x y is even. 2. 70 3. 2 3 4. A C DC 5. isosceles; vertex; base 6. 30 7. 50 8. AA Similarity; 9 9. 140 10. chord 11. 65 12. 565.5 ft2 1. C 2. A 3. D 4. 5. 11. C 12. B 13. A 14. D 15. A 16. B 17. C 18. B 19. D 20. C B B 6. C 7. D 8. C 9. A 10. C Quadrilateral CDEF is a rectangle. 13. 14. E C DF © Glencoe/McGraw-Hill A23 Geometry: Concepts and Applications

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