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Reasoning and Proof Chapter Overview and Pacing Year-long pacing: pages T20–T21. PACING (days) Regular Block LESSON OBJECTIVES Basic/ Average Advanced Basic/ Average Advanced Inductive Reasoning and Conjecture (pp. 62–66) • Make conjectures based on inductive reasoning. • Find counterexamples. 1 1 0.5 0.5 Logic (pp. 67–74) • Determine truth values of conjunctions and disjunctions. • Construct truth tables. 2 1 1 0.5 Conditional Statements (pp. 75–80) • Analyze statements in if-then form. • Write the converse, inverse, and contrapositive of if-then statements. 2 1 1 0.5 (with 2-3 (with 2-3 (with 2-3 (with 2-3 Follow-Up) Follow-Up) Follow-Up) Follow-Up) Deductive Reasoning (pp. 82–88) • Use the Law of Detachment. • Use the Law of Syllogism. Follow-Up: Use a table and deductive reasoning to solve a logic problem. 1 1 0.5 0.5 (with 2-4 (with 2-4 (with 2-3 (with 2-3 Follow-Up) Follow-Up) Follow-Up) Follow-Up) Postulates and Paragraph Proofs (pp. 89–93) • Identify and use basic postulates about points, lines, and planes. • Write paragraph proofs. 1.5 1 1 0.5 Algebraic Proof (pp. 94–100) • Use algebra to write two-column proofs. • Use properties of equality in geometry proofs. 1.5 1 1 0.5 Proving Segment Relationships (pp. 101–106) • Write proofs involving segment addition. • Write proofs involving segment congruence. 2 2 1 1 Proving Angle Relationships (pp. 107–114) • Write proofs involving supplementary and complementary angles. • Write proofs involving congruent and right angles. 2 2 1 1 Study Guide and Practice Test (pp. 115–121) Standardized Test Practice (pp. 122–123) 1 1 0.5 0.5 Chapter Assessment 1 1 0.5 0.5 15 12 8 6 TOTAL An electronic version of this chapter is available on StudentWorksTM. This backpack solution CD-ROM allows students instant access to the Student Edition, lesson worksheet pages, and web resources. 60A Chapter 2 Reasoning and Proof Timesaving Tools ™ All-In-One Planner and Resource Center Chapter Resource Manager See pages T5 and T21. 57–58 59–60 61 62 63–64 65–66 67 68 69–70 71–72 73 74 75–76 77–78 79 80 81–82 83–84 85 86 87–88 89–90 91 92 93–94 95–96 97 98 99–100 101–102 103 104 119 119, 121 120 GCC 19 2-1 2-1 GCC 20 2-2 2-2 2-3 2-3 2-4 2-4 2-5 2-5 2-6 2-6 2-7 2-7 2-8 2-8 SC 3 41–44, 83–86, 93–94 89–90, 101–104 120 Ap plic atio ns* 5-M Tra inute nsp C are heck nci es Int e Cha racti lkb ve oar Ge d om PA Plu SS: s (l T ess utori ons al ) Ass ess me nt Pre req u Wo isite rkb Ski ook lls Enr ich me nt S and tudy Int Guid erv e ent ion (Sk Pra c ills and tice Ave rag e) Rea di Ma ng to the ma Learn tics CHAPTER 2 RESOURCE MASTERS SC 4 81–82, 85–86 Materials protractor 6 patty paper, protractor, paper 105–118, 122–124 *Key to Abbreviations: GCC Graphing Calculator and Computer Masters SC School-to-Career Masters Chapter 2 Reasoning and Proof 60B Mathematical Connections and Background Continuity of Instruction Prior Prior Knowledge Knowledge In algebra, students learned to solve for a variable. In Chapter 1, students learned about points, lines and planes. They were introduced to adjacent and vertical angles, complementary and supplementary angles, and right angles. They can identify congruent segments and angles as well as perpendicular lines. This Chapter In this chapter, students explore methods of reasoning and learn to apply those methods to geometry. They make conjectures, determine the truth values of compound statements, and construct truth tables. They also analyze conditional statements and write related conditionals. The terms postulate and theorem are introduced. Algebraic properties of equality are applied to geometry, enabling students to write formal and informal proofs proving segment and angle relationships. Inductive Reasoning and Conjecture A conjecture is an educated guess based on known information. Examining several specific situations to arrive at a conjecture is called inductive reasoning. Meteorologists use inductive reasoning to predict weather conditions. Just because a conjecture is true in most circumstances does not make it a true conjecture. If just one example contradicts the conjecture, the conjecture is not true. The false example is called a counterexample. Logic A statement is any sentence that is either true or false, but not both. The truth or falsity of a statement is called its truth value. The negation of a statement has the opposite meaning as well as an opposite truth value. This means that if a statement is represented by p, then not p is the negation of the statement. You could also write not p as p. Two or more statements can be joined to form a compound statement. A conjunction is a compound statement formed by joining two or more statements with the word and. The symbol can be used instead of and. Two or more statements can also be joined to form a disjunction. A disjunction is a compound statement formed by joining two or more statements with the word or. You might see the symbol instead of the word or in a disjunction. Conjunctions and disjunctions can be illustrated with Venn diagrams. Truth tables can also be helpful in evaluating the truth values of statements. A truth table will show that a conjunction is true only when both statements are true. A disjunction, on the other hand, is true unless both statements are false. Conditional Statements Future Connections In Chapter 4, students will build on their knowledge of proofs when they investigate triangles. Students will also write proofs in Chapter 7. Thinking logically is a crucial skill for daily living. Logic is used to make informed choices and to examine a statement for truth. 60C Chapter 2 Reasoning and Proof A conditional statement is a statement that can be written in if-then form: if p, then q. The phrase immediately following the word if is called the hypothesis. The phrase immediately following the word then is called the conclusion. An arrow pointing to the right is written between p and q to symbolize an if-then situation. A conditional statement is true in all cases except where the hypothesis is true and the conclusion is false. Related conditionals are statements constructed from an if-then statement. A converse statement is formed by exchanging the hypothesis and the conclusion: if q, then p. An inverse statement is formed by negating both the hypothesis and the conclusion of the original statement: if p, then q. A contrapositive is formed by negating both the hypothesis and the conclusion of the converse statement: if q, then p. Deductive Reasoning Deductive reasoning uses facts, rules, definitions, or properties to reach logical conclusions. A form of deductive reasoning that is used to draw conclusions from true conditional statements is called the Law of Detachment. This law states that if p → q is true and p is true, then q is also true. The Law of Syllogism is another law of logic. It states that if p → q and q → r are true, then p → r is also true. You may see the similarity between this law and the Transitive Property of Equality from algebra. Postulates and Paragraph Proofs In geometry, a postulate is a statement that describes a fundamental relationship between the basic terms of geometry. Postulates are accepted as true without proof. Several postulates based on the relationship among points, lines, and planes were introduced in Chapter 1, but were not labeled as postulates. In this course, you will learn to use various methods to justify the truth of a statement or conjecture. Once a statement or conjecture has been shown to be true, it is called a theorem. A theorem can be used like a definition or postulate to justify that other statements are true. A proof is a logical argument in which each statement you make is supported by a statement that is accepted as true. One type of proof is called a paragraph or informal proof. It is a written explanation of why a conjecture for a given situation is true. A good proof states the theorem or conjecture to be proven. It lists the given information and, if possible, supplies a diagram to illustrate the given information. The proof states what is to be proved and develops a system of deductive reasoning. Algebraic Proof In algebra, you learned to use properties of equality to solve algebraic equations and to verify relationships. These properties can be used to justify each step when solving an equation. A group of algebraic steps used to solve problems form a deductive argument. This argument can be demonstrated by writing the solution to the equation in the first column and listing the property justifying each step in the second column. In geometry, a similar format is used to prove conjectures and theorems. A two-column, or formal, proof contains statements and reasons organized in two columns. Each step is called a statement, and the properties that justify each step are called reasons. Proving Segment Relationships As you learned in Chapter 1, a segment can be measured, and measures can be used in calculations because they are real numbers. One postulate about segments is called the Ruler Postulate. It states that the points on any line or line segment can be paired with real numbers so that, given any two points A and B on a line, A corresponds to 0, and B corresponds to a positive real number. That number is the length of the segment. Another postulate states that if point B lies between points A and C on the same line, AB BC AC. The converse statement holds true as well. The Reflexive, Symmetric, and Transitive Properties of Equality can be used to write proofs about segment congruence. The theorem resulting from the proofs states that congruence of segments is reflexive, symmetric, and transitive. Proving Angle Relationships This lesson introduces postulates and theorems about angle relationships. The Protractor Postulate and a number r between 0 and 180, states, “Given AB there is exactly one ray with endpoint A, extending , such that the measure of the on either side of AB angle formed is r.” The Angle Addition Postulate states that if R is in the interior of PQS, then mPQR mRQS mPQS. If mPQR mRQS mPQS, then R is in the interior of PQS. This postulate can be used with other angle relationships to prove other theorems relating to angles. Some of these theorems relate to supplementary and complementary angles. Another theorem extends the Reflexive, Transitive, and Symmetric properties to angle congruence. There is also a series of theorems about perpendicular lines and right angles. Chapter 2 Reasoning and Proof 60D and Assessment Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters ASSESSMENT INTERVENTION Type Student Edition Teacher Resources Ongoing Prerequisite Skills, pp. 61, 80, 87, 93, 100, 106 Practice Quiz 1, p. 80 Practice Quiz 2, p. 100 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 41–44, 81–86, 89–90, 93–94, 101–104 Quizzes, CRM pp. 119–120 Mid-Chapter Test, CRM p. 121 Study Guide and Intervention, CRM pp. 57–58, 63–64, 69–70, 75–76, 81–82, 87–88, 93–94, 99–100 Mixed Review pp. 66, 74, 80, 93, 100, 106, 114 Cumulative Review, CRM p. 122 Error Analysis Find the Error, pp. 84, 111 Common Misconceptions, p. 76 Find the Error, TWE pp. 84, 111 Unlocking Misconceptions, TWE p. 91 Tips for New Teachers, TWE p. 70 Standardized Test Practice pp. 66, 74, 80, 86, 87, 93, 96, 97, 99, 106, 114, 121, 122 TWE pp. 122–123 Standardized Test Practice, CRM pp. 123–124 Open-Ended Assessment Writing in Math, pp. 66, 74, 79, 86, 93, 99, 106, 114, 123 Open Ended, pp. 63, 71, 78, 84, 91, 97, 103, 111 Standardized Test, p. 123 Modeling: TWE pp. 74, 87, 106 Speaking: TWE pp. 80, 93 Writing: TWE pp. 66, 100, 114 Open-Ended Assessment, CRM p. 117 Chapter Assessment Study Guide, pp. 115–120 Practice Test, p. 121 Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 105–110 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 111–116 Vocabulary Test/Review, CRM p. 118 For more information on Yearly ProgressPro, see p. 2. Geometry Lesson 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 Yearly ProgressPro Skill Lesson Inductive Reasoning and Conjecture Logic Conditional Statements Deductive Reasoning Postulates and Paragraph Proofs Algebraic Proof Proving Segment Relationships Proving Angle Relationships Chapter 2 Reasoning and Proof GeomPASS: Tutorial Plus, Lesson 6 www.geometryonline.com/ self_check_quiz www.geometryonline.com/ extra_examples Standardized Test Practice CD-ROM www.geometryonline.com/ standardized_test ExamView® Pro (see below) MindJogger Videoquizzes www.geometryonline.com/ vocabulary_review www.geometryonline.com/ chapter_test ExamView® Pro Use the networkable ExamView® Pro to: • Create multiple versions of tests. • Create modified tests for Inclusion students. • Edit existing questions and add your own questions. • Use built-in state curriculum correlations to create tests aligned with state standards. • Apply art to your test from a program bank of artwork. For more information on Intervention and Assessment, see pp. T8–T11. 60E Technology/Internet Reading and Writing in Mathematics Glencoe Geometry provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition • Foldables Study Organizer, p. 61 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 63, 71, 78, 84, 91, 97, 103, 111) • Reading Mathematics, p. 81 • Writing in Math questions in every lesson, pp. 66, 74, 79, 86, 93, 99, 106, 114 • Reading Study Tip, p. 75 • WebQuest, p. 65 Teacher Wraparound Edition • Foldables Study Organizer, pp. 61, 115 • Study Notebook suggestions, pp. 64, 72, 78, 81, 84, 88, 91, 97, 104, 111 • Modeling activities, pp. 74, 87, 106 • Speaking activities, pp. 80, 93 • Writing activities, pp. 66, 100, 114 • Differentiated Instruction (Verbal/Linguistic), p. 83 • ELL Resources, pp. 60, 65, 73, 79, 81, 83, 86, 92, 99, 105, 113, 115 Additional Resources • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 2 Resource Masters, pp. vii-viii) • Proof Builder helps students learn and understand theorems and postulates from the chapter. (Chapter 2 Resource Masters, pp. ix–x) • Reading to Learn Mathematics master for each lesson (Chapter 2 Resource Masters, pp. 61, 67, 73, 79, 85, 91, 97, 103) • Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. • Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. • Reading Strategies for the Mathematics Classroom • WebQuest and Project Resources For more information on Reading and Writing in Mathematics, see pp. T6–T7. Many of the vocabulary terms introduced in Chapter 2 can be represented by symbols. Threecolumn notes can be a helpful way for students to organize new vocabulary terms. To reinforce understanding, students can write an explanation of each term in their own words and provide the appropriate symbol. The table at the right shows notes for Lesson 2-2. Students can add on to this sample with other terms from Chapter 2. Term Explanation Symbol negation the opposite of the given statement conjunction a compound statement formed with the word “and” pq disjunction a compound statement formed with the word “or” pq Chapter 2 Reasoning and Proof 60F Reasoning and Proof Notes Have students read over the list of objectives and make a list of any words with which they are not familiar. • Lessons 2-1 through 2-3 Make conjectures, determine whether a statement is true or false, and find counterexamples for statements. • Lesson 2-4 Use deductive reasoning to reach valid conclusions. • Lessons 2-5 and 2-6 Verify algebraic and geometric conjectures using informal and formal proof. • Lessons 2-7 and 2-8 Write proofs involving segment and angle theorems. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson. Key Vocabulary • • • • • inductive reasoning (p. 62) deductive reasoning (p. 82) postulate (p. 89) theorem (p. 90) proof (p. 90) Logic and reasoning are used throughout geometry to solve problems and reach conclusions. There are many professions that rely on reasoning in a variety of situations. Doctors, for example, use reasoning to diagnose and treat patients. You will investigate how doctors use reasoning in Lesson 2-4. Lesson 2-1 2-2 2-3 2-4 2-4 Follow-Up 2-5 2-6 2-7 2-8 NCTM Standards Local Objectives 1, 6, 7, 8, 9, 10 6, 7, 8, 9, 10 3, 6, 7, 8, 9, 10 3, 6, 7, 8, 9, 10 6, 7 3, 6, 7, 8, 9, 10 2, 3, 6, 7, 8, 9, 10 3, 6, 7, 8, 9, 10 3, 6, 7, 8, 9, 10 Key to NCTM Standards: 1=Number & Operations, 2=Algebra, 3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 10=Representation 60 Chapter 2 Reasoning and Proof 60 Chapter 2 Reasoning and Proof B. Busco/Getty Images Vocabulary Builder ELL The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 2 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 2 test. Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 2. For Lesson 2-1 This section provides a review of the basic concepts needed before beginning Chapter 2. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pages 41–44, 81–86, 89–90, 93–94, 101–104. Evaluate Expressions Evaluate each expression for the given value of n. (For review, see page 736.) 1. 3n 2; n 4 10 2. (n 1) n; n 6 13 3. n2 3n; n 3 0 4. 180(n 2); n 5 540 5. n; n 10 50 n(n 3) 6. ; n 8 20 n 2 2 For Lessons 2-6 through 2-8 Solve Equations Prerequisite Skills in the Getting Ready for the Next Lesson section at the end of each exercise set review a skill needed in the next lesson. Solve each equation. (For review, see pages 737 and 738.) 7. 6x 42 4x 21 10. 12 7x x 18 ⫺5 8. 8 3n 2 2n 2 18 1 11. 3x 4 x 5 ⫺ 2 5 For Lesson 2-8 9. 3(y 2) 12 y ⫺9 3 2 12. 2 2x x 2 3 2 Adjacent and Vertical Angles For Exercises 13–14, refer to the figure at the right. (For review, see Lesson 1-5.) 13. If mAGB 4x 7 and mEGD 71, find x. 16 A 14. If mBGC 45, mCGD 8x 4, and mDGE 15x 7, find x. 6 F B C G E Prerequisite Skill 2-3 Evaluating Algebraic Expressions, p. 74 Solving Equations, p. 80 Information from Figures, p. 87 Solving Equations, p. 93 Segment Measures, p. 100 Complementary and Supplementary Angles, p. 106 2-4 2-5 2-6 2-7 2-8 D Reasoning and Proof Make this Foldable to help you organize your notes. Begin with eight sheets of 8 12 ” by 11” grid paper. Staple For Lesson Cut Tabs Cut the bottom of each sheet to form a tabbed book. Stack and staple the eight sheets together to form a booklet. Label Reasoning & Proof 2-1 Label each of the tabs with a lesson number. Add the chapter title to the first tab. 2-2 2-3 2-4 2-5 2-6 2-7 2-8 Reading and Writing As you read and study each lesson, use the corresponding page to write proofs and record examples of when you used logical reasoning in your daily life. Chapter 2 Reasoning and Proof 61 TM For more information about Foldables, see Teaching Mathematics with Foldables. Organization of Data Use this Foldable for student writing about reasoning and proofs. After students make their Foldable, have them label the tabs to correspond to the eight lessons in this chapter. Students use their Foldable to take notes, define terms, record concepts, write statements in if-then form, and write paragraph proofs. On the back of the Foldable, have students record examples of ways in which they use reasoning and proofs in their daily lives. Note how columnists and authors present their reasoning and ways in which they try to prove or disprove their points of view. Chapter 2 Reasoning and Proof 61 Inductive Reasoning and Conjecture Lesson Notes • Make conjectures based on inductive reasoning. 1 Focus 5-Minute Check Transparency 2-1 Use as a quiz or review of Chapter 1. • Find counterexamples. • conjecture • inductive reasoning • counterexample Mathematical Background notes are available for this lesson on p. 60C. can inductive reasoning help predict weather conditions? Ask students: • What are normal temperatures for the month of January? Sample answer: The temperatures in January are usually in the 30s or 40s. (Answers will vary in different parts of the country.) • How do people benefit from the inductive reasoning techniques of meteorologists? Sample answers: People can plan for outdoor events a few days in advance; they can dress appropriately for daily weather conditions and carry weatherrelated items, such as umbrellas, sunglasses, and so on. can inductive reasoning help predict weather conditions? Vocabulary Meteorologists use science and weather patterns to make predictions about future weather conditions. They are able to make accurate educated guesses based on past weather patterns. MAKE CONJECTURES A conjecture is an educated guess based on known information. Examining several specific situations to arrive at a conjecture is called inductive reasoning. Inductive reasoning is reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction. Example 1 Patterns and Conjecture The numbers represented below are called triangular numbers. Make a conjecture about the next triangular number based on the pattern. 1 3 10 15 Observe: Each triangle is formed by adding another row of dots. Find a Pattern: 1 3 2 6 3 10 4 15 5 The numbers increase by 2, 3, 4, and 5. Study Tip Conjectures 6 Conjecture: List your observations and identify patterns before you make a conjecture. The next number will increase by 6. So, it will be 15 6 or 21. In Chapter 1, you learned some basic geometric concepts. These concepts can be used to make conjectures in geometry. 62 Chapter 2 Reasoning and Proof Bob Daemmrich/Stock Boston Resource Manager Workbook and Reproducible Masters Chapter 2 Resource Masters • Study Guide and Intervention, pp. 57–58 • Skills Practice, p. 59 • Practice, p. 60 • Reading to Learn Mathematics, p. 61 • Enrichment, p. 62 Graphing Calculator and Computer Masters, p. 19 Transparencies 5-Minute Check Transparency 2-1 Answer Key Transparencies Technology Interactive Chalkboard Example 2 Geometric Conjecture 2 Teach For points P, Q, and R, PQ ⴝ 9, QR ⴝ 15, and PR ⴝ 12. Make a conjecture and draw a figure to illustrate your conjecture. Given: points P, Q, and R; PQ 9, QR 15, and PR 12 MAKE CONJECTURES Examine the measures of the segments. Since PQ PR QR, the points cannot be collinear. Conjecture: P, Q, and R are noncollinear. In-Class Examples Teaching Tip Tell students to test all fundamental operations, including powers and roots, when they are looking for patterns in a series of numbers. Advise students that sometimes two operations can be used. Q 15 9 P Power Point® 12 R 1 Make a conjecture about the FIND COUNTEREXAMPLES A conjecture based on several observations may be true in most circumstances, but false in others. It takes only one false example to show that a conjecture is not true. The false example is called a counterexample . next number based on the pattern. 2, 4, 12, 48, 240 1440 2 For points L, M, and N, LM 20, MN 6, and LN 14. Make a conjecture and draw a figure to illustrate your conjecture. Example 3 Find a Counterexample FINANCE Find a counterexample for the following statement based on the graph. The rates for CDs are at least 1.5% less than the rates a year ago. Examine the graph. The statement is true for 6-month, 1-year, and 1 2-year CDs. However, the 2 difference in the rate for a 5-year CD is 0.74% less, which is less than 1.5%. The statement is false for a 5-year certificate of deposit. Thus, the change in the 5-year rate is a counterexample to the original statement. Log on for: • Updated data • More on finding counterexamples www.geometryonline.com/ usa_today USA TODAY Snapshots® Latest CD rates L Average certificate of deposit rates as of Wednesday: This week 6-month Last week Year ago 1-year This week Last week Year ago 2 1⁄2-year This week Last week Year ago 5-year This week Last week Year ago N 14 1.80% 1.80% 20 Conjecture: L, M, and N are collinear. 4.55% 2.12% 2.11% M 6 4.64% 2.96% 2.96% FIND COUNTEREXAMPLES 4.74% 4.22% 4.23% 4.96% In-Class Example Power Point® 3 UNEMPLOYMENT Based on Source: Bank Rate Monitor, 800-327-7717, www.bankrate.com Concept Check 1– 3. See p. 123A. USA TODAY 1. Write an example of a conjecture you have made outside of school. 2. Determine whether the following conjecture is always, sometimes, or never true based on the given information. Given: collinear points D, E, and F Conjecture: DE EF DF 3. OPEN ENDED Interactive Chalkboard PowerPoint® Presentations County Write a statement. Then find a counterexample for the statement. www.geometryonline.com/extra_examples Lesson 2-1 Inductive Reasoning and Conjecture 63 This CD-ROM is a customizable Microsoft® PowerPoint® presentation that includes: • Step-by-step, dynamic solutions of each In-Class Example from the Teacher Wraparound Edition • Additional, Try These exercises for each example • The 5-Minute Check Transparencies • Hot links to Glencoe Online Study Tools the table showing unemployment rates for various cities in Kansas, find a counterexample for the following statement. The unemployment rate is highest in the cities with the most people. Shawnee Jefferson Jackson Douglas Osage Wabaunsee Pottawatomie Civilian Rate Labor Force 90,254 9,937 8,915 55,730 10,182 3,575 11,025 3.1% 3.0% 2.8% 3.2% 4.0% 3.0% 2.1% Source: Labor Market Information Services—Kansas Department of Human Resources Osage has only 10,182 people on its civilian labor force, and it has a higher rate of unemployment than Shawnee, which has 90,254 people on its civilian labor force. Lesson 2-1 Inductive Reasoning and Conjecture 63 3 Practice/Apply Study Notebook Guided Practice GUIDED PRACTICE KEY Exercises Examples 4–5 6–7 8–9 1 2 3 Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 2. • include any other item(s) that they find helpful in mastering the skills in this lesson. Organization by Objective • Make Conjectures: 11–28 • Find Counterexamples: 29–36 Odd/Even Assignments Exercises 11–36 are structured so that students practice the same concepts whether they are assigned odd or even problems. 5. 8, 5, 2, 1, 4 7 Make a conjecture based on the given information. Draw a figure to illustrate your conjecture. 6–7. See margin. CD intersect at P. 6. PQ RS and RS TU 7. AB and Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 8. Given: x is an integer. Conjecture: x is negative. False; if x ⴝ 2, then x ⴝ (2) or 2. 9. Given: WXYZ is a rectangle. Conjecture: WX YZ and WZ XY true Application About the Exercises… Make a conjecture about the next item in each sequence. 4. 10. HOUSES Most homes in the northern United States have roofs made with steep angles. In the warmer areas of the southern states, homes often have flat roofs. Make a conjecture about why the roofs are different. Sample answer: Snow will not stick on a roof with a steep angle. ★ indicates increased difficulty Practice and Apply For Exercises See Examples 11–20 21–28 29–36 1 2 3 Make a conjecture about the next item in each sequence. 11. 12. Extra Practice See page 756. 13. 1, 2, 4, 8, 16 32 1 1 1 1 1 16. 1, , , , 2 4 8 16 32 14. 4, 6, 9, 13, 18 24 17. 2, 6, 18, 54 162 Assignment Guide Basic: 11–41 odd, 43–67 Average: 11–41 odd, 43–67 Advanced: 12–40 even, 41–64 (optional: 65–67) Answers 6. PQ TU P Q R S T U 11 1 5 7 15. , 1, , , 3 3 3 3 3 18. 5, 25, 125, 625 3125 Make a conjecture about the number of blocks in the next item of each sequence. 19. 20. 21. Lines ᐉ and m form four right angles. 22. A, B, and C are collinear. 23. ⬔3 and ⬔4 are supplementary. 24. ⬔ABD ⬵ ⬔DBC. 25. ∆PQR is a scalene triangle. 26. HI IJ JK KH 19– 20. See p. 123A. Make a conjecture based on the given information. Draw a figure to illustrate your conjecture. 21 –28. See p. 123A for figures. 21. Lines ᐉ and m are perpendicular. 22. A(2, 11), B(2, 1), C(5, 10) 23. 3 and 4 are a linear pair. is an angle bisector of ABC. 24. BD 25. P(1, 7), Q(6, 2), R(6, 5) 26. HIJK is a square. 27. PQRS is a rectangle. 28. B is a right angle in ABC. PQ SR, QR PS 64 Chapter 2 Reasoning and Proof (AB ) 2 (BC ) 2 (AC)2 7. A, B, C, and D are noncollinear. D A P C 64 B Chapter 2 Reasoning and Proof Differentiated Instruction Naturalist Students can practice brainstorming conjectures and finding counterexamples in nature. For example have students consider the statement, “If plants don’t receive water daily, they will not survive.” A counterexample would be a cactus, which can go weeks without water. Nature topics could include plants, animals, predator/food supply relationships, insects, weather, and so on. You can use scatter plots to make conjectures about the relationships between latitude, longitude, degree distance, and the monthly high temperature. Visit www.geometry online.com/WebQuest to continue work on your WebQuest project. counterexample. p. 57 (shown) p. 58 Inductive Reasoningand and Conjecture Exercises 1. 5, 10, 20, 40 Pattern: Each number is 2 times the previous number. Conjecture: The next number is 80. 2. 1, 10, 100, 1000 Pattern: Each number is 10 times the previous number. Conjecture: The next number is 10,000. 33. Given: n is a real number. Conjecture: n2 is a nonnegative number. true 6 7 8 5 5 5 1 5 3. 1, , , Pattern: Each number is more than the previous number. 9 5 Conjecture: The next number is . Make a conjecture based on the given information. Draw a figure to illustrate your conjecture. 4–7. Sample answers are given. 4. A(1, 1), B(2, 2), C(4, 4) 5. 1 and 2 form a right angle. 1 and 2 are complementary. Points A, B, and C are collinear. y collinear. 35. Given: JK KL LM MJ Conjecture: JKLM forms a square. False; JKLM may not have a right angle. P C (4, 4) B(2, 2) O A(–1, –1) 1 and error, a process of inductive reasoning Structural Formula The average mediumsized piano has about 230 strings. Each string has about 165 pounds of tension. That’s a combined tension of about 18 tons. H C H C H Make a conjecture about the next item in each sequence. 5. ABC is a right angle. H C C H H H H H H H C C C H H H 4. 12, 6, 3, 1.5, 0.75 0.375 6. Point S is between R and T. ⊥ BC BA RS ST RT R A S T C 7. P, Q, R, and S are noncollinear and P Q QR RS SP . 8. ABCD is a parallelogram. AB CD and BC AD. The segments form a square. C3H8 H 1 1 8 16 1 1 2 4 3. 2, 1, , , P Q S R A D B C Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 9. Given: S, T, and U are collinear and ST TU. Conjecture: T is the midpoint of S U . H 10. Given: 1 and 2 are adjacent angles. Conjecture: 1 and 2 form a linear pair. False; 1 and 2 could each measure 60°. 38. Make a conjecture about butane, which is the next compound in the group. Write its structural formula. 39. Write the chemical formula for the 7th compound in the group. C7H16 ★ 40. Develop a rule you could use to find the chemical formula of the nth substance in the alkane group. CnH2n 2 H C H H true 12. ALLERGIES Each spring, Rachel starts sneezing when the pear trees on her street blossom. She reasons that she is allergic to pear trees. Find a counterexample to Rachel’s conjecture. Sample answer: Rachel could be allergic to other types of plants that blossom when the pear trees blossom. Gl NAME ______________________________________________ DATE /M G Hill 60 ____________ Gl PERIOD G _____ Reading 2-1 Readingto to Learn Learn Mathematics Mathematics, p. 61 Inductive Reasoning and Conjecture Pre-Activity 41. CRITICAL THINKING The expression n2 n 41 has a prime value for n 1, n 2, and n 3. Based on this pattern, you might conjecture that this expression always generates a prime number for any positive integral value of n. Try different values of n to test the conjecture. Answer true if you think the conjecture is always true. Answer false and give a counterexample if you think the conjecture is false. false; n 41 www.geometryonline.com/self_check_quiz 11. Given: G H and JK form a right angle and intersect at P. Conjecture: GH ⊥ JK ELL How can inductive reasoning help predict weather conditions? Read the introduction to Lesson 2-1 at the top of page 62 in your textbook. • What kind of weather patterns do you think meteorologists look at to help predict the weather? Sample answer: patterns of high and low temperatures, including heat spells and cold spells; patterns of precipitation, including wet spells and dry spells • What is a factor that might contribute to long-term changes in the weather? Sample answer: global warming due to high usage of fossil fuels Reading the Lesson 1. Explain in your own words the relationship between a conjecture, a counterexample, and inductive reasoning. Lesson 2-1 Inductive Reasoning and Conjecture 65 Mary Kate Denny/PhotoEdit Sample answer: A conjecture is an educated guess based on specific examples or information. A counterexample is an example that shows that a conjecture is false. Inductive reasoning is the process of making a conjecture based on specific examples or information. 2. Make a conjecture about the next item in each sequence. NAME ______________________________________________ DATE Answers 2-1 Enrichment Enrichment, 29. p. 62 Counterexamples b. 1, , , c. 0, 1, 3, 6, 10 15 e. 1, 8, 27, 64 125 g. d. 8, 3, 2, 7 12 f. 1, 2, 4, 8 16 h. a. The sum of two odd integers is even. true b. The product of an odd integer and an even integer is odd. Example False; sample answer: 5 8 40, which is even. 1 x Is the statement 1 true when you replace x with c. The opposite of an integer is a negative integer. False; sample answer: The 1, 2, and 3? Is the statement true for all reals? If possible, find a counterexample. W 1 1 1 1 3 9 27 81 a. 5, 9, 13, 17 21 3. State whether each conjecture is true or false. If the conjecture is false, give a counterexample. 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. 2 1 31. ____________ PERIOD _____ X Y Z opposite of the integer 5 is 5, which is a positive integer. d. The perfect squares (squares of whole numbers) alternate between odd and even. 1 1 1 1 1 1, 1, and 1. But when x , then 2. This counterexample 1 2 3 2 x true shows that the statement is not always true. Helping You Remember 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 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 4. Write a short sentence that can help you remember why it only takes one counterexample to prove that a conjecture is false. Sample answer: True means always true. Lesson 2-1 Inductive Reasoning and Conjecture 65 Lesson 2-1 C H p. 59 and Practice, p. 60 (shown) Inductive Reasoning and Conjecture Make a conjecture based on the given information. Draw a figure to illustrate your conjecture. 5–8. Sample answers are given. Propane C2H6 H R ____________ Gl PERIOD G _____ true 38. Butane will have 4 carbon atoms and 10 hydrogen atoms. H C H Source: www.pianoworld.com H H F Skills Practice, 2-1 Practice (Average) B Ethane H Music Q NAME ______________________________________________ DATE /M G Hill 57 2. 5, 10, 15, 20 25 Alkanes CH4 P E 1. CHEMISTRY For Exercises 38–40, use the following information. Hydrocarbons are molecules composed of only carbon (C) and hydrogen (H) atoms. The simplest hydrocarbons are called alkanes. The first three alkanes are shown below. Chemical Formula D T 37. MUSIC Many people learn to play the piano by ear. This means that they first learned how to play without reading music. What process did they use? trial Methane E and F are congruent. E C Gl Compound Name B W 7. E and F are right angles. ABC and DBE are congruent. A R 2 T x 6. ABC and DBE are vertical angles. 36. Given: noncollinear points R, S, and T S , S T , and R T form a triangle. true Conjecture: R H Example 2 Make a conjecture about the number of small squares in the next figure. Observe a pattern: The sides of the squares have measures 1, 2, and 3 units. Conjecture: For the next figure, the side of the square will be 4 units, so the figure will have 16 small squares. Describe the pattern. Then make a conjecture about the next number in the sequence. counterexample. 34. Given: DE EF . False; D, E, and F do not have to be Conjecture: E is the midpoint of DF H ____________ PERIOD _____ Make Conjectures A conjecture is a guess based on analyzing information or observing a pattern. Making a conjecture after looking at several situations is called inductive reasoning. Example 1 Make a conjecture about the next number in the sequence 1, 3, 9, 27, 81. Analyze the numbers: Notice that each number is a power of 3. 1 3 9 27 81 30 31 32 33 34 Conjecture: The next number will be 35 or 243. 30. Given: m y 10, y 4 Conjecture: m 6 False; if y 7 and m 5, then 7 5 10 and 5 4, but 7 6. 31. Given: points W, X, Y, and Z Conjecture: W, X, Y, and Z are noncollinear. False; see margin for 32. Given: A(4, 8), B(3, 8), C(3, 5) Conjecture: ABC is a right triangle. true NAME ______________________________________________ DATE Study Guide andIntervention Intervention, 2-1 Study Guide and Lesson 2-1 Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 29. Given: 1 and 2 are complementary angles. Conjecture: 1 and 2 form a right angle. False; see margin for 42. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can inductive reasoning help predict weather conditions? 4 Assess Include the following in your answer: • an explanation as to how a conjecture about a weather pattern in the summer might be different from a similar weather pattern in the winter, and • a conjecture about tomorrow’s weather based on your local weather over the past several days. Open-Ended Assessment Writing Ask students to write five conjectures about school rules or activities. Then have students swap papers with a partner and try to come up with a counterexample for each conjecture. An example statement could be: Students must attend school Monday through Friday. A counterexample for this would be a holiday or a snow day. Standardized Test Practice 43. What is the next term in the sequence 1, 1, 2, 3, 5, 8? C A 11 B 12 C 13 D 14 44. ALGEBRA If the average of six numbers is 18 and the average of three of the numbers is 15, then what is the sum of the remaining three numbers? D A 21 B 45 C 53 D 63 Maintain Your Skills Getting Ready for Lesson 2-2 Basic Skill Students will learn about logic statements in Lesson 2-2. They will determine the truth value of various situations. Use Exercises 65–67 to determine your students’ familiarity with determining which elements make a statement true. Answer 42. Sample answer: By past experience, when dark clouds appear, there is a chance of rain. Answers should include the following. • When there is precipitation in the summer, it is usually rain because the temperature is above freezing. When the temperature is below freezing, as in the winter, ice or snow forms. • See students’ work. Mixed Review pentagon, convex, regular hexagon, convex, irregular 48. Yes; the symbol denotes that KJN is a right angle. 49. No; we do not know anything about the angle measures. 50. No; we do not know whether MNP is a right angle. 51. Yes; they form a linear pair. 52. Yes; since the other three angles in rectangle KLPJ are right angles, KLP must also be a right angle. Determine whether each statement can be assumed from the figure. Explain. (Lesson 1-5) 48. KJN is a right angle. 66 heptagon, concave, irregular K L M J P N 49. PLN NLM 50. PNL and MNL are complementary. 51. KLN and MLN are supplementary. 52. KLP is a right angle. Find the coordinates of the midpoint of a segment having the given endpoints. (Lesson 1-3) 53. A B for A(1, 3), B(5, 5) (2, 1) 54. C D for C(4, 1), D(3, 7) (0.5, 4) 55. F G for F(4, 9), G(2, 15) (1, 12) 56. H J for H(5, 2), J(7, 4) (1, 1) 57. K L for K(8, 1.8), L(3, 6.2) (5.5, 2.2) 58. M N for M(1.5, 6), N(4, 3) (2.75, 1.5) Find the value of the variable and MP, if P is between M and N. (Lesson 1-2) 59. MP 7x, PN 3x, PN 24 8; 56 60. MP 2c, PN 9c, PN 63 7; 14 61. MP 4x, PN 5x, MN 36 4; 16 62. MP 6q, PN 6q, MN 60 5; 30 63. MP 4y 3, PN 2y, MN 63 64. MP 2b 7, PN 8b, MN 43 10; 43 Getting Ready for the Next Lesson 66 Chapter 2 Reasoning and Proof Name each polygon by its number of sides and then classify it as convex or concave and regular or irregular. (Lesson 1-6) 45. 46. 47. 5; 3 BASIC SKILL Determine which values in the given replacement set make the inequality true. 66. 12 x 0 13, 14 67. 5x 1 25 5, 6, 7 65. x 2 5 4, 5 {2, 3, 4, 5} {11, 12, 13, 14} {4, 5, 6, 7} Chapter 2 Reasoning and Proof Lesson Notes Logic • Determine truth values of conjunctions and disjunctions. 1 Focus • Construct truth tables. does logic apply to school? Vocabulary • • • • • • • statement truth value negation compound statement conjunction disjunction truth table When you answer true-false questions on a test, you are using a basic principle of logic. For example, refer to the map, and answer true or false. VIRGINIA NORTH CAROLINA Raleigh Ashville Gastonia Charlotte SOUTH CAROLINA Raleigh is a city in North Carolina. Willmington You know that there is only one correct answer, either true or false. Study Tip Statements A mathematical statement with one or more variables is called an open sentence. The truth value of an open sentence cannot be determined until values are assigned to the variables. A statement with only numeric values is a closed sentence. DETERMINE TRUTH VALUES A statement , like the true-false example above, is any sentence that is either true or false, but not both. Unlike a conjecture, we know that a statement is either true or false. The truth or falsity of a statement is called its truth value . Statements are often represented using a letter such as p or q. The statement above can be represented by p. p: Raleigh is a city in North Carolina. This statement is true. The negation of a statement has the opposite meaning as well as an opposite truth value. For example, the negation of the statement above is not p. not p: Raleigh is not a city in North Carolina. In this case, the statement is false. Negation • Words If a statement is represented by p, then not p is the negation of the statement. 5-Minute Check Transparency 2-2 Use as a quiz or review of Lesson 2-1. Mathematical Background notes are available for this lesson on p. 60C. does logic apply to school? Ask students: • Determine whether the following statement is true or false: “South Carolina borders North Carolina, Georgia, and Tennessee.” false • Locate Wilmington on the map of North Carolina. Is this a coastal or inland city? Make a conjecture about whether you might find ocean or lakes in Wilmington given its location on the map. Coastal; sample answer: You would find ocean in Wilmington, North Carolina. • Symbols ~p, read not p Two or more statements can be joined to form a compound statement . Consider the following two statements. p: Raleigh is a city in North Carolina. q: Raleigh is the capital of North Carolina. The two statements can be joined by the word and. p and q: Raleigh is a city in North Carolina, and Raleigh is the capital of North Carolina. Lesson 2-2 Logic 67 Resource Manager Workbook and Reproducible Masters Chapter 2 Resource Masters • Study Guide and Intervention, pp. 63–64 • Skills Practice, p. 65 • Practice, p. 66 • Reading to Learn Mathematics, p. 67 • Enrichment, p. 68 • Assessment, p. 119 Graphing Calculator and Computer Masters, p. 20 Teaching Geometry With Manipulatives Masters, p. 16 Transparencies 5-Minute Check Transparency 2-2 Answer Key Transparencies Technology Interactive Chalkboard Lesson x-x Lesson Title 67 The statement formed by joining p and q is an example of a conjunction. 2 Teach Conjunction • Words DETERMINE TRUTH VALUES In-Class Example A conjunction is a compound statement formed by joining two or more statements with the word and. • Symbols p q, read p and q Power Point® A conjunction is true only when both statements in it are true. Since it is true that Raleigh is in North Carolina and it is the capital, the conjunction is also true. 1 Use the following statements to write a compound statement for each conjunction. Then find its truth value. p: One foot is 14 inches. q: September has 30 days. r: A plane is defined by three noncollinear points. Example 1 Truth Values of Conjunctions Use the following statements to write a compound statement for each conjunction. Then find its truth value. p: January 1 is the first day of the year. q: ⴚ5 ⴙ 11 ⴝ ⴚ6 r: A triangle has three sides. a. p and q One foot is 14 inches, and September has 30 days; false. a. p and q January is the first day of the year, and 5 11 6. p and q is false, because p is true and q is false. b. r p A plane is defined by three noncollinear points, and one foot is 14 inches; false. b. r p A triangle has three sides, and January 1 is the first day of the year. r p is true, because r is true and p is true. c. q r September does not have 30 days, and a plane is defined by three noncollinear points; false. d. p r A foot is not 14 inches, and a plane is defined by three noncollinear points; true. Study Tip Negations The negation of a statement is not necessarily false. It has the opposite truth value of the original statement. c. p and not r January 1 is the first day of the year, and a triangle does not have three sides. p and not r is false, because p is true and not r is false. d. ~q r 5 11 6, and a triangle has three sides q r is true because q is true and r is true. Statements can also be joined by the word or. This type of statement is a disjunction. Consider the following statements. p: Ahmed studies chemistry. q: Ahmed studies literature. p or q: Ahmed studies chemistry, or Ahmed studies literature. Disjunction • Words A disjunction is a compound statement formed by joining two or more statements with the word or. • Symbols p q, read p or q 68 Chapter 2 Reasoning and Proof 68 Chapter 2 Reasoning and Proof A disjunction is true if at least one of the statements is true. In the case of p or q above, the disjunction is true if Ahmed either studies chemistry or literature or both. The disjunction is false only if Ahmed studies neither chemistry nor literature. In-Class Example Power Point® 2 Use the following statements Example 2 Truth Values of Disjunctions Use the following statements to write a compound statement for each disjunction. Then find its truth value. p: 100 ⴜ 5 ⴝ 20 q: The length of a radius of a circle is twice the length of its diameter. r: The sum of the measures of the legs of a right triangle equals the measure of the hypotenuse. a. p or q 100 5 20, or the length of a radius of a circle is twice the length of its diameter. p or q is true because p is true. It does not matter that q is false. b. q r The length of a radius of a circle is twice the length of its diameter, or the sum of the measures of the legs of a right triangle equals the measure of the hypotenuse. q r is false since neither statement is true. Study Tip Venn Diagrams The size of the overlapping region in a Venn Diagram does not indicate how many items fall into that category. Conjunctions can be illustrated with Venn diagrams. Refer to the statement at the beginning of the lesson. The Venn diagram at the right shows that Raleigh (R) is represented by the intersection of the set of cities in North Carolina and the set of state capitals. In other words, Raleigh must be in the set containing cities in North Carolina and in the set of state capitals. to write a compound statement for each disjunction. Then find its truth value. is proper notation for p: AB “line AB.” q: Centimeters are metric units. r: 9 is a prime number. a. p or q B A is proper notation for “line AB,” or centimeters are metric units; true. b. q r Centimeters are metric units, or 9 is a prime number; true. All U.S. Cities U.S. State Capitals R Cities in North Carolina A disjunction can also be illustrated with a Venn diagram. Consider the following statements. p: Jerrica lives in a U.S. state capital. q: Jerrica lives in a North Carolina city. p q: Jerrica lives in a U.S. state capital, or Jerrica lives in a North Carolina city. In the Venn diagrams, the disjunction is represented by the union of the two sets. The union includes all U.S. capitals and all cities in North Carolina. The city in which Jerrica lives could be located in any of the three regions of the union. The three regions represent A U.S. state capitals excluding the capital of North Carolina, B cities in North Carolina excluding the state capital, and C the capital of North Carolina, which is Raleigh. www.geometryonline.com/extra_examples All U.S. Cities U.S. State Capitals A C Cities in North Carolina B Lesson 2-2 Logic 69 Lesson 2-2 Logic 69 In-Class Example Venn diagrams can be used to solve real-world problems involving conjunctions and disjunctions. Power Point® 3 DANCING The Venn diagram Example 3 Use Venn Diagrams shows the number of students enrolled in Monique’s Dance School for tap, jazz, and ballet classes. RECYCLING The Venn diagram shows the number of neighborhoods that have a curbside recycling program for paper or aluminum. Curbside Recycling Tap 28 17 Paper Aluminum 12 46 20 Jazz 43 13 9 25 29 Ballet a. How many neighborhoods recycle both paper and aluminum? The neighborhoods that have paper and aluminum recycling are represented by the intersection of the sets. There are 46 neighborhoods that have paper and aluminum recycling. a. How many students are enrolled in all three classes? 9 b. How many students are enrolled in tap or ballet? 121 b. How many neighborhoods recycle paper or aluminum? The neighborhoods that have paper or aluminum recycling are represented by the union of the sets. There are 12 46 20 or 78 neighborhoods that have paper or aluminum recycling. c. How many students are enrolled in jazz and ballet and not tap? 25 c. How many neighborhoods recycle paper and not aluminum? The neighborhoods that have paper and not aluminum recycling are represented by the nonintersecting portion of the paper region. There are 12 neighborhoods that have paper and not aluminum recycling. TRUTH TABLES Intervention Tell students New that truth tables must display all combinations of Ts and Fs to exhaust all possible outcomes, so for each statement, p, q, and r, they will need to mix the occurrences of true and false. First, they should determine the number of rows they need. Then fill the top half of the p column with Ts and the other half with Fs. For the q column, they can alternate T and F the whole way down. If they need an r column, they can alternate sets of 2 Ts and 2 Fs all the way down, and so on. Assure students that as long as they initially set up the basic structure correctly, they should be able to produce the rest of the table quite easily. 70 Chapter 2 Reasoning and Proof TRUTH TABLES A convenient method for organizing the truth values of statements is to use a truth table . Negation p p T F F T If p is a true statement, then p is a false statement. If p is a false statement, then p is a true statement. Truth tables can also be used to determine truth values of compound statements. Study Tip Conjunction A conjunction is true only when both statements are true. Disjunction p q pq T T T T F F T F T T F F T T F F F F F Tautology p q pq A compound sentence is a tautology if its truth value is always true. For example, “It is snowing or it is not snowing” is a tautology. T T T F F A disjunction is false only when both statements are false. You can use the truth values for negation, conjunction, and disjunction to construct truth tables for more complex compound statements. 70 Chapter 2 Reasoning and Proof Answers 1. The conjunction (p and q) is represented by the intersection of the two circles. 2a. Sample answer: October has 31 days or 5 3 8. 2b. Sample answer: A square has five right angles and the Postal Service does not deliver mail on Sundays. 2c. Sample answer: July 5th is not a national holiday. 3. A conjunction is a compound statement using the word and, while a disjunction is a compound statement using the word or. Example 4 Construct Truth Tables In-Class Example Construct a truth table for each compound statement. a. p q Step 1 Make columns with the headings p, q, q, and p q. Step 2 List the possible combinations of truth values for p and q. Step 3 Use the truth values of q to determine the truth values of q. Step 4 Use the truth values for p and q to write the truth values for p q. p q q p q T T F F T F T T F T F F F F T F 4 Teaching Tip Tell students that they could potentially interchange the columns for p, q, and r in 4c, and as long as they correctly fill in the columns for p q and (p q) r, the overall outcome would be the same. They would still end up with 5 Ts and 3 Fs, but the Ts and Fs would be in a different order. Step 1 Construct a truth table for each compound statement. Step 2 Step 3 a. p q Step 4 b. p q Study Tip Truth Tables Use the Fundamental Counting Principle to determine the number of rows necessary. Power Point® p q p q p q T T F F F T F F T T F T T F T F F T T T b. p (q r) c. (p q) r Make columns for p, q, p q, r, and (p q) r. p q pq r (p q) r T T T T T T F F T T T T T F T T F F F F F T F T T F F F T T F T F F F F F F F F q p p q T F T F F F T T T F T T p T T F F p T T T T F F F F q T F T F T F T F r T T F F T T F F q q r p (q r) F F T T T T F F T T F T F F F T T T F F F T F F c. (p q) r Concept Check 1– 3. See margin. 1. Describe how to interpret the Venn diagram for p q. 2. OPEN ENDED Write a compound statement for each condition. a. a true disjunction b. a false conjunction c. a true statement that includes a negation pq p q 3. Explain the difference between a conjunction and a disjunction. Lesson 2-2 Logic 71 p T T T T F F F F q T F T F T F T F r r p q (p q) r T F T F T F T F F T T T F T T T T F T F T F F F F T T T F T F F Differentiated Instruction Logical/Mathematical Have students examine the relationship between the number of simple statements (p, q, and r) and the number of rows necessary to exhaust all possible combinations in a truth table. Point out that for Example 4a, there are 2 statements and 4 rows; for Example 4c, there are 3 statements and 8 rows. Ask students to form a conjecture about how many rows would be needed for 4, 5, and n statements. Similarly, students can examine the relationship between the number of circles and the number of intersecting areas of a Venn diagram. Lesson 2-2 Logic 71 3 Practice/Apply Study Notebook Guided Practice GUIDED PRACTICE KEY Exercises Examples 4–6 7–9 10–14 15–17 1 2 3 4 Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 2. • include an example of a Venn diagram and an example of a truth table. • include any other item(s) that they find helpful in mastering the skills in this lesson. Odd/Even Assignments Exercises 18–40 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Exercises 48–50 require the Internet or other research materials. Assignment Guide Basic: 19–37 odd, 41–51 odd, 52–73 Average: 19–51 odd, 52–73 Advanced: 18–50 even, 51–52, 54–69 (optional: 70–73) 4. p and q false 7. p or ~q true 6. q r false 9. ~p ~r false 5. p and r true 8. q r true 10. Copy and complete the truth table. p q q p q T T F F T F F T F F T F T T F F Construct a truth table for each compound statement. 11–14. See p. 123A. 11. p q 12. q r 13. ~p r 14. (p q) r Application About the Exercises… Organization by Objective • Determine Truth Values: 18–29, 42–48 • Truth Tables: 30–41 Use the following statements to write a compound statement for each conjunction and disjunction. Then find its truth value. 4–9. See margin for statements. p: 9 ⴙ 5 ⴝ 14 q: February has 30 days. r: A square has four sides. AGRICULTURE For Exercises 15–17, refer to the Venn diagram that represents the states producing more than 100 million bushels of corn or wheat per year. 15. How many states produce more than 100 million bushels of corn? 14 16. How many states produce more than 100 million bushels of wheat? 7 17. How many states produce more than 100 million bushels of corn and wheat? 3 Grain Production Corn Chapter 2 Reasoning and Proof WA MT ID ★ indicates increased difficulty Practice and Apply Use the following statements to write a compound statement for each conjunction and disjunction. Then find its truth value. 18– 29. See p. 123A for statements. For Exercises See Examples 18–29 30–41 42–48 1, 2 4 3 Extra Practice See page 756. p: q: r: s: ⴚ64 ⴝ 8 An equilateral triangle has three congruent sides. 00 An obtuse angle measures greater than 90° and less than 180°. 18. 21. 24. 27. p and q false r and s false p s false s q true 19. 22. 25. 28. p or q true q or r true q r false (p q) s true Copy and complete each truth table. 30. p q p p q T T T F F T F F F F T T 31. T F T T 20. 23. 26. 29. p and r false q and s true r p false s (q and r) true p q p q p q T T F T F F F F T T F T T F F F T T F F Answers 72 ND Source: U.S. Department of Agriculture 72 Chapter 2 Reasoning and Proof 4. 9 5 14 and February has 30 days. 5. 9 5 14 and a square has four sides. 6. February has 30 days and a square has four sides. 7. 9 5 14 or February does not have 30 days. Wheat IA PA IL NE IN MN MO CO SD WI OH KS MI KY 8. February has 30 days or a square has four sides. 9. 9 5 14 or a square does not have four sides. 45. Level of Participation Among 310 Students Sports 95 20 Academic Clubs 60 ★ 32. Copy and complete the truth table. NAME ______________________________________________ DATE p q r pq (p q) r T T T T T F T F T T T T T T T F F T F T F T F F F F F F T T F T F F F T F F F p. 63 Logic(shown) and p. 64 Determine Truth Values A statement is any sentence that is either true or false. The truth or falsity of a statement is its truth value. A statement can be represented by using a letter. For example, Statement p: Chicago is a city in Illinois. The truth value of statement p is true. Several statements can be joined in a compound statement. Construct a truth table for each compound statement. 33– 40. See pp. 123A–123B. 33. q and r 34. p or q 35. p or r 36. p and q 38. p q ★ 39. p (q r) ★ 40. p (q r) 37. q r MUSIC For Exercises 41–44, use the following information. A group of 400 teens were asked what type of Music Preference music they listened to. They could choose among pop, rap, and country. The results are shown in the Venn diagram. Pop 34 Country 175 45 41. How many teens said that they listened to none of 7 these types of music? 42 25 10 42. How many said that they listened to all three types Rap of music? 7 62 42 43. How many said that they listened to only pop and rap music? 25 44. How many teens said that they listened to pop, rap, or country music? 358 Statement p and statement q joined by the word and is a conjunction. Statement p and statement q joined by the word or is a disjunction. Symbols: p q (Read: p and q ) Symbols: p q (Read: p or q) Negation: not p is the negation of the statement p. Symbols: p (Read: not p) The conjunction p q is true only when both p and q are true. The disjunction p q is true if p is true, if q is true, or if both are true. The statements p and p have opposite truth values. Example 1 Write a compound statement for each conjunction. Then find its truth value. p: An elephant is a mammal. q: A square has four right angles. Example 2 Write a compound statement for each disjunction. Then find its truth value. p: A diameter of a circle is twice the radius. q: A rectangle has four equal sides. a. p q Join the statements with and: An elephant is a mammal and a square has four right angles. Both parts of the statement are true so the compound statement is true. a. p q Join the statements p and q with the word or: A diameter of a circle is twice the radius or a rectangle has four equal sides. The first part of the compound statement, p, is true, so the compound statement is true. b. p q p is the statement “An elephant is not a mammal.” Join p and q with the word and: An elephant is not a mammal and a square has four right angles. The first part of the compound statement, p, is false. Therefore the compound statement is false. Lesson 2-2 T ____________ PERIOD _____ Study Guide andIntervention Intervention, 2-2 Study Guide and b. p q Join p and q with the word or: A diameter of a circle is not twice the radius or a rectangle has four equal sides. Neither part of the disjunction is true, so the compound statement is false. Exercises Write a compound statement for each conjunction and disjunction. Then find its truth value. p: 10 8 18 q: September has 30 days. r: A rectangle has four sides. 1. p and q 10 8 18 and September has 30 days; true. 2. p or r 10 8 18 or a rectangle has four sides; true. 3. q or r September has 30 days or a rectangle has four sides; true. 4. q and r September has 30 days and a rectangle does not have four sides; false. NAME ______________________________________________ DATE /M G Hill 63 Gl Skills Practice, p. 65 and 2-2 Practice (Average) Practice, p. 66 (shown) Logic ____________ Gl PERIOD G _____ Use the following statements to write a compound statement for each conjunction and disjunction. Then find its truth value. p: 60 seconds 1 minute q: Congruent supplementary angles each have a measure of 90. r: 12 11 1 1. p q 60 seconds 1 minute and congruent supplementary angles each have a measure of 90; true. 2. q r Congruent supplementary angles each have a measure of 90 or 12 11 1; true. 3. p q 60 seconds 1 minute or congruent supplementary angles each SCHOOL For Exercises 45–47, use the following information. In a school of 310 students, 80 participate in academic clubs, 115 participate in sports, and 20 students participate in both. 45. Make a Venn diagram of the data. See margin. 46. How many students participate in either clubs or sports? 175 47. How many students do not participate in either clubs or sports? 135 have a measure of 90; true. 4. p r 60 seconds 1 minute and 12 11 1; false. Copy and complete each truth table. 5. 6. p q p q p q p q p p q p (p q) T T F T F T T T F F T F T F F F T T T F T F F T T T F F F F T T T F T T T F F F Construct a truth table for each compound statement. RESEARCH For Exercises 48–50, use the Internet or another resource to determine whether each statement about cities in New York is true or false. 48. Albany is not located on the Hudson river. false 49. Either Rochester or Syracuse is located on Lake Ontario. true 50. It is false that Buffalo is located on Lake Erie. false Nationwide, approximately 80% of high school seniors participate in extracurricular activities. Athletics, performing arts, and clubs are the most popular. CRITICAL THINKING For Exercises 51 and 52, use the following information. All members of Team A also belong to Team B, but only some members of Team B also belong to Team C. Teams A and C have no members in common. 51. 2-2 Enrichment Enrichment, B C Solve the alphametic puzzle at the right. F 8, O 3, U 1, R 0 Can you find other solutions to this puzzle? HALF 9703 2. TWO T F F T F T F F F T F F F F T T T T information. The Venn diagram shows the number of students in the band who work after school or on the weekends. Work After School 5 3 Work Weekends 17 NAME ______________________________________________ DATE /M G Hill 66 ____________ Gl PERIOD G _____ ELL How does logic apply to school? How can you use logic to help you answer a multiple-choice question on a Reading the Lesson 73 ____________ PERIOD _____ FOUR ONE F I VE 1. Supply one or two words to complete each sentence. compound a. Two or more statements can be joined to form a statement. b. A statement that is formed by joining two statements with the word or is called a disjunction . truth value . c. The truth or falsity of a statement is called its d. A statement that is formed by joining two statements with the word and is called a conjunction . e. A statement that has the opposite truth value and the opposite meaning from a given negation statement is called the of the statement. 2. Use true or false to complete each sentence. false a. If a statement is true, then its negation is . true b. If a statement is false, then its negation is . true c. If two statements are both true, then their conjunction is true their disjunction is . false d. If two statements are both false, then their conjunction is false their disjunction is . e. If one statement is true and another is false, then their conjunction is false true and their disjunction is . and and 3. Consider the following statements: p: Chicago is the capital of Illinois. q: Sacramento is the capital of California. Write each statement symbolically and then find its truth value. a. Sacramento is not the capital of California. q; false b. Sacramento is the capital of California and Chicago is not the capital of Illinois. q p; true Helping You Remember Find a value for each letter in each alphametic. Sample answers are shown 1. F F F T standardized test if you are not sure of the correct answer? Sample answer: Eliminate the choices that you know are wrong. Then choose the one you think is most likely correct from the ones that are left. 8310 347 8657 N 4, E 7, I 6, and V 5. T T F T Read the introduction to Lesson 2-2 at the top of page 67 in your textbook. An alphametic is a computation puzzle using letters instead of digits. Each letter represents one of the digits 0–9, and two different letters cannot represent the same digit. Some alphametic puzzles have more than one answer. Example F T F Mathematics, p. 67 Logic p. 68 Since R E E, the value of R must be 0. Notice that the thousands digit must be the same in the first addend and the sum. Since the value of I is 9 or less, O must be 4 or less. Use trial and error to find values that work. F T F T F Pre-Activity Letter Puzzles A T T T F T Reading 2-2 Readingto to Learn Learn Mathematics Bill Bachmann/PhotoEdit NAME ______________________________________________ DATE T SCHOOL For Exercises 9 and 10, use the following Gl c. No person that is a member of Team A can be a member of Team C. Lesson 2-2 Logic F 9. How many students work after school and on weekends? 3 b. If a person is not a member of Team B, then the person is not a member of Team A. www.geometryonline.com/self_check_quiz p q p q p q q (p q) T T F 10. How many students work after school or on weekends? 25 51. Draw a Venn diagram to illustrate the situation. See margin. 52. Which of the following statements is true? b a. If a person is a member of Team C, then the person is not a member of Team A. Source: National Center for Education Statistics 8. q (p q) p q q p q q (p q) 4. Prefixes can often help you to remember the meaning of words or to distinguish between similar words. Use your dictionary to find the meanings of the prefixes con and dis and 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. 734 Lesson 2-2 Logic 73 Lesson 2-2 School 7. q (p q) 53. WRITING IN MATH 4 Assess How does logic apply to school? Open-Ended Assessment Modeling Have students model a Venn diagram and a truth table with buttons or chips. For the Venn diagram, students can draw two large overlapping circles on a piece of paper and label them Science and English. Then they can place buttons on the diagram to represent the number of students in the class who like one, the other, or both subjects. Students can also draw a grid and use white buttons for true and black buttons for false to model one of the truth tables in the lesson. Include the following in your answer: • an example of a conjunction using statements about your favorite subject and your favorite extracurricular activity, and • a Venn diagram showing various characteristics of the members of your geometry class (for example, male/female, grade in school, and so on). Standardized Test Practice A Mixed Review 63. 34.4 64. 30.4 Make a conjecture about the next item in each sequence. (Lesson 2-1) 3 3 3 56. 3, 5, 7, 9 11 57. 1, 3, 9, 27 81 58. 6, 3, , 2 4 8 1 59. 17, 13, 9, 5 1 60. 64, 16, 4, 1 61. 5, 15, 45, 135 405 4 COORDINATE GEOMETRY Find the perimeter of each polygon. Round answers to the nearest tenth. (Lesson 1-6) 62. triangle ABC with vertices A(6, 7), B(1, 3), and C(2, 7) 33.1 63. square DEFG with vertices D(10, 9), E(5, 2), F(2, 7), and G(3, 14) 64. quadrilateral HIJK with vertices H(5, 10), I(8, 9), J(5, 5), and K(2, 4) 65. hexagon LMNPQR with vertices L(2, 1), M(4, 5), N(6, 4), P(7, 4), Q(5, 8), and R(3, 7) 29.5 Measure each angle and classify it as right, acute, or obtuse. (Lesson 1-4) 66. ABC 145°, obtuse 67. DBC 55°, acute 68. ABD 90°, right Quiz (Lessons 2-1 and 2-2) is available on p. 119 of the Chapter 2 Resource Masters. D C A B 69. FENCING Michelle wanted to put a fence around her rectangular garden. The front and back measured 35 feet each, and the sides measured 75 feet each. If she wanted to make sure that she had enough feet of fencing, how much should she buy? (Lesson 1-2) 222 ft Answer 74 Chapter 2 Reasoning and Proof C Maintain Your Skills Assessment Options 53. Sample answer: Logic can be used to eliminate false choices on a multiple choice test. Answers should include the following. • Math is my favorite subject and drama club is my favorite activity. • See students’ work. B 54. Which statement about ABC has the same truth value as AB BC? A A mA mC B mA mB C AC BC D AB AC 55. ALGEBRA If the sum of two consecutive even integers is 78, which number is the greater of the two integers? C A 36 B 38 C 40 D 42 Getting Ready for Lesson 2-3 Prerequisite Skill Students will learn about conditional statements in Lesson 2-3. They will substitute the hypothesis and conclusion for the if and then parts of statements. Use Exercises 70–73 to determine your students’ familiarity with substituting numbers for variables in algebraic expressions. Answer the question that was posed at the beginning of the lesson. See margin. Getting Ready for the Next Lesson 74 PREREQUISITE SKILL Evaluate each expression for the given values. (To review evaluating algebraic expressions, see page 736.) 70. 5a 2b if a 4 and b 3 14 72. 4e 3f if e 1 and f 2 ⴚ10 Chapter 2 Reasoning and Proof 71. 4cd 2d if c 5 and d 2 44 73. 3g2 h if g 8 and h 8 184 Lesson Notes Conditional Statements • Analyze statements in if-then form. • Write the converse, inverse, and contrapositive of if-then statements. are conditional statements used in advertisements? Vocabulary • • • • • • • • • conditional statement if-then statement hypothesis conclusion related conditionals converse inverse contrapositive logically equivalent 5-Minute Check Transparency 2-3 Use as a quiz or review of Lesson 2-2. Advertisers often lure consumers into purchasing expensive items by convincing them that they are getting something for free in addition to their purchase. Mathematical Background notes are available for this lesson on p. 60C. - IF-THEN STATEMENTS The statements above are examples of conditional statements. A conditional statement is a statement that can be written in if-then form. The first example above can be rewritten to illustrate this. If you buy a car, then you get $1500 cash back. If-Then Statement • Words An if-then statement is written in the form if p, then q. The phrase immediately following the word if is called the hypothesis , and the phrase immediately following the word then is called the conclusion. • Symbols p → q, read if p then q, or p implies q. Example 1 Identify Hypothesis and Conclusion The word if is not part of the hypothesis. The word then is not part of the conclusion. hypothesis Reading Math are conditional statements used in advertisements? Ask students: • Use the advertisements to answer the following questions: What happens if you buy a new car? How can you get a free phone? You get $1500 cash back; enroll in phone service for one year. • How effective are these types of advertisements? Why? Very effective; people like to get something for free, even if they have to pay for something else. Identify the hypothesis and conclusion of each statement. a. If points A, B, and C lie on line ᐉ, then they are collinear. If points A, B, and C lie on line ᐉ, then they are collinear. Study Tip 1 Focus conclusion Hypothesis: points A, B, and C lie on line ᐉ Conclusion: they are collinear b. The Tigers will play in the tournament if they win their next game. Hypothesis: the Tigers win their next game Conclusion: they will play in the tournament Identifying the hypothesis and conclusion of a statement is helpful when writing statements in if-then form. Lesson 2-3 Conditional Statements 75 Resource Manager Workbook and Reproducible Masters Chapter 2 Resource Masters • Study Guide and Intervention, pp. 69–70 • Skills Practice, p. 71 • Practice, p. 72 • Reading to Learn Mathematics, p. 73 • Enrichment, p. 74 Teaching Geometry With Manipulatives Masters, p. 43 Transparencies 5-Minute Check Transparency 2-3 Real-World Transparency 2 Answer Key Transparencies Technology GeomPASS: Tutorial Plus, Lesson 6 Interactive Chalkboard Lesson x-x Lesson Title 75 Example 2 Write a Conditional in If-Then Form IF-THEN STATEMENTS In-Class Examples Power Point® 1 Identify the hypothesis and conclusion of each statement. a. If a polygon has 6 sides, then it is a hexagon. Hypothesis: a polygon has 6 sides; Conclusion: it is a hexagon Study Tip If-Then Statements When you write a statement in if-then form, identify the condition that causes the result as the hypothesis. The result is the conclusion. b. Perpendicular lines intersect. Sometimes you must add information to a statement. In this case, it is necessary to know that perpendicular lines come in pairs. Hypothesis: two lines are perpendicular Conclusion: they intersect If two lines are perpendicular, then they intersect. b. Tamika will advance to the next level of play if she completes the maze in her computer game. Hypothesis: Tamika completes the maze in her computer game; Conclusion: she will advance to the next level of play. Recall that the truth value of a statement is either true or false. The hypothesis and conclusion of a conditional statement, as well as the conditional statement itself, can also be true or false. 2 Identify the hypothesis and conclusion of each statement. Then write each statement in the if-then form. a. Distance is positive. Hypothesis: a distance is determined; Conclusion: it is positive; If a distance is determined, then it is positive. b. A five-sided polygon is a pentagon. Hypothesis: a polygon has five sides; Conclusion: it is a pentagon; If a polygon has five sides, then it is a pentagon. Teaching Tip Tell students to use parentheses to identify the hypothesis and conclusion in each situation. Explain that if the hypothesis in the situation matches the hypothesis in the original statement, students can mark a T over the parentheses; if not, they can mark an F. They can do the same for the conclusions. 3 Determine the truth value of a. b. c. d. 76 the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Yukon rests for 10 days, and he still has a hurt ankle. false Yukon rests for 3 days, and he still has a hurt ankle. true Yukon rests for 10 days, and he does not have a hurt ankle anymore. true Yukon rests for 7 days, and he does not have a hurt ankle anymore. true Chapter 2 Reasoning and Proof Identify the hypothesis and conclusion of each statement. Then write each statement in if-then form. a. An angle with a measure greater than 90 is an obtuse angle. Hypothesis: an angle has a measure greater than 90 Conclusion: it is an obtuse angle If an angle has a measure greater than 90, then it is an obtuse angle. Example 3 Truth Values of Conditionals SCHOOL Determine the truth value of the following statement for each set of conditions. If you get 100% on your test, then your teacher will give you an A. a. You get 100%; your teacher gives you an A. The hypothesis is true since you got 100%, and the conclusion is true because the teacher gave you an A. Since what the teacher promised is true, the conditional statement is true. Study Tip Common Misconception A true hypothesis does not necessarily mean that a conditional is true. Likewise, a false conclusion does not guarantee that a conditional is false. b. You get 100%; your teacher gives you a B. The hypothesis is true, but the conclusion is false. Because the result is not what was promised, the conditional statement is false. c. You get 98%; your teacher gives you an A. The hypothesis is false, and the conclusion is true. The statement does not say what happens if you do not get 100% on the test. You could still get an A. It is also possible that you get a B. In this case, we cannot say that the statement is false. Thus, the statement is true. d. You get 85%; your teacher gives you a B. As in part c, we cannot say that the statement is false. Therefore, the conditional statement is true. The resulting truth values in Example 3 can be used to create a truth table for conditional statements. Notice that a conditional statement is true in all cases except where the hypothesis is true and the conclusion is false. p q p→q T T T T F F F T T F F T 76 Chapter 2 Reasoning and Proof Teacher to Teacher Nancy Lee Keen Martinsville High School, Martinsville, IN To develop the concept of conditional statements, I made posters of each of the four related conditionals. I wrote the hypotheses on yellow poster board, the conclusions on blue poster board, and NOT on red poster board. As we introduced each type of conditional, we placed the posters in the correct order. CONVERSE, INVERSE, AND CONTRAPOSITIVE Other statements based on a given conditional statement are known as related conditionals . Related Conditionals Statement Formed by Symbols Examples Conditional given hypothesis and conclusion p→q If two angles have the same measure, then they are congruent. Converse exchanging the hypothesis and conclusion of the conditional q→p If two angles are congruent, then they have the same measure. Inverse negating both the hypothesis and conclusion of the conditional p → q If two angles do not have the same measure, then they are not congruent. Contrapositive negating both the hypothesis and conclusion of the converse statement q → p If two angles are not congruent, then they do not have the same measure. Study Tip Contrapositive The relationship of the truth values of a conditional and its contrapositive is known as the Law of Contrapositive. q Conditional p→q Converse q→p Inverse p → q Contrapositive q → p T T T T T T T F F T T F F T T F F T F F T T T T Concept Check Write the converse, inverse, and contrapositive of the statement Linear pairs of angles are supplementary. Determine whether each statement is true or false. If a statement is false, give a counterexample. First, write the conditional in if-then form. Conditional: If two angles form a linear pair, then they are supplementary. The conditional statement is true. Write the converse by switching the hypothesis and conclusion of the conditional. Converse: If two angles are supplementary, then they form a linear pair. The converse is false. ABC and PQR are C supplementary, but are not a linear pair. If two angles do not form a linear pair, then they are not supplementary. The A inverse is false. ABC and PQR do not form a linear pair, but they are supplementary. P 110° B 70° Q R The contrapositive is the negation of the hypothesis and conclusion of the converse. Contrapositive: If two angles are not supplementary, then they do not form a linear pair. The contrapositive is true. www.geometryonline.com/extra_examples Power Point® and contrapositive of the statement All squares are rectangles. Determine whether each statement is true or false. If a statement is false, give a counterexample. Conditional: If a shape is a square, then it is a rectangle. Converse: If a shape is a rectangle, then it is a square. False; a rectangle with 2 and w 4 is not a square. Inverse: If a shape is not a square, then it is not a rectangle. False; a 4-sided polygon with side lengths 2, 2, 4, and 4 is not a square. Contrapositive: If a shape is not a rectangle, then it is not a square. true Example 4 Related Conditionals Inverse: In-Class Examples 4 Write the converse, inverse, If a given conditional is true, the converse and inverse are not necessarily true. However, the contrapositive of a true conditional is always true, and the contrapositive of a false conditional is always false. Likewise, the converse and inverse of a conditional are either both true or both false. Statements with the same truth values are said to be logically equivalent . So, a conditional and its contrapositive are logically equivalent as are the converse and inverse of a conditional. These relationships are summarized below. p CONVERSE, INVERSE, AND CONTRAPOSITIVE In Lesson 2-2, p and q represented simple statements, not necessarily related to one another. In this lesson, they become the hypothesis and conclusion of a conditional statement. Make sure students know that separately, p and q are still simple statements, but they now have an interdependent relationship. Before moving on, students should feel very comfortable identifying the hypothesis and conclusion, determining the truth value of each one separately, and determining their combined truth value in various forms of conditional statements. Lesson 2-3 Conditional Statements 77 Differentiated Instruction Kinesthetic Provide index cards for each student labeled “Hypothesis,” “Conclusion,” and “Implies” (or an arrow pointing to the right). Give each student two cards labeled “Not” in red ink. Ask students to use the cards to form a conditional, a converse, an inverse, and a contrapositive. Students should respond by placing the cards in the correct position and order to reflect the requests. Students can also use the cards to work some examples or exercises in this lesson by writing the parts of conditional statements on corresponding cards. Lesson 2-3 Conditional Statements 77 3 Practice/Apply Concept Check 1 – 3. See margin. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 2. • include a simplified version of the Related Conditionals chart and the truth table on page 77. • include any other item(s) that they find helpful in mastering the skills in this lesson. About the Exercises… Guided Practice GUIDED PRACTICE KEY Exercises Examples 4–6 7–9, 15 10–12 13, 14 1 2 3 4 7. If a pitcher is a 32-ounce pitcher, then it holds a quart of liquid. 8. If two angles are supplementary, then the sum of the measures of the angles is 180. 9. If an angle is formed by perpendicular lines, then it is a right angle. Organization by Objective • If-Then Statements: 16–39 • Converse, Inverse, and Contrapositive: 40–45 Odd/Even Assignments Exercises 16–45 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 17–47 odd, 48–68 Average: 17–47 odd, 48–68 Advanced: 16–48 even, 50–65 (optional: 66–68) All: Quiz 1 (1–5) Answers 1. Writing a conditional in if-then form is helpful so that the hypothesis and conclusion are easily recognizable. 2. Sample answer: If you eat your peas, then you will have dessert. 3. In the inverse, you negate both the hypothesis and the conclusion of the conditional. In the contrapositive, you negate the hypothesis and the conclusion of the converse. 4. H: it rains on Monday; C: I will stay home 5. H: x 3 7; C: x 10 78 Chapter 2 Reasoning and Proof Application 1. Explain why writing a conditional statement in if-then form is helpful. 2. OPEN ENDED Write an example of a conditional statement. 3. Compare and contrast the inverse and contrapositive of a conditional. Identify the hypothesis and conclusion of each statement. 4– 6. See margin. 4. If it rains on Monday, then I will stay home. 5. If x 3 7, then x 10. 6. If a polygon has six sides, then it is a hexagon. Write each statement in if-then form. 7. A 32-ounce pitcher holds a quart of liquid. 8. The sum of the measures of supplementary angles is 180. 9. An angle formed by perpendicular lines is a right angle. Determine the truth value of the following statement for each set of conditions. If you drive faster than 65 miles per hour on the interstate, then you will receive a speeding ticket. 10. You drive 70 miles per hour, and you receive a speeding ticket. true 11. You drive 62 miles per hour, and you do not receive a speeding ticket. true 12. You drive 68 miles per hour, and you do not receive a speeding ticket. false Write the converse, inverse, and contrapositive of each conditional statement. Determine whether each related conditional is true or false. If a statement is false, find a counterexample. 13 –14. See margin. 13. If plants have water, then they will grow. 14. Flying in an airplane is safer than riding in a car. 15. FORESTRY In different regions of the country, different variations of trees dominate the landscape. In Colorado, aspen trees cover high areas of the mountains. In Florida, cypress trees rise from swamps. In Vermont, maple trees are prevalent. Write these conditionals in if-then form. See p. 123B. Practice and Apply For Exercises See Examples 16–21 22–27 28–39 40–45 1 2 3 4 Extra Practice See page 756. Identify the hypothesis and conclusion of each statement. 16 – 21. See p. 123B. 16. If 2x 6 10, then x 2. 17. If you are a teenager, then you are at least 13 years old. 18. If you have a driver’s license, then you are at least 16 years old. 19. If three points lie on a line, then they are collinear. 20. “If a man hasn’t discovered something that he will die for, he isn’t fit to live.” (Martin Luther King, Jr., 1963) 21. If the measure of an angle is between 0 and 90, then the angle is acute. Write each statement in if-then form. 22–27. See p. 123B. 22. Get a free visit with a one-year fitness plan. 23. Math teachers love to solve problems. 24. “I think, therefore I am.” (Descartes) 25. Adjacent angles have a common side. 26. Vertical angles are congruent. 27. Equiangular triangles are equilateral. 78 Chapter 2 Reasoning and Proof 6. H: a polygon has six sides; C: it is a hexagon 13. Converse: If plants grow, then they have water; true. Inverse: If plants do not have water, then they will not grow; true. Contrapositive: If plants do not grow, then they do not have water. False; they may have been killed by overwatering. 14. Converse: If you are safer than riding in a car, then you are flying in an airplane. False; there are other places that are safer than riding in a car. Inverse: If you are not flying in an airplane, then you are not safer than riding in a car. False; there are other places that are safer than riding in a car. Contrapositive: If you are not safer than riding in a car, then you are not flying in an airplane; true. Determine the truth value of the following statement for each set of conditions. If you are over 18 years old, then you vote in all elections. 28. You are 19 years old and you vote. true 29. You are 16 years old and you vote. true NAME ______________________________________________ DATE p. 69 (shown) and p. 70 Conditional Statements If-then Statements An if-then statement is a statement such as “If you are reading this page, then you are studying math.” A statement that can be written in if-then form is called a conditional statement. The phrase immediately following the word if is the hypothesis. The phrase immediately following the word then is the conclusion. A conditional statement can be represented in symbols as p → q, which is read “p implies q” or “if p, then q.” 30. You are 21 years old and do not vote. false Example 1 31. You are 17 years old and do not vote. true Identify the hypothesis and conclusion of the statement. If X R and R S, then X S. hypothesis 32. Your sister is 21 years old and votes. true conclusion Example 2 Identify the hypothesis and conclusion. Write the statement in if-then form. You receive a free pizza with 12 coupons. 33. Your dad is 45 years old and does not vote. false If you have 12 coupons, then you receive a free pizza. hypothesis Identify the hypothesis and conclusion of each statement. 2. If x 8 32, then x 40. H: x 8 32; C: x 40 M A 3. If a polygon has four right angles, then the polygon is a rectangle. H: a polygon has four right angles; C: the polygon is a rectangle Write each statement in if-then form. Lesson 2-3 1. If it is Saturday, then there is no school. H: it is Saturday; C: there is no school 4. All apes love bananas. If an animal is an ape, then it loves bananas. 5. The sum of the measures of complementary angles is 90. If two angles 36. Q lies in plane M . true are complementary, then the sum of their measures is 90. P 37. P, Q, A, and B are coplanar. false 38. AP contains Q. false Q R 6. Collinear points lie on the same line. If points are collinear, then they lie on the same line. B N 39. Planes M and N intersect at RQ . true Determine the truth value of the following statement for each set of conditions. If it does not rain this Saturday, we will have a picnic. 7. It rains this Saturday, and we have a picnic. true 8. It rains this Saturday, and we don’t have a picnic. true 9. It doesn’t rain this Saturday, and we have a picnic. true 10. It doesn’t rain this Saturday, and we don’t have a picnic. false Write the converse, inverse, and contrapositive of each conditional statement. Determine whether each related conditional is true or false. If a statement is false, find a counterexample. 40–45. See p. 123B. 40. If you live in Dallas, then you live in Texas. 41. If you exercise regularly, then you are in good shape. Gl NAME ______________________________________________ DATE /M G Hill 69 Skills Practice, 2-3 Practice (Average) p. 71 and Practice, 72 (shown) Conditionalp. Statements ____________ Gl PERIOD G _____ Identify the hypothesis and conclusion of each statement. 1. If 3x 4 5, then x 3. H: 3x 4 5; C: x 3 2. If you take a class in television broadcasting, then you will film a sporting event. H: you take a class in television broadcasting; C: you will film a sporting event 42. The sum of two complementary angles is 90. 43. All rectangles are quadrilaterals. Write each statement in if-then form. 3. “Those who do not remember the past are condemned to repeat it.” (George Santayana) If you do not remember the past, then you are condemned to repeat it. 44. All right angles measure 90. 4. Adjacent angles share a common vertex and a common side. 45. Acute angles have measures less than 90. If two angles are adjacent, then they share a common vertex and a common side. SEASONS For Exercises 46 and 47, use the following information. Due to the movement of Earth around the sun, summer days in Alaska have more hours of daylight than darkness, and winter days have more hours of darkness than daylight. 46. Write two true conditional statements in if-then form for summer days and winter days in Alaska. Seasons conclusion Exercises In the figure, P, Q, and R are collinear, P and A lie in plane M , and Q and B lie in plane N . Determine the truth value of each statement. 34. P, Q, and R lie in plane M . true 35. QB lies in plane N . true 46. Sample answer: In Alaska, if it is summer, then there are more hours of daylight than darkness. In Alaska, if it is winter, then there are more hours of darkness than daylight. ____________ PERIOD _____ Study Guide andIntervention Intervention, 2-3 Study Guide and At the poles, sunlight may shine continuously for six months during spring and summer, but never rises more than 23.5° above the horizon. During the other six months of the year, the poles are in darkness. 47. Write the converse of the two true conditional statements. State whether each is true or false. If a statement is false, find a counterexample. See p. 123B. Source: U.S. Geological Survey 49. WRITING IN MATH Determine the truth value of the following statement for each set of conditions. If DVD players are on sale for less than $100, then you buy one. 5. DVD players are on sale for $95 and you buy one. true 6. DVD players are on sale for $100 and you do not buy one. true 7. DVD players are not on sale for under $100 and you do not buy one. true 8. Write the converse, inverse, and contrapositive of the conditional statement. Determine whether each statement is true or false. If a statement is false, find a counterexample. If (8) 2 0, then 8 0. Converse: If 8 0, then (8)2 0; true. Inverse: If (8)2 0, then 8 0; true. Contrapositive: If 8 0, then (8)2 0; false. SUMMER CAMP For Exercises 9 and 10, use the following information. Older campers who attend Woodland Falls Camp are expected to work. Campers who are juniors wait on tables. 9. Write a conditional statement in if-then form. 48. CRITICAL THINKING Write a false conditional statement. Is it possible to insert the word not into your conditional to make it true? If so, write the true conditional. Answer the question that was posed at the beginning of the lesson. See margin. How are conditional statements used in advertisements? Sample answer: If you are a junior, then you wait on tables. 10. Write the converse of your conditional statement. If you wait on tables, then you are a junior. Gl NAME ______________________________________________ DATE /M G Hill 72 ____________ Gl PERIOD G _____ Reading 2-3 Readingto to Learn Learn Mathematics Mathematics, p. 73 Conditional Statements Pre-Activity ELL How are conditional statements used in advertisements? Read the introduction to Lesson 2-3 at the top of page 75 in your textbook. Does the second advertising statement in the introduction mean that you will not get a free phone if you sign a contract for only six months of service? Explain your answer. No; it only tells you what happens if you sign up for one year. Include the following in your answer: • an example of a conditional statement in if-then form, and • an example of a conditional statement that is not in if-then form. 1. Identify the hypothesis and conclusion of each statement. a. If you are a registered voter, then you are at least 18 years old. Hypothesis: you are a registered voter; Conclusion: you are at least 18 years old b. If two integers are even, their product is even. Hypothesis: two integers are even; Conclusion: their product is even Lesson 2-3 Conditional Statements 79 Galen Rowell/CORBIS Answer 49. Conditional statements can be used to describe how to get a discount, rebate, or refund. Sample answers should include the following. • If you are not 100% satisfied, then return the product for a full refund. • Wearing a seatbelt reduces the risk of injuries. NAME ______________________________________________ DATE 2-3 Enrichment Enrichment, ____________ PERIOD _____ p. 74 2. Complete each sentence. a. The statement that is formed by replacing both the hypothesis and the conclusion of a inverse conditional with their negations is the . b. The statement that is formed by exchanging the hypothesis and conclusion of a converse conditional is the . 3. Consider the following statement: You live in North America if you live in the United States. a. Write this conditional statement in if-then form and give its truth value. If the statement is false, give a counterexample. If you live in the United States, then you live in North America; false: You live in Hawaii. b. Write the inverse of the given conditional statement in if-then form and give its truth value. If the statement is false, give a counterexample. If you do not live in the United States, then you do not live in North America; false; sample answer: You live in Mexico. Venn Diagrams A type of drawing called a Venn diagram can be useful in explaining conditional statements. A Venn diagram uses circles to represent sets of objects. c. Write the contrapositive of the given conditional statement in if-then form and give its truth value. If the statement is false, give a counterexample. If you do not live Consider the statement “All rabbits have long ears.” To make a Venn diagram for this statement, a large circle is drawn to represent all animals with long ears. Then a smaller circle is drawn inside the first to represent all rabbits. The Venn diagram shows that every rabbit is included in the group of long-eared animals. d. Write the converse of the given conditional statement in if-then form and give its truth value. If the statement is false, give a counterexample. If you live in North animals with long ears rabbits The set of rabbits is called a subset of the set of long-eared animals. The Venn diagram can also explain how to write the statement, “All rabbits have long ears,” in if-then form. Every rabbit is in the group of long-eared animals, so if an animal is a rabbit, then it has long ears. in North America, then you do not live in the United States; false: You live in Hawaii. America, then you live in the United States; false; sample answer: You live in Canada. Helping You Remember 4. When working with a conditional statement and its three related conditionals, what is an easy way to remember which statements are logically equivalent to each other? Sample answer: The two statements whose names contain verse (the converse and the inverse) are a logically equivalent pair. The other two (the original conditional and the contrapositive) are the other logically equivalent pair. For each statement, draw a Venn diagram. Then write the sentence in if-then form. 1 Every dog has long hair 2 All rational numbers are real Lesson 2-3 Conditional Statements 79 Lesson 2-3 www.geometryonline.com/self_check_quiz Reading the Lesson Standardized Test Practice 4 Assess 50. Which statement has the same truth value as the following statement? C If Ava and Willow are classmates, then they go to the same school. A If Ava and Willow go to the same school, then they are classmates. B If Ava and Willow are not classmates, then they do not go to the same school. C If Ava and Willow do not go to the same school, then they are not classmates. D If Ava and Willow go to the same school, then they are not classmates. Open-Ended Assessment Speaking Students can practice their speaking skills by identifying parts of statements and translating statements into the converse, inverse, and contrapositive aloud. 51. ALGEBRA In a history class with 32 students, the ratio of girls to boys is 5 to 3. How many more girls are there than boys? B A 2 B 8 C 12 D 20 Getting Ready for Lesson 2-4 Maintain Your Skills Mixed Review Prerequisite Skill Students will learn about deductive reasoning in Lesson 2-4. They will apply concepts of solving equations to deductive-reasoning techniques. Use Exercises 66–68 to determine your students’ familiarity with solving equations. Use the following statements to write a compound statement for each conjunction and disjunction. Then find its truth value. (Lesson 2-2) 52–57. See margin. p: George Washington was the first president of the United States. q: A hexagon has five sides. r: 60 ⴛ 3 ⴝ 18 53. q r false 54. p q true 52. p q false 55. ~q r true 56. p ~q true 57. ~p ~r false Make a conjecture based on the given information. Draw a figure to illustrate your conjecture. (Lesson 2-1) 58–61. See margin for sample figures. 58. ABCD is a rectangle. 59. In FGH, mF 45, mG 67, mH 68. 60. J(3, 2), K(1, 8), L(5, 2) 61. In PQR, mPQR 90 PQR is a right angle. 58. AB CD; AD BC 59. The sum of the measures of the angles in a triangle is 180. 60. JKL has two sides congruent. Assessment Options Practice Quiz 1 The quiz provides students with a brief review of the concepts and skills in Lessons 2-1 through 2-3. Lesson numbers are given to the right of the exercises or instruction lines so students can review concepts not yet mastered. Use the Distance Formula to find the distance between each pair of points. (Lesson 1-3) 62. C(2, 1), D(0, 3) 20 4.5 64. P(3, 1), Q(2, 3) 29 5.4 Getting Ready for the Next Lesson 63. J(3, 5), K(1, 0) 41 6.4 65. R(1, 7), S(4, 3) 125 11.2 PREREQUISITE SKILL Identify the operation used to change Equation (1) to Equation (2). (To review solving equations, see pages 737 and 738.) 66–68. See margin. 1 66. (1) 3x 4 5x 8 67. (1) (a 5) 12 68. (1) 8p 24 2 (2) 3x 5x 12 (2) a 5 24 (2) p 3 Answers P ractice Quiz 1 52. George Washington was the first president of the United States and a hexagon has 5 sides. 53. A hexagon has five sides or 60 3 18. 54. George Washington was the first president of the United States or a hexagon has five sides. 55. A hexagon doesn’t have five sides or 60 3 18. 56. George Washington was the first president of the United States and a hexagon doesn’t have five sides. 57. George Washington was not the first president of the United States and 60 3 18. 58. D C Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. Lessons 2-1 through 2-3 (Lesson 2-1) 1. Given: WX XY Conjecture: W, X, and Y are collinear. 1 and 2 are complementary. 2 and 3 are complementary. Conjecture: m1 m3 true 2. Given: False; see p. 123B for counterexample. Construct a truth table for each compound statement. (Lesson 2-2) 3–4. See p. 123B. 3. ~p q 4. p (q r) 5. Write the converse, inverse, and contrapositive of the following conditional statement. Determine whether each related conditional is true or false. If a statement is false, find a counterexample. (Lesson 2-3) If two angles are adjacent, then the angles have a common vertex. See p. 123C. 80 Chapter 2 Reasoning and Proof 59. G 60. F 61. y K 67 P 45 68 Q H A B L J O 80 Chapter 2 Reasoning and Proof x R 66. Subtract 4 from each side. 67. Multiply each side by 2. 68. Divide each side by 8. Reading Mathematics Biconditional Statements Getting Started Ashley began a new summer job, earning $10 an hour. If she works over 40 hours a week, she earns time and a half, or $15 an hour. If she earns $15 an hour, she has worked over 40 hours a week. Explain that true biconditional statements are extremely helpful for writing proofs because they can be used forwards or backwards. Tell students, however, that they are not as common as regular conditionals, and students will need to be wary and thorough when they are determining whether a biconditional is true or false. p: Ashley earns $15 an hour q: Ashley works over 40 hours a week p → q: q → p: If Ashley earns $15 an hour, she has worked over 40 hours a week. If Ashley works over 40 hours a week, she earns $15 an hour. In this case, both the conditional and its converse are true. The conjunction of the two statements is called a biconditional . Biconditional Statement • Words A biconditional statement is the conjunction of a conditional and its converse. • Symbols (p → q) (q → p) is written (p ↔ q) and read p if and only if q. Teach If and only if can be abbreviated iff. Biconditional Statements Students can also get some extra practice writing the inverse and contrapositive of each biconditional statement. So, the biconditional statement is as follows. p ↔ q: Ashley earns $15 an hour if and only if she works over 40 hours a week. Examples Write each biconditional as a conditional and its converse. Then determine whether the biconditional is true or false. If false, give a counterexample. a. Two angle measures are complements if and only if their sum is 90. Conditional: If two angle measures are complements, then their sum is 90. Converse: If the sum of two angle measures is 90, then they are complements. Both the conditional and the converse are true, so the biconditional is true. Assess b. x 9 iff x 0 Study Notebook Conditional: If x 9, then x 0. Converse: If x 0, then x 9. The conditional is true, but the converse is not. Let x 2. Then 2 0 but 2 ⬎ 9. So, the biconditional is false. Ask students to summarize what they have learned about biconditional statements. Reading to Learn 1 –5. See margin. Write each biconditional as a conditional and its converse. Then determine whether the biconditional is true or false. If false, give a counterexample. 1. A calculator will run if and only if it has batteries. 2. Two lines intersect if and only if they are not vertical. 3. Two angles are congruent if and only if they have the same measure. 4. 3x 4 20 iff x 7. 5. A line is a segment bisector if and only if it intersects the segment at its midpoint. Reading Mathematics Biconditional Statements 81 ELL English Language Learners may benefit from writing key concepts from this activity in their Study Notebooks in their native language and then in English. Answers 1. Conditional: If a calculator runs, then it has batteries. Converse: If a calculator has batteries, then it will run. False; a calculator may be solar powered. 2. Conditional: If two lines intersect, then they are not vertical. Converse: If two lines are not vertical, then they intersect. False; two parallel horizontal lines will not intersect. 3. Conditional: If two angles are congruent, then they have the same measure. Converse: If two angles have the same measure, then they are congruent. true 4. Conditional: If 3x 4 20, then x 7. Converse: If x 7, then 3x 4 20. False; 3x 4 17 when x 7. 5. Conditional: If a line is a segment bisector, then it intersects the segment at its midpoint. Converse: If a line intersects a segment at its midpoint, then it is a segment bisector. true Reading Mathematics Biconditional Statements 81 Lesson Notes 1 Focus 5-Minute Check Transparency 2-4 Use as a quiz or review of Lesson 2-3. Deductive Reasoning • Use the Law of Detachment. • Use the Law of Syllogism. Mathematical Background notes are available for this lesson on p. 60D. does deductive reasoning apply to health? Ask students: • If you have a mass of 57.8 kg, what dose will a doctor give you based on the chart? 350 mg • What might happen if a patient used inductive reasoning to form a conjecture about the dose of an antidepressant based on the chart above? Is this a safe method for health situations? Sample answer: The dose of the antidepressant might be much less than that of the antibiotic for the same weight, so the patient could potentially overdose on the antidepressant; no. does deductive reasoning apply to health? Vocabulary • deductive reasoning • Law of Detachment • Law of Syllogism Weigh t (kg) 10–20 20–30 30–40 40–50 50–60 60–70 When you are ill, your doctor may prescribe an antibiotic to help you get better. Doctors may use a dose chart like the one shown to determine the correct amount of medicine you should take. Dose (mg) 150 200 250 300 350 400 LAW OF DETACHMENT The process that doctors use to determine the amount of medicine a patient should take is called deductive reasoning . Unlike inductive reasoning, which uses examples to make a conjecture, deductive reasoning uses facts, rules, definitions, or properties to reach logical conclusions. A form of deductive reasoning that is used to draw conclusions from true conditional statements is called the Law of Detachment . Law of Detachment • Words If p → q is true and p is true, then q is also true. • Symbols [(p → q) p] → q Study Tip Validity When you apply the Law of Detachment, make sure that the conditional is true before you test the validity of the conclusion. Example 1 Determine Valid Conclusions The following is a true conditional. Determine whether each conclusion is valid based on the given information. Explain your reasoning. If a ray is an angle bisector, then it divides the angle into two congruent angles. bisects ABC. a. Given: BD Conclusion: ABD CBD A D B is the bisector The hypothesis states that BD of ABC. Since the conditional is true and the hypothesis is true, the conclusion is valid. C b. Given: PQT RQS P and QT are angle bisectors. Conclusion: QS Knowing that a conditional statement and its conclusion are true does not make the hypothesis true. An angle bisector divides an angle into two separate congruent angles. In this case, the given angles are not separated by one ray. Instead, they overlap. The conclusion is not valid. 82 Chapter 2 Reasoning and Proof Resource Manager Workbook and Reproducible Masters Chapter 2 Resource Masters • Study Guide and Intervention, pp. 75–76 • Skills Practice, p. 77 • Practice, p. 78 • Reading to Learn Mathematics, p. 79 • Enrichment, p. 80 • Assessment, pp. 119, 121 School-to-Career Masters, p. 3 Teaching Geometry With Manipulatives Masters, p. 47 Transparencies 5-Minute Check Transparency 2-4 Answer Key Transparencies Technology Interactive Chalkboard Q S T R LAW OF SYLLOGISM Another law of logic is the Law of Syllogism. It is similar to the Transitive Property of Equality. Law of Syllogism • Words If p → q and q → r are true, then p → r is also true. 2 Teach LAW OF DETACHMENT In-Class Example • Symbols [(p → q) (q → r)] → (p → r) Power Point® 1 The following is a true Study Tip Conditional Statements Label the hypotheses and conclusions of a series of statements before applying the Law of Syllogism. Example 2 Determine Valid Conclusions From Two Conditionals CHEMISTRY Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements. a. (1) If the symbol of a substance is Pb, then it is lead. (2) The atomic number of lead is 82. Let p, q, and r represent the parts of the statement. p: the symbol of a substance is Pb q: it is lead r: the atomic number is 82 Statement (1): p → q Statement (2): q → r Since the given statements are true, use the Law of Syllogism to conclude p → r. That is, If the symbol of a substance is Pb, then its atomic number is 82. b. (1) Water can be represented by H2O. (2) Hydrogen (H) and oxygen (O) are in the atmosphere. There is no valid conclusion. While both statements are true, the conclusion of each statement is not used as the hypothesis of the other. conditional. Determine whether each conclusion is valid based on the given information. Explain your reasoning. If two segments are congruent and the second segment is congruent to a third segment, then the first segment is also congruent to the third segment. X UV ; U V RT a. Given: W Conclusion: WX RT true b. Given: UV ; W X RT Conclusion: WX UV and V RT false U LAW OF SYLLOGISM In-Class Example Example 3 Analyze Conclusions 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) Vertical angles are congruent. (2) If two angles are congruent, then their measures are equal. (3) If two angles are vertical, then their measures are equal. p: two angles are vertical q: they are congruent r: their measures are equal Statement (3) is a valid conclusion by the Law of Syllogism. b. (1) If a figure is a square, then it is a polygon. (2) Figure A is a polygon. (3) Figure A is a square. Statement (1) is true, but statement (3) does not follow from statement (2). Not all polygons are squares. Statement (3) is invalid. www.geometryonline.com/extra_examples Lesson 2-4 Deductive Reasoning 83 Differentiated Instruction Power Point® 2 PROM Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements. a. (1) If Salline attends the prom, she will go with Mark. (2) Mark is a 17-year-old student. not valid b. (1) If Mel and his date eat at the Peddler Steakhouse before going to the prom, they will miss the senior march. (2) The Peddler Steakhouse stays open until 10 P.M. not valid In-Class Example 3 is on p. 84. ELL Verbal/Linguistic Have students write a paragraph to explain and provide an example for the Law of Detachment. Repeat for the Law of Syllogism. Then students can write another paragraph to point out similarities and differences between the two laws. They can place their written explanations in their study notebooks. Lesson 2-4 Deductive Reasoning 83 In-Class Example Power Point® 3 Determine whether statement Concept Check (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) If the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle. (2) For XYZ, (XY)2 (YZ)2 (ZX)2. (3) XYZ is a right triangle. Law of Detachment b. (1) If Ling wants to participate in the wrestling competition, he will have to meet an extra three times a week to practice. (2) If Ling adds anything extra to his weekly schedule, he cannot take karate lessons. (3) If Ling wants to participate in the wrestling competition, he cannot take karate lessons. Law of Syllogism 1. OPEN ENDED Write an example to illustrate the correct use of the Law of Detachment. 1–3. See margin. 2. Explain how the Transitive Property of Equality is similar to the Law of Syllogism. 3. FIND THE ERROR An article in a magazine states that if you get seasick, then you will get dizzy. It also says that if you get seasick, you will get an upset stomach. Suzanne says that this means that if you get dizzy, then you will get an upset stomach. Lakeisha says that she is wrong. Who is correct? Explain. Guided Practice GUIDED PRACTICE KEY Exercises Examples 4, 5 6, 7 8, 9 1 2 3 Determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. If two angles are vertical angles, then they are congruent. 4. Given: A and B are vertical angles. Conclusion: A B valid 5. Given: C D Conclusion: C and D are vertical angles. Invalid; congruent angles do not have to be vertical. Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements. If a valid conclusion is possible, write it. If not, write no conclusion. 6. If you are 18 years old, you are in college. You are in college. no conclusion 7. The midpoint of a segment divides it into two segments with equal measures. 7. The midpoint divides a segment into two congruent segments. If two segments are congruent, then their measures are equal. 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. 8. valid; Law of Syllogism 8. (1) If Molly arrives at school at 7:30 A.M., she will get help in math. (2) If Molly gets help in math, then she will pass her math test. (3) If Molly arrives at school at 7:30 A.M., then she will pass her math test. 9. (1) Right angles are congruent. (2) X Y (3) X and Y are right angles. invalid 3 Practice/Apply Application Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 2. • include any other item(s) that they find helpful in mastering the skills in this lesson. INSURANCE For Exercises 10 and 11, use the following information. An insurance company advertised the following monthly rates for life insurance. If you are a: Premium for $30,000 Coverage Premium for $50,000 Coverage Female, age 35 $14.35 $19.00 Male, age 35 $16.50 $21.63 Female, age 45 $21.63 $25.85 Male, age 45 $23.75 $28.90 10. If Ann is 35 years old and she wants to purchase $30,000 of insurance from this company, then what is her premium? $14.35 11. If Terry paid $21.63 for life insurance, can you conclude that Terry is 35? Explain. No; Terry could be a man or a woman. She could be 45 and have purchased $30,000 of life insurance. 84 Chapter 2 Reasoning and Proof FIND THE ERROR Explain that if you have p → q and p → r, you cannot assume q → r, as Suzanne tries to do. Tell students that the same hypothesis can imply two different conclusions, but the conclusions are not related to one another. 84 Chapter 2 Reasoning and Proof Answers 1. Sample answer: a: If it is rainy, the game will be cancelled. b: It is rainy. c: The game will be cancelled. 2. Transitive Property of Equality: a b and b c implies a c. Law of Syllogism: a implies b and b implies c implies a implies c. Each statement establishes a relationship between a and c through their relationships to b. 3. Lakeisha; if you are dizzy, that does not necessarily mean that you are seasick and thus have an upset stomach. Practice and Apply For Exercises See Examples 12–19 20–23 24–29 1 2 3 Extra Practice See page 757. 13. Valid; since 5 and 7 are odd, the Law of Detachment indicates that their sum is even. 14. Valid; since 11 and 23 are odd, the Law of Detachment indicates that their sum is even. 16. Valid; A, B, and C are noncollinear, and by definition three noncollinear points determine a plane. 17. Invalid; E, F, and G are not necessarily noncollinear. 18. Invalid; the hypothesis is false as there are only two points. 19. Valid; the vertices of a triangle are noncollinear, and therefore determine a plane. 21. If the measure of an angle is less than 90, then it is not obtuse. 22. If X is the midpoint of YZ, then YX XZ. For Exercises 12–19, determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. If two numbers are odd, then their sum is even. 12. Given: The sum of two numbers is 22. Conclusion: The two numbers are odd. invalid; 10 12 22 13. Given: The numbers are 5 and 7. Conclusion: The sum is even. 14. Given: 11 and 23 are added together. Conclusion: The sum of 11 and 23 is even. 15. Given: The numbers are 2 and 6. Conclusion: The sum is odd. Invalid; the sum is even. If three points are noncollinear, then they determine a plane. 16. Given: A, B, and C are noncollinear. Conclusion: A, B, and C determine a plane. About the Exercises… Organization by Objective • Law of Detachment: 12–19 • Law of Syllogism: 20–29 Odd/Even Assignments Exercises 12–29 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Exercise 31 requires the Internet or other research materials. Assignment Guide 17. Given: E, F, and G lie in plane M. Conclusion: E, F, and G are noncollinear. Basic: 13–31 odd, 32–58 Average: 13–31 odd, 32–58 Advanced: 12–30 even, 32, 34–55 (optional: 56–58) 18. Given: P and Q lie on a line. Conclusion: P and Q determine a plane. 19. Given: XYZ Conclusion: X, Y, and Z determine a plane. Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements. If a valid conclusion is possible, write it. If not, write no conclusion. 20. If you spend money on it, then it is a business. If you spend money on it, then it is fun. no conclusion 21. If the measure of an angle is less than 90, then it is acute. If an angle is acute, then it is not obtuse. 22. If X is the midpoint of segment YZ, then YX XZ. If the measures of two segments are equal, then they are congruent. 23. If two lines intersect to form a right angle, then they are perpendicular. Lines ᐉ and m are perpendicular. no conclusion 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. 24. (1) In-line skaters live dangerously. (2) If you live dangerously, then you like to dance. (3) If you are an in-line skater, then you like to dance. yes; Law of Syllogism 25. (1) If the measure of an angle is greater than 90, then it is obtuse. (2) mABC > 90 (3) ABC is obtuse. yes; Law of Detachment 26. (1) Vertical angles are congruent. (2) 3 4 (3) 3 and 4 are vertical angles. invalid 27. (1) If an angle is obtuse, then it cannot be acute. (2) A is obtuse. (3) A cannot be acute. yes; Law of Detachment Lesson 2-4 Deductive Reasoning 85 Lesson 2-4 Deductive Reasoning 85 NAME ______________________________________________ DATE 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. 28. (1) If you drive safely, then you can avoid accidents. (2) Tika drives safely. (3) Tika can avoid accidents. yes; Law of Detachment 29. (1) If you are a customer, then you are always right. (2) If you are a teenager, then you are always right. (3) If you are a teenager, then you are a customer. invalid ____________ PERIOD _____ Study Guide andIntervention Intervention, 2-4 Study Guide and p. 75 (shown) and p. 76 Deductive Reasoning Law of Detachment Deductive reasoning is the process of using facts, rules, definitions, or properties to reach conclusions. One form of deductive reasoning that draws conclusions from a true conditional p → q and a true statement p is called the Law of Detachment. Law of Detachment If p → q is true and p is true, then q is true. Symbols [(p → q)] p] → q 30. then he could hear the grating noise of the fish canneries Example The statement If two angles are supplementary to the same angle, then they are congruent is a true conditional. Determine whether each conclusion is valid based on the given information. Explain your reasoning. a. Given: A and C are supplementary to B. Conclusion: A is congruent to C. The statement A and C are supplementary to B is the hypothesis of the conditional. Therefore, by the Law of Detachment, the conclusion is true. F E G B H A D C J 30. LITERATURE John Steinbeck, a Pulitzer Prize winning author, lived in Monterey, California, for part of his life. In 1945, he published the book, Cannery Row, about many of his local working-class heroes from Monterey. If you visited Cannery Row in Monterey during the 1940s, then you could hear the grating noise of the fish canneries. Write a valid conclusion to the following hypothesis. If John Steinbeck lived in Monterey in 1941, . . . b. Given: A is congruent to C. Conclusion: A and C are supplementary to B. The statement A is congruent to C is not the hypothesis of the conditional, so the Law of Detachment cannot be used. The conclusion is not valid. Exercises Determine whether each conclusion is valid based on the true conditional given. If not, write invalid. Explain your reasoning. If two angles are complementary to the same angle, then the angles are congruent. The given statement is the hypothesis of the conditional statement. Since the conditional is true, the conclusion A C is true. 2. Given: A C Conclusion: A and C are complements of B. The given statement is not the hypothesis of the conditional. Therefore, the conclusion is invalid. 3. Given: E and F are complementary to G. Conclusion: E and F are vertical angles. While the given statement is the hypothesis of the conditional statement, the statement that E and F are vertical angles is not the conclusion of the conditional. The conclusion is invalid. Gl NAME ______________________________________________ DATE /M G Hill 75 Skills Practice, 2-4 Practice (Average) p. 77 and Practice, p. 78 (shown) Deductive Reasoning ____________ Gl PERIOD G _____ Determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. If a point is the midpoint of a segment, then it divides the segment into two congruent segments. Lesson 2-4 1. Given: A and C are complementary to B. Conclusion: A is congruent to C. 31. SPORTS In the 2002 Winter Olympics, Canadian speed skater Catriona Le May Doan won her second Olympic title in 500-meter speed skating. Ms. Doan was in the last heat for the second round of that race. Use the two true conditional statements to reach a valid conclusion about Ms. Doan’s 2002 competition. (1) If Catriona Le May Doan skated her second 500 meters in 37.45 seconds, then she would beat the time of Germany’s Monique Garbrecht-Enfeldt. (2) If Ms. Doan beat the time of Monique Garbrecht-Enfeldt, then she would win the race. If Catriona Le May Doan skated her second 500 meters in Literature The Pulitzer Prize is awarded annually for outstanding contributions in the fields of journalism, literature, drama, and music. 37.45 seconds, then she would win the race. Online Research Data Update Use the Internet or another resource to find the winning times for other Olympic events. Write statements using these times that can lead to a valid conclusion. Visit www.geometryonline.com/data_update to learn more. Source: www.pulitzer.org . 1. Given: R is the midpoint of QS Conclusion: QR RS Valid; since R is the midpoint of Q S , the Law of Detachment indicates into two congruent segments. that it divides QS 2. Given: AB BC Conclusion: B divides A C into two congruent segments. Invalid; the points A, B, and C may not be collinear, and if they are not, C . then B will not be the midpoint of A Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements. If a valid conclusion is possible, write it. 3. If two angles form a linear pair, then the two angles are supplementary. If two angles are supplementary, then the sum of their measures is 180. If two angles form a linear pair, then the sum of their measures is 180. 32. Sample answer: Stacey assumed that the conditional statement was true. 32. CRITICAL THINKING An advertisement states that “If you like to ski, then you’ll love Snow Mountain Resort.” Stacey likes to ski, but when she went to Snow Mountain Resort, she did not like it very much. If you know that Stacey saw the ad, explain how her reasoning was flawed. 4. If a hurricane is Category 5, then winds are greater than 155 miles per hour. If winds are greater than 155 miles per hour, then trees, shrubs, and signs are blown down. 33. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See p. 123C. How does deductive reasoning apply to health? If a hurricane is Category 5, then trees, shrubs, and signs are blown down. 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. 5. (1) If a whole number is even, then its square is divisible by 4. (2) The number I am thinking of is an even whole number. (3) The square of the number I am thinking of is divisible by 4. Include the following in your answer: • an explanation of how doctors may use deductive reasoning to prescribe medicine, and • an example of a doctor’s uses of deductive reasoning to diagnose an illness, such as strep throat or chickenpox. yes; Law of Detachment 6. (1) If the football team wins its homecoming game, then Conrad will attend the school dance the following Friday. (2) Conrad attends the school dance on Friday. (3) The football team won the homecoming game. invalid 7. BIOLOGY If an organism is a parasite, then it survives by living on or in a host organism. If a parasite lives in or on a host organism, then it harms its host. What conclusion can you draw if a virus is a parasite? If a virus is a parasite, then it harms its host. Gl NAME ______________________________________________ DATE /M G Hill 78 ____________ Gl PERIOD G _____ Standardized Test Practice Reading 2-4 Readingto to Learn Learn Mathematics Mathematics, p. 79 Deductive Reasoning Pre-Activity ELL How does deductive reasoning apply to health? Read the introduction to Lesson 2-4 at the top of page 82 in your textbook. Suppose a doctor wants to use the dose chart in your textbook to prescribe an antibiotic, but the only scale in her office gives weights in pounds. How can she use the fact that 1 kilogram is about 2.2 pounds to determine the correct dose for a patient? Sample answer: The doctor can divide the patient’s weight in pounds by 2.2 to find the equivalent mass in kilograms. She can then use the dose chart. 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 t e a. s u 2. conjunction of s and u g b. [(s → t) s] → t 3. converse of s → t h c. s → u 4. disjunction of s and u a d. u → s 5. Law of Detachment b e. t 6. contrapositive of s → t j f. [(u → t) (t → s)] → (u → s) 7. inverse of s → u c g. s u 8. contrapositive of s → u d h. t → s 9. Law of Syllogism f i. t 10. negation of t i j. t → s 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 U and do not intersect. (3) Lines and m are parallel. yes; Law of Detachment c. (1) Perpendicular lines intersect to form four right angles. (2) A, B, C, and D are four right angles. (3) A, B, C, and D are formed by intersecting perpendicular lines. invalid 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? 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. 86 Chapter 2 Reasoning and Proof 86 34. Based on the following statements, which statement must be true? C I If Yasahiro is an athlete and he gets paid, then he is a professional athlete. II Yasahiro is not a professional athlete. III Yasahiro is an athlete. A Yasahiro is an athlete and he gets paid. B Yasahiro is a professional athlete or he gets paid. C Yasahiro does not get paid. D Yasahiro is not an athlete. Chapter 2 Reasoning and Proof AP/Wide World Photos NAME ______________________________________________ DATE 2-4 Enrichment Enrichment, ____________ PERIOD _____ p. 80 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. C l ’ l ll 2. (1) If you buy Tuff Cote luggage, it will survive airline travel. (2) Justin’s luggage survived airline travel. C l h ff C 35. ALGEBRA At a restaurant, a diner uses a coupon for 15% off the cost of one meal. If the diner orders a meal regularly priced at $16 and leaves a tip of 20% of the discounted meal, how much does she pay in total? B A $15.64 B $16.32 C $16.80 D $18.72 Maintain Your Skills Mixed Review 36. If you try Casa Fiesta, then you’re looking for a fast, easy way to add some fun to your family’s menu. 37. They are a fast, easy way to add fun to your family’s menu. 38. No; the conclusion is implied. 44. Sample answer: KHJ and DHG 45. Sample answer: JHK and DHK 46. congruent, adjacent, supplementary, linear pair ADVERTISING For Exercises 36–38, use the following information. (Lesson 2-3) Advertising writers frequently use if-then statements to relay a message and promote their product. An ad for a type of Mexican food reads, If you’re looking for a fast, easy way to add some fun to your family’s menu, try Casa Fiesta. 36. Write the converse of the conditional. 37. What do you think the advertiser wants people to conclude about Casa Fiesta products? 38. Does the advertisement say that Casa Fiesta adds fun to your family’s menu? Construct a truth table for each compound statement. (Lesson 2-2) 39–42. See p. 123C. 39. q r 40. ~p r 41. p (q r) 42. p (~q r) For Exercises 43–47, refer to the figure at the right. (Lesson 1-5) 43. Which angle is complementary to FDG? HDG 44. Name a pair of vertical angles. 45. Name a pair of angles that are noncongruent and supplementary. 46. Identify FDH and CDH as congruent, adjacent, vertical, complementary, supplementary, and/or a linear pair. C C K? Explain. Yes, slashes 47. Can you assume that D on the segments indicate that they are congruent. F G H K J Use the Pythagorean Theorem to find the distance between each pair of points. (Lesson 1-3) 49. C(4, 2), D(2, 6) 10 48. A(1, 5), B(2, 9) 5 50. F(7, 4), G(1, 0) 7.2 52 51. M(5, 0), N(4, 7) 11.4 130 For Exercises 52–55, draw and label a figure for each relationship. 52. FG lies in plane M and contains point H. 52–55. See margin. 53. Lines r and s intersect at point W. 54. Line contains P and Q, but does not contain R. 55. Planes A and B intersect in line n. Getting Ready for the Next Lesson 56–58. See margin. Quiz (Lessons 2-3 and 2-4) is available on p. 119 of the Chapter 2 Resource Masters. Mid-Chapter Test (Lessons 2-1 through 2-4) is available on p. 121 of the Chapter 2 Resource Masters. 5 6 M 1 C N 2 B www.geometryonline.com/self_check_quiz Lesson 2-4 Deductive Reasoning 87 Answers 52. M H Prerequisite Skill Students will learn about postulates and paragraph proofs in Lesson 2-5. They will extract information from figures to test postulates and write proofs. Use Exercises 56–58 to determine your students’ familiarity with information from figures. Assessment Options A F Modeling Students can use shapes of paper to model the Law of Detachment and the Law of Syllogism. For example, you could provide students with two yellow squares of laminated construction paper marked with p, two blue triangles marked with q, and two red circles marked with r. Students can physically arrange the shapes to represent the symbolic relationships of the two laws. They could also secure these shapes in their study notebooks for a colorful reminder of these laws. (Lesson 1-1) PREREQUISITE SKILL Write what you can assume about the segments or angles listed for each figure. (To review information from figures, see Lesson 1-5.) 56. A M, C M, C N, B N 57. 1, 2 58. 4, 5, 6 4 Open-Ended Assessment Getting Ready for Lesson 2-5 D C 4 Assess 53. r W G s 54. Q P R 55. A n B 56. Sample answer: C M AM , CN B N , AM CM, CN BN, M is midpoint of A C , N is midpoint of B C . 57. Sample answer: 1 and 2 are complementary, m1 m2 90. 58. Sample answer: 4 and 5 are supplementary, m4 m5 180, 5 and 6 are supplementary, m5 m6 180, 4 6, m4 m6. Lesson 2-4 Deductive Reasoning 87 Geometry Activity A Follow-Up of Lesson 2-4 A Follow-Up of Lesson 2-4 Matrix Logic Getting Started Deductive reasoning can be used in problem-solving situations. One method of solving problems uses a table. This method is called matrix logic . You could provide students with a sheet of three blank logic tables in which they would fill in the titles of the rows and columns, or ask students to use a ruler to draw the tables as they go along. Example GEOLOGY On a recent test, Rashaun was given five different mineral samples to identify, along with the chart at right. Rashaun observed the following. • Sample C is brown. • Samples B and E are harder than glass. • Samples D and E are red. Objective Apply deductive reasoning by using matrix logic tables to solve problems. Color Hardness Biotite Mineral brown or black softer than glass Halite white softer than glass Hematite red softer than glass Feldspar white, pink, or green harder than glass Jaspar red harder than glass Identify each of the samples. Teach Make a table to organize the information. Mark each false condition with an ⫻ and each true condition with a ⻫. The first observation is that Sample C is brown. Only one of the minerals, biotite, is brown, so place a check in the box that corresponds to biotite and Sample C. Then place an ⫻ in each of the other boxes in the same column and row. • When students are considering the second observation, tell them they can think in terms of boxes to mark out or boxes to leave open. The only two minerals that are harder than glass are feldspar and jasper, so these boxes are left open in the columns B and E. Then they can place an X in the rest of the boxes in these columns. • For Exercise 2, advise students to draw a model with six adjacent boxes representing the apartments. After marking all obvious information, students can use the model to place each person in the correct apartment. The second observation is that Samples B and E are harder than glass. Place an ⫻ in each box for minerals that are softer than glass. The third observation is that Samples D and E are red. Mark the boxes accordingly. Notice that Sample E has an ⫻ in all but one box. Place a check mark in the remaining box, and an ⫻ in all other boxes in that row. Ask students to summarize what they have learned about using deductive reasoning in problemsolving situations. 88 Chapter 2 Reasoning and Proof B C D E ⫻ ⫻ ⫻ ⫻ ⻫ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ Halite Hematite Feldspar Jaspar A B C D E ⫻ ⫻ ⫻ ⫻ ⻫ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ Halite Hematite Feldspar Jaspar ⻫ ⫻ ⫻ ⫻ ⻫ ⫻ ⫻ ⫻ ⫻ ⫻ 2. Six friends live in consecutive apartments on the same side of their apartment building. Anita lives in apartment C. Kelli’s apartment is just past Scott’s. Anita’s closest neighbors are Eric and Ava. Scott’s apartment is not A through D. Eric’s apartment is before Ava’s. If Roberto lives in one of the apartments, who lives in which apartment? A, Roberto; B, Eric; C, Anita; D, Ava; E, Scott; F, Kelli Chapter 2 Reasoning and Proof Teaching Geometry with Manipulatives • p. 46 (student recording sheet) ⻫ Biotite 1. Nate, John, and Nick just began after-school jobs. One works at a veterinarian’s office, one at a computer store, and one at a restaurant. Nate buys computer games on the way to work. Nick is allergic to cat hair. John receives free meals at his job. Who works at which job? Nate, veterinarian’s office; John, restaurant; Nick, computer store 88 ⫻ ⫻ Sample Exercises Resource Manager Study Notebook A Biotite Then complete the table. Sample A is Halite, Sample B is Feldspar, Sample C is Biotite, Sample D is Hematite, and Sample E is Jaspar. Assess In Exercises 1 and 2 students practice their thinking and deductive reasoning skills by working more logic tables. Sample ⻫ ⫻ ⫻ ⻫ Postulates and Paragraph Proofs Lesson Notes • Identify and use basic postulates about points, lines, and planes. 1 Focus • Write paragraph proofs. were postulates used by the founding fathers of the United States? Vocabulary • • • • • • postulate axiom theorem proof paragraph proof informal proof 5-Minute Check Transparency 2-5 Use as a quiz or review of Lesson 2-4. U.S. Supreme Court Justice William Douglas stated “The First Amendment makes confidence in the common sense of our people and in the maturity of their judgment the great postulate of our democracy.” The writers of the constitution assumed that citizens would act and speak with common sense and maturity. Some statements in geometry also must be assumed or accepted as true. Mathematical Background notes are available for this lesson on p. 60D. POINTS, LINES, AND PLANES In geometry, a postulate, or axiom , is a statement that describes a fundamental relationship between the basic terms of geometry. Postulates are accepted as true. The basic ideas about points, lines, and planes can be stated as postulates. Postulates 2.1 2.2 Through any two points, there is exactly one line. Through any three points not on the same line, there is exactly one plane. Example 1 Points and Lines COMPUTERS Jessica is setting up a network for her father’s business. There are five computers in his office. Each computer needs to be connected to every other computer. How many connections does Jessica need to make? Study Tip Explore There are five computers, and each is connected to four others. Drawing Diagrams Plan Draw a diagram to illustrate the solution. Solve Let noncollinear points A, B, C, D, and E represent the five computers. Connect each point with every other point. Then, count the number of segments. When listing segments, start with one vertex and draw all of the segments from that vertex. Then move on to the other vertices until all possible segments have been drawn. Between every two points there is exactly one segment. So, the connection between computer A and computer B is the same as the connection between computer B and computer A. For the five points, ten segments can be drawn. were postulates used by the founding fathers of the United States? Ask students: • How would you interpret the words of William Douglas? Sample answer: The constitution assumes that people have the maturity to handle the responsibilities of democracy. • Do you think the founding fathers set up a test to find out whether or not people do have common sense and maturity? Sample answer: No; they assumed that people did. 2 Teach A E B POINTS, LINES, AND PLANES In-Class Example D C Examine A B , A C , A D , A E , B C , B D , BE , C D , C E , and D E each represent a connection between two computers. So there will be ten connections among the five computers. Power Point® 1 SNOW CRYSTALS Some snow crystals are shaped like regular hexagons. How many lines must be drawn to interconnect all vertices of a hexagonal snow crystal? 15 Lesson 2-5 Postulates and Paragraph Proofs 89 Jeff Hunter/Getty Images Resource Manager Workbook and Reproducible Masters Chapter 2 Resource Masters • Study Guide and Intervention, pp. 81–82 • Skills Practice, p. 83 • Practice, p. 84 • Reading to Learn Mathematics, p. 85 • Enrichment, p. 86 Teaching Geometry With Manipulatives Masters, p. 8 Transparencies 5-Minute Check Transparency 2-5 Answer Key Transparencies Technology Interactive Chalkboard Lesson x-x Lesson Title 89 In-Class Example There are other postulates that are based on relationships among points, lines, and planes. Power Point® Teaching Tip Tell students that most postulates are very obvious and make very good sense, but they do not have a formal proof behind them. Nonetheless, students are to accept them as true and use them to prove other statements and theorems. Postulates 2 Determine whether each 2.3 2.4 2.5 A line contains at least two points. 2.6 2.7 If two lines intersect, then their intersection is exactly one point. A plane contains at least three points not on the same line. If two points lie in a plane, then the entire line containing those points lies in that plane. If two planes intersect, then their intersection is a line. statement is always, sometimes, or never true. Explain. and EF a. If plane T contains EF contains point G, then plane T contains point G. Always; Postulate 2.5 states that if two points lie in a plane, then the entire line containing those points lies in the plane. b. For XY, if X lies in plane Q and Y lies in plane R , then plane Q intersects plane R . Sometimes; planes Q and R can can intersect be parallel, and XY both planes. c. GH contains three noncollinear points. Never; noncollinear points do not lie on the same line by definition. Building on Prior Knowledge Students learned basic principles about points, lines, and planes in Chapter 1. In this lesson, they will revisit those concepts in the form of postulates that they can use to write informal proofs and paragraph proofs. Answers 1. Deductive reasoning is used to support claims that are made in a proof. 2. Example 2 Use Postulates Determine whether each statement is always, sometimes, or never true. Explain. a. If points A, B, and C lie in plane M, then they are collinear. Sometimes; A, B, and C do not necessarily have to be collinear to lie in plane M. b. There is exactly one plane that contains noncollinear points P, Q, and R. Always; Postulate 2.2 states that through any three noncollinear points, there is exactly one plane. c. There are at least two lines through points M and N. Never; Postulate 2.1 states that through any two points, there is exactly one line. PARAGRAPH PROOFS Undefined terms, definitions, postulates, and algebraic properties of equality are used to prove that other statements or conjectures are true. Once a statement or conjecture has been shown to be true, it is called a theorem, and it can be used like a definition or postulate to justify that other statements are true. Study Tip Proofs Before writing a proof, you should have a plan. One strategy is to work backward. Start with what you want to prove, and work backward step by step until you reach the given information. You will study and use various methods to verify or prove statements and conjectures in geometry. A proof is a logical argument in which each statement you make is supported by a statement that is accepted as true. One type of proof is called a paragraph proof or informal proof . In this type of proof, you write a paragraph to explain why a conjecture for a given situation is true. Proofs Five essential parts of a good proof: • State the theorem or conjecture to be proven. • List the given information. • If possible, draw a diagram to illustrate the given information. • State what is to be proved. • Develop a system of deductive reasoning. In Lesson 1-2, you learned the relationship between segments formed by the midpoint of a segment. This statement can be proven, and the result stated as a theorem. 90 Chapter 2 Reasoning and Proof Differentiated Instruction 3. postulates, theorems, algebraic properties, definitions 90 Chapter 2 Reasoning and Proof Intrapersonal Tell students to read quietly over the postulates and examples in this lesson and note the differences in the postulate statements and the statements they are to write proofs for. Advise students to go through the text and their study notebooks to compile a list of useful information they could use to write the proofs in this lesson. Example 3 Write a Paragraph Proof PARAGRAPH PROOFS Given that M is the midpoint of PQ , write a paragraph proof to show that —– –— PM MQ. . Given: M is the midpoint of PQ Q Prove: M MQ . P In-Class Example 3 Given CD, AC intersecting M From the definition of midpoint of a segment, PM MQ. This means that PM and M Q have the same measure. By the definition of congruence, if two segments have the same measure, then they MQ . are congruent. Thus, PM write a paragraph proof to show that A, C, and D determine a plane. and CD must intersect at C AC because if two lines intersect, then their intersection is exactly and one point. Point A is on AC point D is on CD . Therefore points A and D are not collinear. Therefore ACD is a plane as it contains three points not on the same line. P Once a conjecture has been proven true, it can be stated as a theorem and used in other proofs. The conjecture in Example 3 is known as the Midpoint Theorem. Theorem 2.8 Midpoint Theorem If M is the midpoint of , AB then AM MB . Concept Check Guided Practice GUIDED PRACTICE KEY Exercises Examples 4–5, 11 6 7–10 1 2 3 7. definition of collinear. 8. Through any three points not on the same line, there is exactly one plane. 9. Through any two points, there is exactly one line. Application 3 Practice/Apply 1. Explain how deductive reasoning is used in a proof. 1–3. See margin. 2. OPEN ENDED Draw figures to illustrate Postulates 2.6 and 2.7. 3. List the types of reasons that can be used for justification in a proof. Determine the number of segments that can be drawn connecting each pair of points. 6 5. 15 4. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 2. • include any other item(s) that they find helpful in mastering the skills in this lesson. 6. Determine whether the following statement is always, sometimes, or never true. Explain. See p. 123C. The intersection of three planes is two lines. are In the figure, BD and BR BD . State the in plane P, and W is on postulate or definition that can be used to show each statement is true. 7. B, D, and W are collinear. 8. E, B, and R are coplanar. 9. R and W are collinear. E R D B P 10. PROOF In the figure at the right, P is the R and ST , and Q R ST . midpoint of Q Write a paragraph proof to show that PQ PT. See p. 123C. W Q About the Exercises… T Organization by Objective • Points, Lines, and Planes: 12–21 • Paragraph Proofs: 22–28 P S R 11. DANCING Six students are participating in a dance to celebrate the opening of a new community center. The students, each connected to each of the other students with wide colored ribbons, will move in a circular motion. How many ribbons are needed? 15 ribbons www.geometryonline.com/extra_examples Power Point® Odd/Even Assignments Exercises 12–27 are structured so that students practice the same concepts whether they are assigned odd or even problems. Lesson 2-5 Postulates and Paragraph Proofs 91 Unlocking Misconceptions Assignment Guide Writing Proofs Explain to students that a common mistake in writing proofs is skipping a step or assuming a step that should be included in the proof. Sometimes, the missed step can be quite obvious, but it still has to be included. Tell students to make a habit of listing each piece of information with a separate explanation for each and to avoid using two reasons or postulates for the same statement when they are writing proofs. Basic: 13–19 odd, 23–31 odd, 33–48 Average: 13–31 odd, 33–48 Advanced: 12–30 even, 31–42 (optional: 43–48) Lesson 2-5 Postulates and Paragraph Proofs 91 ★ indicates increased difficulty NAME ______________________________________________ DATE ____________ PERIOD _____ Practice and Apply Study Guide andIntervention Intervention, 2-5 Study Guide and p. 81 (shown) and Proofs p. 82 Postulates and Paragraph Determine the number of segments that can be drawn connecting each pair of points. Points, Lines, and Planes In geometry, a postulate is a statement that is accepted as true. Postulates describe fundamental relationships in geometry. Postulate: Postulate: Postulate: Postulate: Postulate: Postulate: Postulate: Through any two points, there is exactly one line. Through any three points not on the same line, there is exactly one plane. A line contains at least two points. A plane contains at least three points not on the same line. If two points lie in a plane, then the line containing those points lies in the plane. If two lines intersect, then their intersection is exactly one point. If two planes intersect, then their intersection is a line. For Exercises See Examples 12–15 16–21 22–28 1 2 3 Example Determine whether each statement is always, sometimes, or never true. a. There is exactly one plane that contains points A, B, and C. Sometimes; if A, B, and C are collinear, they are contained in many planes. If they are noncollinear, then they are contained in exactly one plane. 6 12. 10 13. Extra Practice See page 757. b. Points E and F are contained in exactly one line. Always; the first postulate states that there is exactly one line through any two points. 15 14. c. Two lines intersect in two distinct points M and N. Never; the intersection of two lines is one point. 21 15. Exercises Use postulates to determine whether each statement is always, sometimes, or never true. 1. A line contains exactly one point. never 2. Noncollinear points R, S, and T are contained in exactly one plane. always 3. Any two lines and Determine whether the following statements are always, sometimes, or never true. Explain. 16–21. See p. 123C. 16. Three points determine a plane. m intersect. sometimes 4. If points G and H are contained in plane M, then G H is perpendicular to plane M. never 5. Planes R and S intersect in point T. never 6. If points A, B, and C are noncollinear, then segments A B , BC , and C A are contained in exactly one plane. always In the figure, A C and D E are in plane Q and A C || D E . State the postulate that can be used to show each statement is true. any three points not on the same line, there is exactly one plane. Q C B Lesson 2-5 A E D G 8. BE lies in plane Q. If two points lie in a plane, then the line containing those points lies in the plane. NAME ______________________________________________ DATE /M G Hill 81 Gl 17. Points G and H are in plane X. Any point collinear with G and H is in plane X . F 7. Exactly one plane contains points F, B, and E. Through Skills Practice, p. 83 and 2-5 Practice (Average) Practice, (shown) Postulates p. and 84 Paragraph Proofs 18. The intersection of two planes can be a point. 19. Points S, T, and U determine three lines. ____________ Gl PERIOD G _____ ★ 20. Points A and B lie in at least one plane. ★ 21. If line lies in plane P and line m lies in plane Q , then lines and m lie in plane R. Determine the number of line segments that can be drawn connecting each pair of points. 21 1. 2. 28 In the figure at the right, AC and BD lie BY and CX lie in plane K. in plane J, and State the postulate that can be used to show each statement is true. 22–27. See p. 123C. 22. C and D are collinear. 23. XB lies in plane K. Determine whether the following statements are always, sometimes, or never true. Explain. 3. The intersection of two planes contains at least two points. Always; the intersection of two planes is a line, and a line contains at least two points. 4. If three planes have a point in common, then they have a whole line in common. Sometimes; they might have only that single point in common. lie in plane A . State the postulate In the figure, line m and TQ that can be used to show that each statement is true. S 24. Points A, C, and X are coplanar. 25. AD lies in plane J. m A T m lie in the same plane. Postulate 2.5: If two points lie in a plane, then the entire line containing those points lies in that plane. 5. L, T, and line Q L m and S T intersect at T. Postulate 2.6: If two lines intersect, then their intersection is exactly one point. 6. Line 7. In the figure, E is the midpoint of A B and C D , and AB CD. Write a paragraph proof to prove that A E ED . C A E Given: E is the midpoint of A B and CD AB CD D Prove: AE ED Proof: Since E is the midpoint of A B and C D , we know by the Midpoint Theorem, that A E EB and C E ED . By the definition of congruent B 1 1 segments, AE EB AB and CE ED CD. Since AB CD, 2 2 1 1 AB CD by the Multiplication Property. So AE ED, and by the 2 2 definition of congruent segments, A E ED . Detective 27. Points Y, D, and C are coplanar. A police detective gathers facts and collects evidence for use in criminal cases. The facts and evidence are used together to prove a suspect’s guilt in court. 28. PROOF Point C is the midpoint of A B and B is the midpoint of C D . Prove that C BD . See margin. A the plane that contains A, B, and C and the plane that contains B, C, and D NAME ______________________________________________ DATE /M G Hill 84 ____________ Gl PERIOD G _____ Reading 2-5 Readingto to Learn Learn MathematicsELL Mathematics, p. 85 Postulates and Paragraph Proofs Pre-Activity How are postulates used by the founding fathers of the United States? Read the introduction to Lesson 2-5 at the top of page 89 in your textbook. Postulates are often described as statements that are so basic and so clearly correct that people will be willing to accept them as true without asking for evidence or proof. Give a statement about numbers that you think most people would accept as true without evidence. Sample answer: Every Online Research For information about a career as a detective, visit: www.geometryonline. com/careers 1. Determine whether each of the following is a correct or incorrect statement of a geometric postulate. If the statement is incorrect, replace the underlined words to make the statement correct. incorrect; a. A plane contains at least two points that do not lie on the same line. three points d. A line contains at least one point . incorrect; two points incorrect; e. If two lines are parallel , then their intersection is exactly one point. intersect f. Through any two points, there is at most one line. incorrect; exactly 92 Chapter 2 Reasoning and Proof NAME ______________________________________________ DATE a. If two planes intersect, their intersection is a line. always c. There is exactly one plane that contains three collinear points. never; Sample answer: There are infinitely many planes if the three points are collinear, but only one plane if the points are noncollinear. d. If two lines intersect, their intersection is one point. always 3. Use the walls, floor, and ceiling of your classroom to describe a model for each of the following geometric situations. 2-5 Enrichment Enrichment, ____________ PERIOD _____ p. 86 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. a. two planes that intersect in a line Sample answer: two adjacent walls that intersect at an edge of both walls in the corner of the room b. two planes that do not intersect Sample answer: the ceiling and the floor (or two opposite walls) c. three planes that intersect in a point Sample answer: the floor (or ceiling) and two adjacent walls that intersect at a corner of the floor (or ceiling) Helping You Remember 4. A good way to remember a new mathematical term is to relate it to a word you already know. Explain how the idea of a mathematical theorem is related to the idea of a scientific theory. Sample answer: Scientists do experiments to prove theories; mathematicians use deductive reasoning to prove theorems. Both processes involve using evidence to show that certain statements are true. 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 O 92 Chapter 2 Reasoning and Proof X arguments, which is a speech that uses deductive reasoning, in court cases. Spencer Grant/PhotoEdit 2. Determine whether each statement is always, sometimes, or never true. If the statement is not always true, explain why. b. The midpoint of a segment divides the segment into two congruent segments. always J 29. MODELS Faith’s teacher asked her to make a figure showing the number of lines and planes formed from four points that are noncollinear and noncoplanar. Faith decided to make a mobile of straws, pipe cleaners, and colored sheets of tissue paper. She plans to glue the paper to the straws and connect the straws together to form a group of connected planes. How many planes and lines will she have? She will have 4 different planes and 6 lines. incorrect; c. Through any four points not on the same line, there is exactly one plane. three points D C B 30. CAREERS Many professions use deductive reasoning and paragraph proofs. For example, a police officer uses deductive reasoning investigating a traffic accident and then writes the findings in a report. List a profession, and describe how it can use paragraph proofs. Sample answer: Lawyers make final number is equal to itself. Reading the Lesson b. If two planes intersect, then the intersection is a line. correct A 26. X and Y are collinear. 8. LOGIC Points A, B, and C are not collinear. Points B, C, and D are not collinear. Points A, B, C, and D are not coplanar. Describe two planes that intersect in line BC. Gl K Y Nancy Olivia Mario X X X X X Kenji 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 31. CRITICAL THINKING You know that three noncollinear points lie in a single plane. In Exercise 29, you found the number of planes defined by four noncollinear points. What are the least and greatest number of planes defined by five noncollinear points? one, ten 4 Assess Open-Ended Assessment 32. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How are postulates used in literature? Include the following in your answer: • an example of a postulate in historic United States’ documents, and • an example of a postulate in mathematics. Standardized Test Practice Getting Ready for Lesson 2-6 33. Which statement cannot be true? C A A plane can be determined using three noncollinear points. B Two lines intersect at exactly one point. C At least two lines can contain the same two points. D A midpoint divides a segment into two congruent segments. Prerequisite Skill Students will learn about algebraic proof in Lesson 2-6. Use Exercises 43–48 to determine your students’ familiarity with solving equations. 34. ALGEBRA For all values of x, (8x 4 2x2 3x 5) (2x 4 x 3 3x 5) A Answers A 6x 4 x 3 2x2 10. B 6x 4 3x2 6x 10. C 6x 4 x 3 2x2 6x. D 6x 4 3x 2. Maintain Your Skills Mixed Review 35. 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. (Lesson 2-4) yes; Law of Detachment (1) Part-time jobs require 20 hours of work per week. (2) Jamie has a part-time job. (3) Jamie works 20 hours per week. Write the converse, inverse, and contrapositive of each conditional statement. Determine whether each related conditional is true or false. If a statement is false, find a counterexample. (Lesson 2-3) 36–37. See margin. 36. If you have access to the Internet at your house, then you have a computer. 37. If ABC is a right triangle, one of its angle measures is greater than 90. 38. BIOLOGY Use a Venn diagram to illustrate the following statement. If an animal is a butterfly, then it is an arthropod. (Lesson 2-2) See p. 123C. Use the Distance Formula to find the distance between each pair of points. (Lesson 1-3) Getting Ready for the Next Lesson Speaking Have students choose a paragraph proof they wrote and explain each step aloud. 4.1 17 P(8, 2), Q(1, 3) 106 10.3 39. D(3, 3), F(4, 1) 40. M(0, 2), N(5, 5) 41. 42. R(5, 12), S(2, 1) 5.8 34 13.0 170 PREREQUISITE SKILL Solve each equation. (To review solving equations, see pages 737 and 738.) y 6 43. m 17 8 25 44. 3y 57 19 45. 12 14 12 46. t 3 27 ⴚ24 47. 8n 39 41 10 48. 6x 33 0 www.geometryonline.com/self_check_quiz 11 2 Lesson 2-5 Postulates and Paragraph Proofs 93 28. Given: C is the midpoint of AB . B is the midpoint of C D . Prove: AC BD Proof: We are given that C is the midpoint of AB , and B is the midpoint of CD . By the definition of midpoint AC CB and C B BD . Using the definition of congruent segments, AC CB, and CB BD. AC BD by the Transitive Property of Equality. Thus, A C BD by the definition of congruent segments. 32. Sample answer: The forms and structures of different types of writing are accepted as valid, such as the structure of a poem. Answers should include the following. • The Declaration of Independence, “We hold these truths to be self-evident, …” • Through any two points, there is exactly one line. 36. Converse: If you have a computer, then you have access to the Internet at your house. False; you can have a computer and not have access to the Internet. Inverse: If you do not have access to the Internet at your house, then you do not have a computer. False; it is possible to not have access to the Internet and still have a computer. Contrapositive: If you do not have a computer, then you do not have access to the Internet at your house. False; you could have Internet access through your television or wireless phone. 37. Converse: If ABC has an angle with measure greater than 90, then ABC is a right triangle. False; the triangle would be obtuse. Inverse: If ABC is not a right triangle, none of its angle measures are greater than 90. False; it could be an obtuse triangle. Contrapositive: If ABC does not have an angle measure greater than 90, ABC is not a right triangle. False; mABC could still be 90 and ABC be a right triangle. Lesson 2-5 Postulates and Paragraph Proofs 93 Lesson Notes 1 Focus 5-Minute Check Transparency 2-6 Use as a quiz or review of Lesson 2-5. Algebraic Proof • Use algebra to write two-column proofs. • Use properties of equality in geometry proofs. • deductive argument • two-column proof • formal proof Mathematical Background notes are available for this lesson on p. 60D. is mathematical evidence similar to evidence in law? Ask students: • In math, what one thing do you need to prove a statement is false? What evidence can a lawyer use to prove that someone is innocent (or has been falsely accused)? a counterexample; sample answer: an alibi • How does the use of evidence in law differ from its use in math? Sample answer: Lawyers present evidence to sway opinions, sometimes without knowing the truth. A mathematician presents evidence to prove factual statements. is mathematical evidence similar to evidence in law? Vocabulary Lawyers develop their cases using logical arguments based on evidence to lead a jury to a conclusion favorable to their case. At the end of a trial, a lawyer will make closing remarks summarizing the evidence and testimony that they feel proves their case. These closing arguments are similar to a proof in mathematics. ALGEBRAIC PROOF Algebra is a system with sets of numbers, operations, and properties that allow you to perform algebraic operations. Properties of Equality for Real Numbers Study Tip Commutative and Associative Properties Throughout this text, we shall assume the Commutative and Associative Properties for addition and multiplication. Reflexive Property For every number a, a a. Symmetric Property For all numbers a and b, if a b, then b a. Transitive Property For all numbers a, b, and c, if a b and b c, then a c. Addition and Subtraction Properties For all numbers a, b, and c, if a b, then a c b c and a c b c. Multiplication and Division Properties For all numbers a, b, and c, if a b, then a c b c a b and if c 0, c c. Substitution Property For all numbers a and b, if a b, then a may be replaced by b in any equation or expression Distributive Property For all numbers a, b, and c, a(b c) ab ac. The properties of equality can be used to justify each step when solving an equation. A group of algebraic steps used to solve problems form a deductive argument . Example 1 Verify Algebraic Relationships Solve 3(x 2) 42. Algebraic Steps 3(x 2) 42 3x 6 42 3x 6 6 42 6 3x 48 3x 48 3 3 x 16 94 Properties Original equation Distributive Property Addition Property Substitution Property Division Property Substitution Property Chapter 2 Reasoning and Proof Bob Daemmrich/The Image Works Resource Manager Workbook and Reproducible Masters Chapter 2 Resource Masters • Study Guide and Intervention, pp. 87–88 • Skills Practice, p. 89 • Practice, p. 90 • Reading to Learn Mathematics, p. 91 • Enrichment, p. 92 • Assessment, p. 120 Prerequisite Skills Workbook, pp. 41–44, 83–86, 93–94 Teaching Geometry With Manipulatives Masters, p. 8 Transparencies 5-Minute Check Transparency 2-6 Answer Key Transparencies Technology Interactive Chalkboard Multimedia Applications: Virtual Activities Example 1 is a proof of the conditional statement If 5x 3(x 2) 42, then x 6. Notice that the column on the left is a step-by-step process that leads to a solution. The column on the right contains the reason for each statement. In geometry, a similar format is used to prove conjectures and theorems. A two-column proof , or formal proof , contains statements and reasons organized in two columns. In a two-column proof, each step is called a statement, and the properties that justify each step are called reasons. Write a Two-Column Proof Example 2 a. If 3x 1, then x 2 5 3 Study Tip Mental Math If your teacher permits you to do so, some steps may be eliminated by performing mental calculations. For example, in part a of Example 2, statements 4 and 6 could be omitted. Then the reason for statements 5 would be Addition Property and Division Property for statement 7. Reasons 5 1. 3 x 1 3 5 2. 3x 3 1 3 1. Given 3. 3x 5 1 4. 3x 5 5 1 5 5. 3x 6 3. Substitution 4. Addition Property 5. Substitution 3x 6 6. 6. Division Property 7. x 2 7. Substitution 3 b. Given: 2. Distributive Property 3 3. 4. 5. 6. 7. 7 2n 8 n 7 2n n 8 n n 7n8 7n787 n 1 1. Given 2. Multiplication Property 3. 4. 5. 6. 7. Distributive Property Addition Property Substitution Subtraction Property Substitution 1 n 8. 8. Division Property 9. n 1 9. Substitution 1 1 Power Point® 2(5 3a) 4(a 7) 92. 2(5 3a) 4(a 7) 92 (Original eqn.) 10 6a 4a 28 92 (Distr. Prop.) 18 10a 92 (Subst. Prop.) 18 10a 18 92 18 (Add. Prop.) 10a 110 (Subst. Prop.) 110 10a (Div. Prop.) 10 10 Teaching Tip Reasons 7 1 1. n 4 n 2 2 7 1 2. 2 n 2 4 n 2 2 In-Class Examples a 11 (Subst. Prop.) 7 1 n 4 n 2 2 Prove: n 1 Proof: Statements ALGEBRAIC PROOFS 1 Solve Write a two-column proof. Statements 2 Teach Explain that since students may have different preferences when solving algebraic equations, their proofs might vary slightly from the examples. For example, one student may distribute a variable first, while another uses addition or subtraction. Assure students that as long as they use properties of equality appropriately, their proofs will be correct. 2 Write a two-column proof for each of the following. 7d 3 4 a. If 6, then d 3. GEOMETRIC PROOF Since geometry also uses variables, numbers, and operations, many of the properties of equality used in algebra are also true in geometry. For example, segment measures and angle measures are real numbers, so properties from algebra can be used to discuss their relationships. Some examples of these applications are shown below. Property Segments Angles Reflexive AB AB m1 m1 Symmetric If AB CD, then CD AB. If m1 m2, then m2 m1. Transitive If AB CD and CD EF, then AB EF. If m1 m2 and m2 m3, then m1 m3. www.geometryonline.com/extra_examples Lesson 2-6 Algebraic Proof 95 Statements (Reasons) 7d 3 4 7d 3 2. 4 4(6) (Mult. Prop.) 4 1. 6 (Given) 3. 7d 3 24 (Substitution) 4. 7d 3 3 24 3 (Subtr. Prop.) 5. 7d 21 (Substitution) 7d 7 21 7 6. (Div. Prop.) 7. d 3 (Substitution) (continued on the next page) Lesson 2-6 Algebraic Proofs 95 Standardized Example Test Practice 9 p 11 b. If 3p , then 5 10 10 p 10 p 9 11 2. 10 3p 10 10 5 10 11 10 If AB CD EF , and C D , then which of the following is a valid conclusion? 1. 3p (Given) (Mult. Prop.) 3. 30p 18 11 p (Distr. Prop.) 4. 30p p 18 11 p p (Subtr. Prop.) 5. 29p 18 11 (Substitution) 6. 29p 18 18 11 18 (Add. Prop.) 7. 29p 29 (Substitution) 29p 29 Test-Taking Tip More than one statement may be correct. Work through each problem completely before indicating your answer. 29 29 9. p 1 (Substitution) GEOMETRIC PROOFS Power Point® ST , then which of the RP following is a valid conclusion? B I. GH JK RP II. PR TS III. GH JK ST RP A I only B I and II C I and III D I, II, and III Chapter 2 Reasoning and Proof B D A I only B I and II C I and III D I, II, and III E F Read the Test Item Determine whether the statements are true based on the given information. In Example 3, each conclusion was justified using a definition or property. This process is used in geometry to verify and prove statements. Example 4 Geometric Proof TIME On a clock, the angle formed by the hands at 2:00 is a 60 angle. If the angle formed at 2:00 is congruent to the angle formed at 10:00, prove that the angle at 10:00 is a 60 angle. Given: 4 SEA LIFE A starfish has five 96 C I AB CD and CD EF II AB EF III AB EF Because Statements I, II, and III are true, choice D is correct. 3 If GH JK ST and arms. If the length of arm 1 is 22 cm, and arm 1 is congruent to arm 2, and arm 2 is congruent to arm 3, prove that arm 3 has length 22 cm. We are given arm 1 arm 2 and arm 2 arm 3, so by the definition of congruence, the measure of arm 1 the measure of arm 2 and the measure of arm 2 the measure of arm 3. By the Transitive Property of Equality, we know that the measure of arm 1 the measure of arm 3. We can then substitute 22 cm for the measure of arm 1 to prove that the measure of arm 3 is 22 cm. A Solve the Test Item Statement I: Examine the given information, AB CD EF and C D . From the definition of CD EF congruent segments, if A B and C D , then AB CD and CD EF. Thus, Statement I is true. Statement II: B E F. By the definition of congruent segments, if AB EF, then A Statement II is true also. Statement III: If AB CD and CD EF, then AB EF by the Transitive Property. Thus, Statement III is true. 8. (Div. Prop.) In-Class Examples Justify Geometric Relationships Multiple-Choice Test Item p 1. Statements (Reasons) 9 5 3 Prove: m2 60 2 10 m10 60 Proof: Statements 1. m2 60 2 10 2. m2 m10 3. 60 m10 4. m10 60 96 Reasons 1. Given 2. Definition of congruent angles 3. Substitution 4. Symmetric Property Chapter 2 Reasoning and Proof Aaron Haupt Differentiated Instruction Interpersonal Let groups of students work one or two selected problems from Exercises 24–29 on p. 98. Stipulate that each group member should contribute at least one step of the proof. Encourage groups to brainstorm beforehand to determine the properties they will use and the order they will use them in. Allow the groups to check and compare their proofs when they are done to see if any two groups found different ways to prove the same statement. Concept Check Guided Practice GUIDED PRACTICE KEY Exercises Examples 4–7 8 9, 10, 12 11 1 3 2 4 1. OPEN ENDED Write a statement that illustrates the Substitution Property of Equality. 1–2. See margin. 2. Describe the parts of a two-column proof. 3. State the part of a conditional that is related to the Given statement of a proof. What part is related to the Prove statement? hypothesis; conclusion State the property that justifies each statement. 5 2 4. If 2x 5, then x Division Property x 2 5. If 7, then x 14. Multiplication Property 6. If x 5 and b 5, then x b. Substitution Property 7. If XY AB WZ AB, then XY WZ. Addition Property x 2 8. Solve 4x 7 11. List the property that justifies each step. See margin. 9. Complete the following proof. 2 3 Given: 5 x 1 Prove: x 6 3 Practice/Apply Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 2. • include a sample algebraic proof and a sample geometric proof. • include any other item(s) that they find helpful in mastering the skills in this lesson. Proof: Statements a. ? 5 2x 1 3 2 b. 35 x 3(1) 3 c. 15 2x 3 d. ? 2x 12 e. x 6 Reasons a. Given b. About the Exercises… Mult. Prop. ? c. ? Dist. Prop. d. Subtraction Prop. e. ? Div. Prop. Organization by Objective • Algebraic Proofs: 14–21 • Geometric Proofs: 22–29 Write a two-column proof. 10–12. See pp. 123C–123D. 10. Prove that if 25 7(y 3) 5y, then 2 y. 11. If rectangle ABCD has side lengths AD 3 and AB 10, then AC BD. 12. The Pythagorean Theorem states that in a right triangle ABC, c2 a2 b2. PROOF 2 b2. Prove that a c Standardized Test Practice 13. ALGEBRA If 8 x = 12, then 4 x = A 28 B 24 ? . C C 0 D 4 Practice and Apply For Exercises See Examples 15, 16, 20 14, 17–19, 21 22–27 28, 29 1 2 3 4 Extra Practice See page 757. State the property that justifies each statement. 14. If mA mB and mB mC, mA mC. Trans. Prop. 15. If HJ 5 20, then HJ 15. Subt. Prop. 16. If XY 20 YW and XY 20 DT, then YW DT. Substitution 17. If m1 m2 90 and m2 m3, then m1 m3 90. Substitution 1 2 1 2 18. If AB EF, then AB EF. Div. or Mult. Prop. Odd/Even Assignments Exercises 14–29 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 15–31 odd, 32–33, 35–51 Average: 15–31 odd, 32–33, 35–51 Advanced: 14–30 even, 32–48 (optional: 49–51) All: Quiz 2 (1–5) x 19. AB AB Reflexive Property Lesson 2-6 Algebraic Proof 97 Answers 1. Sample answer: If x 2 and x y 6, then 2 y 6. 2. given and prove statements and two columns, one of statements and one of reasons 8. Given: 4x 7 11 2 Prove: x 4 Proof: Statements (Reasons) x 2 1. 4x 7 11 (Given) 2x 2. 2 4x 7 2(11) (Mult. Prop.) 3. x 8x 14 22 (Dist.Prop.) 4. 9x 14 22 (Substitution) 5. 9x 36 (Add. Prop.) 6. x 4 (Div. Prop.) Lesson 2-6 Algebraic Proofs 97 Answers 20. If 2x 5, which property can be used to support the statement 2x 3 5? 32. Given: Ek hf W 21. Which property allows you to state m4 m5, if m4 35 and m5 35? 3 2 Dist. Prop. Ek W h Substitution Prove: f 1 2 1 2 22. If AB CD, which property can be used to justify the statement AB CD? Proof: Statements (Reasons) 1. Ek hf W (Given) 2. Ek W hf (Subt. Prop.) Div. or Mult. Prop. 23. Which property could be used to support the statement EF JK, given that EF GH and GH JK? Transitive Prop. Complete each proof. Ek W h Ek W 4. f (Sym. Prop.) h 24. Given: 3. f (Div. Prop.) 3x 5 7 2 x3 Prove: Proof: Statements 36. Sample answer: Lawyers use evidence and testimony as reasons for justifying statements and actions. All of the evidence and testimony are linked together to prove a lawyer’s case, much as in a proof in mathematics. Answers should include the following. • Evidence is used to verify facts from witnesses or materials. • Postulates, theorems, definitions, and properties can be used to justify statements made in mathematics. Reasons 3x 5 a. 7 2 b. c. d. e. a. 3x 5 ? 2 2(7) 2 3x 5 14 3x 9 ? x3 ? Given b. Mult. Prop. ? Substitution c. d. ? Subt. Prop. e. Div. Prop. 1 25. Given: 2x 7 x 2 3 Prove: x 3 Proof: Statements a. ? 2x 7 1x 2 3 1 b. ? 3(2x 7) 3 x 2 3 c. 6x 21 x 6 d. ? 5x 21 6 e. 5x 15 ? x3 f. A gymnast exhibits kinetic energy when performing on the balance beam. The movements and flips show the energy that is being displayed while the gymnast is moving. a. Given b. Mult. Prop. ? Dist. Prop. c. d. Subt. Prop. e. ? Add. Prop. f. Div. Prop. Write a two-column proof. 26–31. See p. 123D. PROOF Physics Reasons 1 2 7 2 26. If 4 a a, then a 1. 1 2 3 2 13 4 27. If 2y 8, then y . 2 3 28. If m 9, then m 18. 29. If 5 z 1, then z 6. 30. If XZ ZY, XZ 4x 1, and ZY 6x 13, then x 7. 31. If mACB mABC, then mXCA mYBA. X 4x 1 Z A Source: www.infoplease.com 6x 13 X C B Y Y 32. PHYSICS Kinetic energy is the energy of motion. The formula for kinetic energy is Ek h f W, where h represents Planck’s Constant, f represents the frequency of its photon, and W represents the work function of the material being used. Solve this formula for f and justify each step. See margin. 98 Chapter 2 Reasoning and Proof Duomo/CORBIS 98 Chapter 2 Reasoning and Proof 33. GARDENING Areas in the southwest and southeast have cool but mild winters. In these areas, many people plant pansies in October so that they have flowers outside year-round. In the arrangement of pansies shown, the walkway divides the two sections of pansies into four beds that are the same size. If mACB mDCE, what could you conclude about the relationship among ACB, DCE, ECF, and ACG? All of the angle measures A G B NAME ______________________________________________ DATE The following properties of algebra can be used to justify the steps when solving an algebraic equation. C Property Statement Reflexive For every number a, a a. Symmetric For all numbers a and b, if a b then b a. Transitive For all numbers a, b, and c, if a b and b c then a c. Addition and Subtraction For all numbers a, b, and c, if a b then a c b c and a c b c. Multiplication and Division For all numbers a, b, and c, if a b then a c b c, and if c 0 then . Substitution For all numbers a and b, if a b then a may be replaced by b in any equation or expression. D E Example Solve 6x 2(x 1) 30. Algebraic Steps 6x 2(x 1) 30 6x 2x 2 30 8x 2 30 8x 2 2 30 2 8x 32 CRITICAL THINKING For Exercises 34 and 35, use the following information. Below is a family tree of the Gibbs family. Clara, Carol, Cynthia, and Cheryl are all daughters of Lucy. Because they are sisters, they have a transitive and symmetric relationship. That is, Clara is a sister of Carol, Carol is a sister of Cynthia, so Clara is a sister of Cynthia. Properties Given Distributive Property Substitution Addition Property Substitution 8x 32 8 8 Division Property x4 Substitution Exercises Complete each proof. 4x 6 2. Given: 4x 8 x 2 Prove: x 2 1. Given: 9 2 Prove: x 3 Statements 4x 6 2 4x 6 b. 2 2(9) 2 a. 9 Lucy Cynthia Carol Chris Diane Kevin Dierdre Reasons Statements Reasons a. Given a. 4x 8 x 2 a. Given b. Mult. Prop. b. 4x 8 x x2x b. Subtr. Prop. c. 3x 8 2 c. Substitution c. 4x 6 18 c. Subst. d. 4x 6 6 18 6 d. Subtr. Prop. e. 4x 12 e. Substitution d. 3x 8 8 f. f. Div. Prop. 28 e. 3x 6 g. x 3 g. Substitution f. 12 4 4x 4 Cheryl Gl Michael 3x 3 d. Subtr. Prop. e. Substitution 6 3 f. Div. Prop. g. x 2 NAME ______________________________________________ DATE /M G Hill 87 Skills Practice, 2-6 Practice (Average) Steven b c For all numbers a, b, and c, a(b c) ab ac. Distributive F a c Lesson 2-6 p. 87 (shown) and p. 88 Algebraic Proof Algebraic Proof would be equal. Clara ____________ PERIOD _____ Study Guide andIntervention Intervention, 2-6 Study Guide and p. 89 and Practice, p. 90 (shown) Algebraic Proof g. Substitution ____________ Gl PERIOD G _____ PROOF Write a two-column proof. Cyle Standardized Test Practice Allycia Maria 1. If mABC mCBD 90, mABC 3x 5, x1 2 34. What other relationships in a family have reflexive, symmetric, or transitive relationships? Explain why. Remember that the child or children of each person are listed beneath that person’s name. Consider relationships such as first cousin, ancestor or descendent, aunt or uncle, sibling, or any other relationship. 1. mABC mCBD 90 mABC 3x 5 x1 mCBD 35. Construct your family tree on one or both sides of your family and identify the reflexive, symmetric, or transitive relationships. See students’ work. 2 x1 2 2. 3x 5 90 36. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How is mathematical evidence similar to evidence in law? Include the following in your answer: • a description of how evidence is used to influence jurors’ conclusions in court, and • a description of the evidence used to make conclusions in mathematics. 37. In PQR, mP m Q and mR 2(mQ). Find mP if mP mQ + mR 180. B A C 30 B 60 D D and mCBD , then x 27. Given: mABC mCBD 90 mABC 3x 5 x1 m CBD 2 Prove: x 27 Proof: Statements x1 2 C B A Reasons 1. Given 2. Substitution Property 3. (2)(3x 5) (2) (2)90 3. Multiplication Property 4. 5. 6. 7. 10 x 1 180 9 180 9 9 180 9 189 4. 5. 6. 7. 8. 189 7 8. Division Property 9. x 27 9. Substitution Property 6x 7x 7x 7x 7x 7 Substitution Property Substitution Property Addition Property Substitution Property 2. FINANCE The formula for simple interest is I prt, where I is interest, p is principal, r is rate, and t is time. Solve the formula for r and justify each step. Given: I prt I Prove: r pt Proof: Statements 1. I prt prt I 2. pt pt I 3. r pt I 4. r pt Reasons 1. Given 2. Division Property 3. Substitution Property 4. Symmetric Property P Gl NAME ______________________________________________ DATE /M G Hill 90 ____________ Gl PERIOD G _____ Reading 2-6 Readingto to Learn Learn Mathematics Mathematics, p. 91 Algebraic Proof 45 Pre-Activity 90 R ELL How is mathematical evidence similar to evidence in law? Read the introduction to Lesson 2-6 at the top of page 94 in your textbook. Q What are some of the things that lawyers might use in presenting their closing arguments to a trial jury in addition to evidence gathered prior to the trial and testimony heard during the trial? Sample answer: They might tell the jury about laws related to the case, court rulings, and precedents set by earlier trials. Reading the Lesson 38. ALGEBRA If 4 more than x is 5 less than y, what is x in terms of y? B A y1 B y9 y9 C www.geometryonline.com/self_check_quiz D y5 Lesson 2-6 Algebraic Proof 99 1. Name the property illustrated by each statement. a. If a 4.75 and 4.75 b, then a b. Transitive Property of Equality b. If x y, then x 8 y 8. Addition Property of Equality c. 5(12 19) 5 12 5 19 Distributive Property Substitution Property d. If x 5, then x may be replaced with 5 in any equation or expression. of Equality e. If x y, then 8x 8y. Multiplication Property of Equality f. If x 23.45, then 23.45 x. Symmetric Property of Equality 7 g. If 5x 7, then x . Division Property of Equality 5 h. If x 12, then x 3 9. Subtraction Property of Equality NAME ______________________________________________ DATE 2-6 Enrichment Enrichment, ____________ PERIOD _____ p. 92 2. Give the reason for each statement in the following two-column proof. Given: 5(n 3) 4(2n 7) 14 Prove: n 9 Statements Reasons 1. 5(n 3) 4(2n 7) 14 1. Given 2. 5n 15 8n 28 14 2. Distributive Property 3. 5n 15 8n 42 3. Substitution Property Symmetric, Reflexive, and Transitive Properties 4. 5n 15 15 8n 42 15 4. Addition Property Equality has three important properties. 5. 5n 8n 27 Reflexive Symmetric Transitive aa If a b, then b a. If a b and b c, then a c. Other relations have some of the same properties. Consider the relation “is next to” for objects labeled X, Y, and Z. Which of the properties listed above are true for this relation? X is next to X. False If X is next to Y, then Y is next to X. True If X is next to Y and Y is next to Z, then X is next to Z. False Only the symmetric property is true for the relation “is next to.” For each relation, state which properties (symmetric, reflexive, transitive) are true. 5. Substitution Property 6. 5n 8n 8n 27 8n 6. Subtraction Property 7. 3n 27 7. Substitution Property 3n 3 27 3 8. 8. Division Property 9. n 9 9. Substitution Property Helping You Remember 3. A good way to remember mathematical terms is to relate them to words you already know. Give an everyday word that is related in meaning to the mathematical term reflexive and explain how this word can help you to remember the Reflexive Property and to distinguish it from the Symmetric and Transitive Properties. Sample answer: Reflection: If you look at your reflection, you see yourself. The Reflexive Property says that every number is equal to itself. The Reflexive Property involves only one number, while the Symmetric and Transitive Properties each involve two or three numbers. Lesson 2-6 Algebraic Proofs 99 Lesson 2-6 34. Sample answers are: Michael has a symmetric relationship of first cousin with Chris, Kevin, Diane, Dierdre, and Steven. Diane, Dierdre, and Steve have a symmetric and transitive relationship of sibling. Any direct line from bottom to top has a transitive descendent relationship. Ryan 4 Assess Maintain Your Skills Mixed Review Open-Ended Assessment Writing Select some statements to prove and write them on the board. Have different volunteers come up to the board and write one statement and reason to advance the proof until the students have proven the original statement. Getting Ready for Lesson 2-7 Prerequisite Skill Students will learn about proving segment relationships in Lesson 2-7. They will learn about segment addition and will use segment measures to prove segment congruence. Use Exercises 51–53 to determine your students’ familiarity with segment measures. (Lesson 2-5) 40. Valid; since 24 is divisible by 6, the Law of Detachment says it is divisible by 3. 41. Invalid; 27 6 4.5, which is not an integer. 42. Valid; since 85 is not divisible by 3, the contrapositive of the statement and the Law of Detachment say that 85 is not divisible by 6. Getting Ready for the Next Lesson Assessment Options Practice Quiz 2 The quiz provides students with a brief review of the concepts and skills in Lessons 2-4 through 2-6. Lesson numbers are given to the right of the exercises or instruction lines so students can review concepts not yet mastered. Quiz (Lessons 2-5 and 2-6) is available on p. 120 of the Chapter 2 Resource Masters. Answers 43. If people are happy, then they rarely correct their faults. 44. If you don’t know where you are going, then you will probably end up somewhere else. 45. If a person is a champion, then the person is afraid of losing. 46. If we would have new knowledge, then we must get a whole new world of questions. 100 Chapter 2 Reasoning and Proof 39. CONSTRUCTION There are four buildings on the Medfield High School Campus, no three of which stand in a straight line. How many sidewalks need to be built so that each building is directly connected to every other building? 6 Determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. A number is divisible by 3 if it is divisible by 6. (Lesson 2-4) 40. Given: 24 is divisible by 6. Conclusion: 24 is divisible by 3. 41. Given: 27 is divisible by 3. Conclusion: 27 is divisible by 6. 42. Given: 85 is not divisible by 3. Conclusion: 85 is not divisible by 6. Write each statement in if-then form. (Lesson 2-3) 43–46. See margin. 43. “Happy people rarely correct their faults.” (La Rochefoucauld) 44. “If you don’t know where you are going, you will probably end up somewhere else.” (Laurence Peters) 45. “A champion is afraid of losing.” (Billie Jean King) 46. “If we would have new knowledge, we must get a whole new world of questions.” (Susanne K. Langer) Find the precision for each measurement. (Lesson 1-2) 47. 13 feet 48. 5.9 meters 49. 74 inches 1 ft 0.05 m 0.5 in. 2 PREREQUISITE SKILL Find the measure of each segment. 50. 3.1 kilometers 0.05 km (To review segment measures, see Lesson 1-2.) 51. K L 11 52. Q S 28 25 51 K L P Q P ractice Quiz 2 9 38 23 14 J 53. W Z 47 W R S X Y Z Lessons 2-4 through 2-6 1. 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. (Lesson 2-4) (1) If n is an integer, then n is a real number. (2) n is a real number. (3) n is an integer. invalid In the figure at the right, A, B, and C are collinear. Points A, B, C, and D lie in plane N. State the postulate or theorem that can be used to show each statement is true. (Lesson 2-5) 2–4. See margin. 2. A, B, and D determine plane N. AC at B. 3. BE intersects 4. lies in plane N. 5. PROOF E A D If 2(n 3) 5 3(n 1), prove that n 2. (Lesson 2-6) See p. 123E. 100 Chapter 2 Reasoning and Proof Answers Practice Quiz 2 2. Through any three points not on the same line, there is exactly one plane. 3. If two lines intersect, then their intersection is exactly one point. 4. If two points lie in a plane, then the entire line containing those points lies in that plane. N B C Proving Segment Relationships • Write proofs involving segment addition. 1 Focus • Write proofs involving segment congruence. can segment relationships be used for travel? San Diego When leaving San Diego, the pilot said that the flight would be about 360 miles to Phoenix before continuing on to Dallas. When the plane left Phoenix, the pilot said that the flight would be flying about 1070 miles to Dallas. 5-Minute Check Transparency 2-7 Use as a quiz or review of Lesson 2-6. Phoenix Dallas Mathematical Background notes are available for this lesson on p. 60D. 1/2 inch = 400 mi. SEGMENT ADDITION In Lesson 1-2, you measured segments with a ruler by placing the mark for zero on one endpoint, then finding the distance to the other endpoint. This illustrates the Ruler Postulate. Postulate 2.8 Ruler Postulate The points on any line or line segment can be paired with real numbers so that, given any two points A and B on a line, A corresponds to zero, and B corresponds to a positive real number. A B 0 The Ruler Postulate can be used to further investigate line segments. Adding Segment Measures Construct a Figure Segment Measures • Use The Geometer’s Sketchpad to C . construct A C . • Place point B on A • Find AB, BC, and AC. AB = 1.79 cm BC = 3.21 cm AC = 5.00 cm A Analyze the Model Lesson Notes B can segment relationships be used for travel? Ask students: • Use a ruler to measure the length in millimeters from San Diego to Phoenix and from Phoenix to Dallas. about 11 mm; about 32 mm • Given the pilot’s information, how many miles will Janelle be flying from San Diego to Dallas? 1430 • How are segment lengths helpful for air travel? Sample answer: Pilots can use segment lengths to calculate distance, flight time and necessary fuel requirements. C 1. What is the sum AB BC? 2. Move B. Find AB, BC and AC. What is the sum of AB BC? 3. Repeat moving B, measuring the segments, and finding the sum AB BC three times. Record your results. 1–3. See students’ work. The sum AB BC Make a Conjecture should always equal AC. 4. What is true about the relationship of AB, BC, and AC ? AB BC AC 5. Is it possible to place B on A C so that this relationship is not true? no Lesson 2-7 Proving Segment Relationships 101 Resource Manager Workbook and Reproducible Masters Chapter 2 Resource Masters • Study Guide and Intervention, pp. 93–94 • Skills Practice, p. 95 • Practice, p. 96 • Reading to Learn Mathematics, p. 97 • Enrichment, p. 98 School-to-Career Masters, p. 4 Prerequisite Skills Workbook, pp. 89–90, 101–104 Teaching Geometry With Manipulatives Masters, p. 8 Transparencies 5-Minute Check Transparency 2-7 Answer Key Transparencies Technology Interactive Chalkboard Lesson x-x Lesson Title 101 Study Tip 2 Teach Betweenness SEGMENT ADDITION In-Class Example Power Point® 1 Prove the following. Use the figure from Example 1 in the Student Edition. Given: PR QS Prove: PQ RS Statements (Reasons) 1. PR QS (Given) 2. PR QR QS QR (Subtr. Prop.) 3. PR QR PQ; QS QR RS (Seg. Add. Post.) 4. PQ RS (Substitution) In general, the definition of between is that B is between A and C if A, B, and C are collinear and AB BC AC. Examine the measures AB, BC, and AC in the Geometry Activity. Notice that wherever B is placed between A and C, AB BC AC. This suggests the following postulate. Postulate 2.9 Segment Addition Postulate If B is between BC AB A and C, then AB BC AC. A B If AB BC AC, then B is between A and C. C AC Example 1 Proof With Segment Addition Prove the following. Given: PQ RS Prove: PR QS P Q R S Proof: Statements Reasons 1. PQ RS 2. PQ QR QR RS 3. PQ QR PR QR RS QS 4. PR QS 1. Given 2. Addition Property 3. Segment Addition Postulate 4. Substitution Teaching Tip Tell students that with each new lesson, they are accumulating more postulates and theorems that they can use for writing proofs. Encourage students to practice using these concepts as much as possible before moving on to the next lesson to strengthen their ability to recall important facts for proof-writing skills. SEGMENT CONGRUENCE In Lesson 2-5, you learned that once a theorem is proved, it can be used in proofs of other theorems. One theorem we can prove is similar to properties of equality from algebra. Segment Congruence Theorem 2.2 Congruence of segments is reflexive, symmetric, and transitive. Reflexive Property A AB B Symmetric Property If AB CD CD AB , then . Transitive Property If AB CD CD EF AB EF , and , then . You will prove the first two properties in Exercises 10 and 24. Proof Given: Transitive Property of Congruence N M N PQ P Q P RS Prove: RS M N M Proof: Method 1 Paragraph Proof Q R PQ and P Q RS , MN PQ and PQ RS by the Since MN definition of congruent segments. By the Transitive Property RS by the definition of of Equality, MN RS. Thus, MN congruent segments. 102 Chapter 2 Reasoning and Proof Geometry Software Investigation Adding Segment Measures Have students repeat the activity for different lengths of AC . Students can also construct AC vertically and at various diagonals. Tell students that this activity provides several examples to substantiate the Segment Addition Postulate. 102 Chapter 2 Reasoning and Proof S Two-Column Proof Method 2 Statements 1. 2. 3. 4. SEGMENT CONGRUENCE Reasons M N PQ , P Q R S MN PQ, PQ RS MN RS N RS M 1. 2. 3. 4. In-Class Example Given Definition of congruent segments Transitive Property Definition of congruent segments 2 Prove the following. 3 cm Y W Example 2 Proof With Segment Congruence J 1. 2. 3. 4. 5. Concept Check Guided Practice GUIDED PRACTICE KEY Exercises 6 4, 5, 7–10 Examples 1 2 K G Reasons 1. 2. 3. 4. 5. Given Transitive Property Given Transitive Property Symmetric Property 1. Sample answer: The distance from Cleveland to Chicago is the same as the distance from Cleveland to Chicago. 2. Sample answer: If AB XY and X Y PQ , then AB PQ . If A, B, and C are collinear and AB BC AC, then B is between A and C. Justify each statement with a property of equality or a property of congruence. Y XY Reflexive 4. X 5. If G H MN , then MN G H . Symmetric 6. If AB AC CB, then AB AC CB. Subtraction Prove: Y S T Q X RT P S Q P Statements Reasons a. ? , ? b. PQ RS, QS ST c. PS PQ QS, RT RS ST d. ? PQ QS RS ST e. ? PS RT RT S f. P a. Given b. ? Def. of segs. c. ? Segment Addition Post. d. Addition Property e. Substitution f. ? Def. of segs. www.geometryonline.com/extra_examples P B A R Proof: 7a. PQ RS , QS ST Answers 1. Choose two cities from a United States road map. Describe the distance between the cities using the Reflexive Property. See margin. 2. OPEN ENDED Draw three congruent segments, and illustrate the Transitive Property using these segments. See margin. 3. Describe how to determine whether a point B is between points A and C. 7. Copy and complete the proof. RS ST Q , Q S Given: P X Given: WY YZ Z XZ Y Z X WX Prove: WX WY Proof: Statements (Reasons) 1. WY YZ (Given) 2. WY Y Z (Def. of Segs.) 3. YZ XZ; X Z WX (Given) 4. WY WX (Trans. Prop.) 5. WX WY (Symmetric) L H JK KL , K L HJ HJ JK J GH H GH JK H JK G Z 3 cm The theorems about segment congruence can be used to prove segment relationships. Prove the following. KL , H J GH , KL HJ Given: JK Prove: G H JK Proof: Statements Power Point® Lesson 2-7 Proving Segment Relationships 103 C Squared Studios/PhotoDisc (t) file photo (b) Differentiated Instruction Visual/Spatial When students are first examining figures to determine the steps necessary for writing proofs, encourage them to use their spatial skills to locate obvious and hidden congruent segments and parts of segments that qualify for the Segment Addition Postulate. Advise students to use the given information to mark the figures so they can easily refer to the relationships in the figures while they are writing their proofs. Lesson 2-7 Proving Segment Relationships 103 For Exercises 8–10, write a two-column proof. 8–9. See p. 123E. 8. Given: AP CP 9. Given: H I TU P B DP J H TV Prove: A B CD Prove: IJ UV PROOF 3 Practice/Apply A D Study Notebook P Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 2. • include a sample proof using segment addition and one using segment congruence. • include any other item(s) that they find helpful in mastering the skills in this lesson. About the Exercises… Odd/Even Assignments Exercises 12–23 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 13–27 odd, 29–45 Average: 13–27 odd, 29–45 Advanced: 12–26 even, 27–39 (optional: 40–45) I B C T J U V 10. Symmetric Property of Congruence (Theorem 2.2) See margin. Application 11. GEOGRAPHY Aberdeen in South Dakota and Helena, Miles City, and Missoula, all in Montana, are connected in a straight line by interstate highways. Missoula is 499 miles from Miles City and 972 miles from Aberdeen. Aberdeen is 473 miles from Miles City and 860 miles from Helena. Between which cities does Helena lie? Helena is between Missoula and Miles City. Practice and Apply For Exercises See Examples 14, 16, 17 12, 13, 15, 18–24 1 2 Extra Practice Organization by Objective • Segment Addition: 14, 16, 17 • Segment Congruence: 12, 13, 15, 18–24 H See page 758. Justify each statement with a property of equality or a property of congruence. LM , then LM JK . Symmetric 12. If JK 13. If AB 14 and CD 14, then AB CD. Substitution 14. If W, X, and Y are collinear, in that order, then WY WX XY. Segment Addition 15. If M N PQ and P Q RS , then MN RS . Transitive 16. If EF TU and GH VW, then EF GH TU VW. Addition 17. If JK MN JK QR, then MN QR. Subtraction B 18. Copy and complete the proof. D CE , D B EB Given: A Prove: Proof: Statements 18d. AB AD DB, CB CE EB a. b. c. d. e. f. D A Reasons D CE , DB EB ? A AD CE, DB EB AD DB CE EB ? AB CB A B CB Write a two-column proof. 19 – 20. See p. 123E. PROOF 19. If X Y WZ and W Z AB , 20. If AB AC and P C QB , AB . then AP AQ . then XY W C B A P Y B Z A 104 Chapter 2 Reasoning and Proof C D Proof: Statements (Reasons) 1. AB CD (Given) 2. AB CD (Def. of segs.) 3. CD AB (Symmetric Prop.) 4. CD AB (Def. of segs.) 104 Q X 10. Given: A B CD Prove: C D AB A Chapter 2 Reasoning and Proof C a. Given b. ? Def. of segs. c. ? Add. Prop. d. Segment Addition Postulate e. ? Substitution f. ? Def. of segs. B Answer E AB CB 21. Copy and complete the proof. Y ZX Given: W A is the midpoint of WY . . A is the midpoint of ZX ZA Prove: WA Proof: Statements W NAME ______________________________________________ DATE p. 93 (shown) and p. 94 Proving Segment Relationships X A Y Reasons Two basic postulates for working with segments and lengths are the Ruler Postulate, which establishes number lines, and the Segment Addition Postulate, which describes what it means for one point to be between two other points. Ruler Postulate The points on any line or line segment can be paired with real numbers so that, given any two points A and B on a line, A corresponds to zero and B corresponds to a positive real number. Segment Addition Postulate B is between A and C if and only if AB BC AC. Example Write a two-column proof. Given: Q is the midpoint of P R . . R is the midpoint of QS Prove: PR QS Given ? a. Segment Addition b. ? Def. of segs. c. Definition of midpoint d. ? Segment Addition Post. e. ? Substitution f. ? Substitution g. ? Substitution h. Division Property i. ? Def. of segs. For Exercises 22–24, write a two-column proof. 22–24. See p. 123E. 22. If L M PN and X M XN , 23. If AB BC, PX . then AC 2BC. then LX S Statements Reasons 1. Given 2. Definition of midpoint 3. Given 4. Definition of midpoint 5. Addition Property 6. Segment Addition Postulate 7. Substitution Exercises Complete each proof. 1. Given: BC DE Prove: AB DE AC Statements a. BC DE A 2. Given: Q is between P and R, R is between Q and S, PR QS. Prove: PQ RS C B E D Reasons a. Given b. AB BC AC b. Seg. Add. Post. c. AB DE AC Gl c. Substitution Q Reasons a. Q is between P and R. b.PQ QR PR c. R is between Q and S. d. QR RS QS e. PR QS f. PQ QR QR RS g. PQ QR QR QR RS QR h. PQ RS a. Given NAME ______________________________________________ DATE /M G Hill 93 p. 95 and Practice, p. 96Relationships (shown) Proving Segment S R P Statements Skills Practice, 2-7 Practice (Average) P b. Seg. Add. Post. c. Given d. Seg. Add. Post. e. Given f. Substitution g. Subtraction Prop. h. Substitution ____________ Gl PERIOD G _____ Complete the following proof. L M A X B 1. Given: A B DE B is the midpoint of A C . E is the midpoint of D F . C EF Prove: B Proof: C N A B D C E F Statements Reasons DE a. A B a. Given B is the midpoint of A C . E is the midpoint of D F . 24. Reflexive Property of Congruence (Theorem 2.2) 25. DESIGN The front of a building has a DE and C is triangular window. If AB D , prove that A C CE . the midpoint of B b. AB DE b. Definition of segments c. AB BC c. Definition of Midpoint DE EF F d. AC AB BC d. Segment Addition Postulate DF DE EF e. AB BC DE EF See p. 123E. A B C D f. Substitution Property g. AB BC AB AB EF AB g. Subtraction Property h. BC EF h. Substitution Property EF i. B C i. Definition of segments 2. TRAVEL Refer to the figure. DeAnne knows that the Grayson distance from Grayson to Apex is the same as the distance G from Redding to Pine Bluff. Prove that the distance from Grayson to Redding is equal to the distance from Apex to Pine Bluff. E Given: G A RP Prove: G R AP Proof: Statements 26. LIGHTING The light fixture in Gerrard Hall of the University of North Carolina is shown at B EF and B C DE , prove that the right. If A DF C . See p. 123F. A B A E D e. Substitution Property f. AB BC AB EF 1. 2. 3. 4. 5. 6. F C Gl A G GA GA GR GR GR RP RP AR AR RP GA AR, AP AR RP AP AP www.geometryonline.com/self_check_quiz L M N NAME ______________________________________________ DATE /M G Hill 96 ____________ Gl PERIOD G _____ • Before leaving home, a passenger used a road atlas to determine that the distance between San Diego and Dallas is about 1350 miles. Why is the flying distance greater than that? Sample answer: Phoenix is not T on a straight line between San Diego and Dallas, so the stop added to the distance traveled. A nonstop flight would have been shorter. P Reading the Lesson O NAME ______________________________________________ DATE 2-7 Enrichment Enrichment, ____________ PERIOD _____ p. 98 1 2 L L L I O N E A N 8 A J 13 C Q O 16 P U L A N 4 R R 6 I E C 9 G D 12 O I 10 S E 11 G M A T I C R D E A P 17 C O E N B T P A Y 7 U P S E T R E P 14 L G 5 C M T V E V E E N E M R D T I 15 N R 1. If E is between Y and S, which of the following statements are always true? B, E A. YS ES YE B. YS ES YE C. YE ES D. YE ES YS E. SE EY SY F. E is the midpoint of Y S . 2. Give the reason for each statement in the following two-column proof. Given: C is the midpoint of B D . D is the midpoint of CE . Prove: BD CE Statements Reasons 1. C is the midpoint of B D . Geometry Crossword Puzzle O ELL How can segment relationships be used for travel? • What is the total distance that the plane will fly to get from San Diego to Dallas? 1430 mi S (t)David Madison/Getty Images, (b)Dan Sears C P Read the introduction to Lesson 2-7 at the top of page 101 in your textbook. Lesson 2-7 Proving Segment Relationships 105 3 Pine Bluff R Given Definition of segments Addition Property Segment Addition Postulate Substitution Property Definition of segments Mathematics, p. 97 Proving Segment Relationships RT 27. CRITICAL THINKING Given that L N , T QO , L Q NO , M P NO , S is the R midpoint of R T , M is the midpoint of L N , and P R , list three statements that is the midpoint of QO you could prove using the postulates, theorems, and definitions that you have learned. See p. 123F. Q A Reasons 1. 2. 3. 4. 5. 6. Reading 2-7 Readingto to Learn Learn Mathematics Pre-Activity Apex Redding L D N I U E A R T G I T N I C N A H N E T C U A N T T X A R T R N 18 A L N G L A B C D E 1. Given 2. BC CD 2. Definition of midpoint 3. D is the midpoint of CE . 3. Given 4. CD DE 4. Definition of midpoint 5. BC DE 5. Transitive Property of Equality 6. BC CD CD DE 7. BC CD BD CD DE CE 6. Addition Property of Equality 7. Segment Addition Postulate 8. BD CE 8. Substitution Property 9. B D CE 9. Def. of segments Helping You Remember 3. One way to keep the names of related postulates straight in your mind is to associate something in the name of the postulate with the content of the postulate. How can you use this idea to distinguish between the Ruler Postulate and the Segment Addition Postulate? Sample answer: There are two words in “Ruler Postulate” and three words in “Segment Addition Postulate.” The statement of the Ruler Postulate mentions two points, and the statement of the Segment Addition Postulate mentions three points. E A Lesson 2-7 Proving Segment Relationships 105 Lesson 2-7 Source: www.pella.com R R . 1. Q is the midpoint of P 2. PQ QR 3. R is the midpoint of Q S . 4. QR RS 5. PQ QR QR RS 6. PQ QR PR, QR RS QS 7. PR QS PROOF Windows come in many different shapes and sizes. Some commonly used shapes for windows are circles, rectangles, squares, triangles, pentagons, and octagons. Q P Lesson 2-7 Z a. W Y ZX A is the midpoint of W Y . X . A is the midpoint of Z b. WY ZX c. ? WA AY, ZA AX d. WY WA AY, ZX ZA AX e. WA AY ZA AX f. WA WA ZA ZA g. 2WA = 2ZA h. ? WA ZA A ZA i. W Design ____________ PERIOD _____ Study Guide andIntervention Intervention, 2-7 Study Guide and 28. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can segment relationships be used for travel? 4 Assess Include the following in your answer: • an explanation of how a passenger can use the distances the pilot announced to find the total distance from San Diego to Dallas, and • an explanation of why the Segment Addition Postulate may or may not be useful when traveling. Open-Ended Assessment Modeling Make a “Reasons/Statements” board with Velcro in positions where you could place given information, statements and reasons. Create three or four proofs using segment addition and segment congruence, and write the given information and each statement and reason on a separate rectangular piece of poster board (large enough to read from the back of the classroom). Affix Velcro to the back of the boards so they can be easily placed on the R/S board. Place the given information at the top of the R/S board. Have students select each statement, match it with its corresponding reason and place it in the correct order on the R/S board. Standardized Test Practice Answers 28. Sample answer: You can use segment addition to find the total distance between two destinations by adding the distances of various points in between. Answers should include the following. • A passenger can add the distance from San Diego to Phoenix and the distance from Phoenix to Dallas to find the distance from San Diego to Dallas. • The Segment Addition Postulate can be useful if you are traveling in a straight line. 106 Chapter 2 Reasoning and Proof A 1 2 B 1 C 2 D 2 1 3 14 4 1 12 4 A B 12 4 C D 1 2 30. GRID IN A refreshment stand sells a large tub of popcorn for twice the price of a box of popcorn. If 60 tubs were sold for a total of $150 and the total popcorn sales were $275, how many boxes of popcorn were sold? 100 Maintain Your Skills Mixed Review State the property that justifies each statement. (Lesson 2-6) 31. If mP mQ 110 and mR 110, then mP mQ mR. Substitution 32. If x(y z) a, then xy xz = a. Dist. Prop. 33. If n 17 39, then n 56. Add. Prop. 34. If cv md and md 15, then cv 15. Trans. Prop. Determine whether the following statements are always, sometimes, or never true. Explain. (Lesson 2-5) 35–38. See margin for explanations. 35. A midpoint divides a segment into two noncongruent segments. never 36. Three lines intersect at a single point. sometimes 37. The intersection of two planes forms a line. always 38. Three single points determine three lines. sometimes Getting Ready for Lesson 2-8 Prerequisite Skill In Lesson 2-8, students will apply properties of supplementary and complementary angles to prove angle relationships. Use Exercises 4045 to determine your students’ familiarity with complementary and supplementary angles. and Q is 29. If P is the midpoint of BC D , what is PQ? B the midpoint of A D 39. If the perimeter of rectangle ABCD is 44 centimeters, find x and the dimensions of the rectangle. (Lesson 1-6) A (x 6) cm 3; 9 cm by 13 cm C (2x 7) cm Getting Ready for the Next Lesson B PREREQUISITE SKILL Find x. (To review complementary and supplementary angles, see Lesson 1-5.) 30 40. 2x ° 43. 15 41. 42. x° 22 (3x 2)˚ 2x ° x° 4x ° 45 x° 45. 25 44. 5 3x ° 26x ° 10x ° 106 Chapter 2 Reasoning and Proof 35. The midpoint of a segment divides it into two congruent segments. 36. If the lines have a common intersection point, then it is a single point. 37. If two planes intersect, they intersect in a line. 38. If the points are noncollinear, then they lie on three distinct lines. (4x 10)˚ (3x 5)˚ 1/16/2003 1:06 PM T_Maria_Manko 107-114 GEO C2L8- Lesson Notes Proving Angle Relationships • Write proofs involving supplementary and complementary angles. 1 Focus • Write proofs involving congruent and right angles. do scissors illustrate supplementary angles? 5-Minute Check Transparency 2-8 Use as a quiz or review of Lesson 2-7. Notice that when a pair of scissors is opened, the angle formed by the two blades, 1, and the angle formed by a blade and a handle, 2, are a linear pair. Likewise, the angle formed by a blade and a handle, 2, and the angle formed by the two handles, 3, also forms a linear pair. 2 1 Mathematical Background notes are available for this lesson on p. 60D. 3 SUPPLEMENTARY AND COMPLEMENTARY ANGLES Recall that when you measure angles with a protractor, you position the protractor so that one of the rays aligns with zero degrees and then determine the position of the second ray. This illustrates the Protractor Postulate. Postulate 2.10 80 70 110 100 90 100 80 110 70 12 0 60 13 0 50 14 0 40 30 30 15 0 15 170 10 r° A 170 10 20 160 160 20 B In Lesson 2-7, you learned about the Segment Addition Postulate. A similar relationship exists between the measures of angles. A D B 60 0 12 40 14 0 50 0 13 0 and a number r Protractor Postulate Given AB between 0 and 180, there is exactly one ray with , such endpoint A, extending on either side of AB that the measure of the angle formed is r. do scissors illustrate supplementary angles? Ask students: • In the figure, label 4 vertical to 2 and name all pairs of supplementary angles. 1 and 2, 2 and 3, 3 and 4, 4 and 1 • Use a protractor to measure angles 1 and 2. What is the sum of these two measures? about 40°; about 140°; 180°. • Will the same angles still form linear pairs if the scissors were opened wider? narrower? yes; yes Postulate 2.11 C Angle Addition Postulate If R is in the interior of PQS, then mPQR mRQS mPQS. If mPQR mRQS mPQS, then R is in the interior of PQS. P R Q S History The Grand Union flag was the first flag used by the colonial United States that resembles the current flag. It was made up of thirteen stripes with the flag of Great Britain in the corner. Source: www.usflag.org Example 1 Angle Addition HISTORY The Grand Union Flag at the left contains several angles. If mABD 44 and mABC 88, find mDBC. mABD mDBC mABC Angle Addition Postulate 44 mDBC 88 mABD 44, mABC 88 mDBC 44 Subtraction Property Lesson 2-8 Proving Angle Relationships 107 (t)C Squared Studios, (b)file photo Resource Manager Workbook and Reproducible Masters Chapter 2 Resource Masters • Study Guide and Intervention, pp. 99–100 • Skills Practice, p. 101 • Practice, p. 102 • Reading to Learn Mathematics, p. 103 • Enrichment, p. 104 • Assessment, p. 120 Prerequisite Skills Workbook, pp. 81–82, 85–86 Teaching Geometry With Manipulatives Masters, pp. 8, 16, 48 Transparencies 5-Minute Check Transparency 2-8 Answer Key Transparencies Technology Interactive Chalkboard Lesson x-x Lesson Title 107 The Angle Addition Postulate can be used with other angle relationships to provide additional theorems relating to angles. 2 Teach SUPPLEMENTARY AND COMPLEMENTARY ANGLES In-Class Examples Power Point® Theorems Study Tip 2.3 Look Back To review supplementary and complementary angles, see Lesson 1-5. 1 TIME At 4 o’ clock, the angle a linear pair, then they are supplementary angles. 1 2 m1 m2 180 2.4 between the hour and minute hands of a clock is 120°. If the second hand stops where it bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands? They are both 60° by the definition of angle bisector and the Angle Addition Postulate. Supplement Theorem If two angles form Complement Theorem If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles. 1 2 m1 m2 90 You will prove Theorems 2.3 and 2.4 in Exercises 10 and 11. Example 2 Supplementary Angles If 1 and 2 form a linear pair and m 2 67, find m1. m1 m2 180 Supplement Theorem 1 m1 67 180 m2 67 2 m1 113 Subtraction Property 2 If 1 and 2 form a linear pair and m2 166, find m1. 14 CONGRUENT AND RIGHT ANGLES The properties of algebra that applied to the congruence of segments and the equality of their measures also hold true for the congruence of angles and the equality of their measures. Theorem 2.5 Congruence of angles is reflexive, symmetric, and transitive. Reflexive Property 1 1 Symmetric Property If 1 2, then 2 1. Transitive Property If 1 2, and 2 3, then 1 3. You will prove the Reflexive and Transitive Properties of Angle Congruence in Exercises 26 and 27. TEACHING TIP The Symmetric Property is often assumed in proofs to condense the number of steps in a proof. The rigor of proof is left up to the teacher, but we will assume symmetric property statements in future chapters. Proof Symmetric Property of Congruence Given: A B Prove: B A Paragraph Proof: A B We are given A B. By the definition of congruent angles, mA mB. Using the Symmetric Property, mB mA. Thus, B A by the definition of congruent angles. Algebraic properties can be applied to prove theorems for congruence relationships involving supplementary and complementary angles. 108 Chapter 2 Reasoning and Proof Differentiated Instruction Auditory/Musical Ask students to close their books. Read Theorems 2.3–2.13 aloud for students one by one. After each one, ask students to discuss how they know the theorem is true and how they might use the theorem in a proof. 108 Chapter 2 Reasoning and Proof CONGRUENT AND RIGHT ANGLES Theorems 2.6 Angles supplementary to the same angle or to congruent angles are congruent. Abbreviation: suppl. to same or are . Example: 2.7 3 1 form a linear pair, and m3 m1 180. Prove that 3 and 4 are congruent. 1 2 1 2 3 If m1 m2 90 and m2 m3 90, then 1 3. You will prove Theorem 2.6 in Exercise 6. Proof Theorem 2.7 Given: 1 and 3 are complementary. 2 and 3 are complementary. Prove: 1 2 Proof: Statements 1. 1 and 3 are complementary. 2 and 3 are complementary. 2. m1 m3 90 m2 m3 90 3. m1 m3 m2 m3 4. m3 m3 5. m1 m2 6. 1 2 Power Point® 3 In the figure, 1 and 4 If m1 m2 180 and m2 m3 180, then 1 3. Angles complementary to the same angle or to congruent angles are congruent. Abbreviation: compl. to same or are . Example: In-Class Example 2 2 1 3 Reasons 1. Given 4 3 Statements (Reasons) 1. m3 m1 180; 1 and 4 form a linear pair. (Given) 2. 1 and 4 are supplementary. (Linear pairs are suppl.) 3. 3 and 1 are supplementary. (Def. of suppl. ) 4. 3 4 ( supplementary to same are .) 2. Definition of complementary angles 3. 4. 5. 6. Substitution Reflective Property Subtraction Property Definition of congruent angles Example 3 Use Supplementary Angles In the figure, 1 and 2 form a linear pair and 2 and 3 form a linear pair. Prove 1 that 1 and 3 are congruent. 4 2 Given: 1 and 2 form a linear pair. 3 2 and 3 form a linear pair. Prove: 1 3 Proof: Statements Reasons 1. 1 and 2 form a linear pair. 1. Given 2 and 3 form a linear pair. 2. 1 and 2 are supplementary. 2. Supplement Theorem 2 and 3 are supplementary. 3. 1 3 3. suppl. to same or are . www.geometryonline.com/extra_examples Lesson 2-8 Proving Angle Relationships 109 Lesson 2-8 Proving Angle Relationships 109 In-Class Example Note that in Example 3, 1 and 3 are vertical angles. The conclusion in the example is a proof for the following theorem. Power Point® Teaching Tip Tell students to read problems carefully so they can be sure to provide the information requested. For this example, point out that students are to find angle measures, not just the value of the variable; however, they have to use the value of the variable to find the answer. 4 If 1 and 2 are vertical angles and m1 d 32 and m2 175 2d, find m1 and m2. 37; 37 Study Tip Theorem 2.8 Look Back Vertical Angles Theorem If two angles are To review vertical angles, see Lesson 1-5. vertical angles, then they are congruent. Abbreviation: Vert. are . 2 1 3 4 1 3 and 2 4 Example 4 Vertical Angles If 1 and 2 are vertical angles and m1 x and m2 228 3x, find m1 and m2. Vertical Angles Theorem 1 2 m1 m2 x 228 3x 4x 228 x 57 m1 x Definition of congruent angles Substitution Add 3x to each side. Divide each side by 4. m2 m1 57 57 The theorems you have learned can be applied to right angles. You can create right angles and investigate congruent angles by paper folding. Right Angles Make a Model • Fold the paper so that one corner is folded downward. • Fold along the crease so that the top edge meets the side edge. • Unfold the paper and measure each of the angles formed. • Repeat the activity three more times. Analyze the Model 1. The lines are perpendicular. 1. What do you notice about the lines formed? 2. What do you notice about each pair of adjacent angles? They are congruent and they form linear pairs. 3. What are the measures of the angles formed? 90 Make a Conjecture 4. They form right angles. 4. What is true about perpendicular lines? 5. What is true about all right angles? They all measure 90 and are congruent. The following theorems support the conjectures you made in the Geometry Activity. 110 Chapter 2 Reasoning and Proof Geometry Activity Materials: paper, protractor • When students are repeating the activity, tell them to use different folds from the right and the left sides of the paper each time. • Ask students what they notice about each pair of vertical angles (they are congruent and form right angles). 110 Chapter 2 Reasoning and Proof Theorems Right Angles 2.9 Perpendicular lines intersect to form four right angles. 2.10 All right angles are congruent. 2.11 Perpendicular lines form congruent adjacent angles. 2.12 If two angles are congruent and supplementary, then each angle is a right angle. 2.13 Concept Check If two congruent angles form a linear pair, then they are right angles. 1. FIND THE ERROR Tomas and Jacob wrote equations involving the angle measures shown. Tomas m∆ABE + m∆EBC = m∆ABC E F C Jacob m∆ABE + m∆FBC = m∆ABC B A Who is correct? Explain your reasoning. 1–2. See margin. 2. OPEN ENDED Draw three congruent angles. Use these angles to illustrate the Transitive Property for angle congruence. Guided Practice GUIDED PRACTICE KEY Exercises Examples 4 5 6 7–11 1 2 3 4 Find the measure of each numbered angle. 3. m1 65 4. 6 and 8 are complementary. m8 47 1 5. m11 x 4, m12 2x 5 2 6 7 11 12 8 m11 59, m12 121 m2 65 m6 43, m7 90 6. PROOF Given: Copy and complete the proof of Theorem 2.6. 1 and 2 are supplementary. 3 and 4 are supplementary. 1 1 4 2 3 Prove: Proof: Statements a. 1 and 2 are supplementary. 3 and 4 are supplementary. 1 4 b. m1 m2 180 m3 m4 180 c. m1 m2 m3 m4 d. m1 m4 e. m2 m3 f. 2 3 2 3 4 Reasons Given a. ? b. ? Def. of suppl. c. d. e. f. ? ? ? ? Substitution Def. of Subtr. Prop. Def. of Lesson 2-8 Proving Angle Relationships 111 Answers 1. Tomas; Jacob’s answer left out the part of ABC represented by EBF. 2. Sample answer: If 1 2 and 2 3, then 1 3. 3 Practice/Apply Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 2. • include an example each of a proof involving supplementary, complementary, congruent, and right angles. • include any other item(s) that they find helpful in mastering the skills in this lesson. FIND THE ERROR Explain that when two angle measures are added using the Angle Addition Postulate, they must share a common ray. Students can note that the common ray in Tomas’s answer , and a combination of these is BE letters appears in both angles that are being added (ABE and EBC). About the Exercises… Organization by Objective • Supplementary and Complementary Angles: 16–18 • Congruent and Right Angles: 19–39 Odd/Even Assignments Exercises 20–36 and 42–43 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide 1 2 3 Basic: 17–41 odd, 42–55 Average: 17–41 odd, 42–55 Advanced: 16–42 even, 44–55 Lesson 2-8 Proving Angle Relationships 111 Write a two-column proof. See margin. VX bisects WVY. bisects XVZ. VY 7. PROOF Given: Answers bisects WVY, 7. Given: VX VY bisects XVZ. Prove: WVX YVZ W X V WVX YVZ Prove: Y Z W Determine whether the following statements are always, sometimes, or never true. 8. Two angles that are nonadjacent are ? vertical. sometimes 9. Two angles that are congruent are ? complementary to the same angle. X V Y sometimes Z Proof: Statements (Reasons) bisects WVY; VY bisects 1.VX XVZ. (Given) 2.WVX XVY (Def. of bisector) 3.XVY YVZ (Def. of bisector) 4.WVX YVZ (Tran. Prop.) 10. Given: Two angles form a linear pair. Prove: The angles are supplementary 1 2 Paragraph Proof: When two angles form a linear pair, the resulting angle is a straight angle whose measure is 180. By definition, two angles are supplementary if the sum of their measures is 180. By the Angle Addition Postulate, m1 m2 180. Thus, if two angles form a linear pair, then the angles are supplementary. 11. Given: ABC is a right angle. Prove: 1 and 2 are complementary angles. Application ALGEBRA For Exercises 12–15, use the following information. 1 and X are complementary, 2 and X are complementary, m1 2n 2, and m2 n 32. 12. Find n. 30 14. What is m2? 62 1 2 X 13. Find m1. 62 15. Find mX. 28 Practice and Apply For Exercises See Examples 16–18 19–24 25–39 1, 2 4 3 Find the measure of each numbered angle. 16. m2 = 67 m1 113 17. m3 = 38 m4 52 18. 7 and 8 are complementary. 5 8 and m6 = 29. 1 3 2 4 Extra Practice 6 1 5 18. m5 61, m7 29, m8 61 19. m9 2x 4, m10 2x 4 20. m11 4x, m12 2x 6 10 9 11 22. m15 x, m16 6x 290 15 16 m15 58, m16 58 Proof: Statements (Reasons) 1.ABC is a right angle. (Given) 2.mABC 90 (Def. of rt. ) 3.mABC m1 m2 ( Add. Post.) 4.90 m1 m2 (Subst.) 5.1 and 2 are complementary angles. (Def. of comp. ) Chapter 2 Reasoning and Proof 112 Chapter 2 Reasoning and Proof 8 21. m13 2x 94, m14 7x 49 13 12 14 m9 86, m10 94 2 A 7 See page 758. m11 124, m12 56 23. m17 2x 7, m18 x 30 m13 112, m14 112 24. m19 100 20x, m20 20x m19 140, m20 40 C B 112 Write a proof for each theorem. 10–11. See margin. 10. Supplement Theorem 11. Complement Theorem PROOF 17 18 m17 53, m18 53 20 19 25. Prove that congruence of angles is reflexive. See p. 123F. 26. Write a proof of the Transitive Property of Angle Congruence. See p. 123F. NAME ______________________________________________ DATE ____________ PERIOD _____ Study Guide andIntervention Intervention, 2-8 Study Guide and p. 99 (shown) and p. 100 Proving Angle Relationships Supplementary and Complementary Angles There are two basic postulates for working with angles. The Protractor Postulate assigns numbers to angle measures, and the Angle Addition Postulate relates parts of an angle to the whole angle. Protractor Postulate and a number r between 0 and 180, there is exactly one ray Given AB , such that the measure with endpoint A, extending on either side of AB of the angle formed is r. Angle Addition Postulate R is in the interior of PQS if and only if mPQR mRQS mPQS. R Supplement Theorem If two angles form a linear pair, then they are supplementary angles. If 1 and 2 form a linear pair, then m1 m2 180. Complement Theorem Example 1 If 1 and 2 form a linear pair and m2 115, find m1. D R A Interstate highways that run from north to south are odd-numbered with the lowest numbers in the west. East-west interstates are even-numbered, and begin in the south. B W 1 S T m1 m2 90 m1 20 90 m1 70 Suppl. Theorem Substitution Subtraction Prop. Compl. Theorem Substitution Subtraction Prop. 1. 2. T P 7 Y X 8 Q 8 R U S m7 5x 5, m8 x 5 m7 155, m8 25 7 6 Z 3. A 13 5 V F W m5 5x, m6 4x 6, m7 10x, m8 12x 12 H J C m11 110, m12 110, m13 70 m5 30, m6 30, m7 60, m8 60 NAME ______________________________________________ DATE /M G Hill 99 Skills Practice, 2-8 Practice (Average) ____________ Gl PERIOD G _____ p. 101 and Practice, p. Relationships 102 (shown) Proving Angle Find the measure of each numbered angle. 1. m1 x 10 m2 3x 18 2. m4 2x 5 m5 4x 13 3. m6 7x 24 m7 5x 14 6 3 5 1 T U 7 2 m1 48, m2 132 W S 12 m11 11x, m12 10x 10 4 C 11 m3 90, m4 31, m5 59 m6 109, m7 109 Determine whether the following statements are always, sometimes, or never true. 4. Two angles that are supplementary are complementary. never 5. Complementary angles are congruent. sometimes 40. RIVERS Tributaries of rivers sometimes form a linear pair of angles when they meet the main river. The Yellowstone River forms the linear pair 1 and 2 with the Missouri River. If m1 is 28, find m2. 152 1 6. Write a two-column proof. Given: 1 and 2 form a linear pair. 2 and 3 are supplementary. Prove: 1 3 2 Proof: Statements 1. 1 and 2 form a linear pair. 2 and 3 are supplementary. 2. 1 and 2 are supplementary. 3. 1 3 Source: www.infoplease.com 41. HIGHWAYS Near the city of Hopewell, Virginia, Route 10 runs perpendicular to Interstate 95 and Interstate 295. Show that the angles at the intersections of Route 10 with Interstate 95 and Interstate 295 are congruent. 95 1 2 2. Supplement Theorem 3. suppl. to the same or are . Barton Rd Tryon St Olive Tree Lane Route 10 NAME ______________________________________________ DATE /M G Hill 102 Gl ____________ Gl PERIOD G _____ Reading 2-8 Readingto to Learn Learn Mathematics Mathematics, p. 103 Proving Angle Relationships Pre-Activity ELL How do scissors illustrate supplementary angles? Read the introduction to Lesson 2-8 at the top of page 107 in your textbook. Is it possible to open a pair of scissors so that the angles formed by the two blades, a blade and a handle, and the two handles, are all congruent? If so, explain how this could happen. Sample answer: Yes; open the scissors so that the two blades are perpendicular. Then all the angles will be right angles and will be congruent. Reading the Lesson 1. Complete each sentence to form a statement that is always true. 2 3 1 a. If two angles form a linear pair, then they are adjacent and supplementary . 4 b. If two angles are complementary to the same angle, then they are Lesson 2-8 Proving Angle Relationships 113 (l)Richard Pasley/Stock Boston, (r)Sam Abell/National Geographic Image Collection NAME ______________________________________________ DATE 2-8 Enrichment Enrichment, ____________ PERIOD _____ . . e. If two angles are congruent and supplementary, then each angle is a(n) right angle. Perpendicular lines form congruent adjacent angles. g. “Every angle is congruent to itself” is a statement of the of angle congruence. Reflexive Property h. If two congruent angles form a linear pair, then the measure of each angle is p. 104 90 . i. If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary . 2. Determine whether each statement is always, sometimes, or never true. a. Supplementary angles are congruent. sometimes b. If two angles form a linear pair, they are complementary. never c. Two vertical angles are supplementary. sometimes d. Two adjacent angles form a linear pair. sometimes e. Two vertical angles form a linear pair. never f. Complementary angles are congruent. sometimes g. Two angles that are congruent to the same angle are congruent to each other. always h. Complementary angles are adjacent angles. sometimes Bisecting a Hidden Angle The vertex of BAD at the right is hidden in a region. Within the region, you are not allowed to use a compass. Can you bisect the angle? C B P Q Helping You Remember A D E Follow these instructions to bisect BAD. congruent mDBC c. If D is a point in the interior of ABC, then mABC mABD and a number x between 0 and 180 , there is exactly one ray d. Given RS with endpoint R, extended on either side of RS, such that the measure of the angle formed is x. f. 42. m1 m4 90; m1 m2 m3 m4 180 m1 m1 m4 m4 180 2(m1) 2(m4) 180 2(m1 m4) 180 m1 m4 90 Reasons 1. Given 7. STREETS Refer to the figure. Barton Road and Olive Tree Lane form a right angle at their intersection. Tryon Street forms a 57° angle with Olive Tree Lane. What is the measure of the acute angle Tryon Street forms with Barton Road? 33 42. CRITICAL THINKING What conclusion can you make about the sum of m1 and m4 if m1 m2 and m3 m4? Explain. See margin. Answer 3 295 Because the lines are perpendicular, the angles formed are right angles. All right angles are congruent. Therefore, 1 is congruent to 2. www.geometryonline.com/self_check_quiz 1 2 Lesson 2-8 Highways H Example 2 If 1 and 2 form a right angle and m2 20, find m1. P Y Z 4 G Exercises Gl R C J 3 Find the measure of each numbered angle. Write a two-column proof. 38–39. See p. 123G. 38. Given: ABD YXZ 39. Given: mRSW mTSU Prove: CBD WXZ Prove: mRST mWSU X B F 2 N m1 m2 180 m1 115 180 m1 65 PROOF W 2 1 If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles. ⊥ GH , then m3 m4 90. If GF 2 1 33. 34. 35. 36. 37. D A M Use the figure to write a proof of each theorem. 33–37. See p. 123F. Theorem 2.9 Theorem 2.10 m 1 2 Theorem 2.11 3 4 Theorem 2.12 Theorem 2.13 S The two postulates can be used to prove the following two theorems. Q PROOF P Q Lesson 2-8 Determine whether the following statements are always, sometimes, or never true. 27. Two angles that are complementary ? form a right angle. sometimes 28. Two angles that are vertical are ? nonadjacent. always 29. Two angles that form a right angle are ? complementary. always 30. Two angles that form a linear pair are ? congruent. sometimes 31. Two angles that are supplementary are ? congruent. sometimes 32. Two angles that form a linear pair are ? supplementary. always 3. A good way to remember something is to explain it to someone else. Suppose that a classmate thinks that two angles can only be vertical angles if one angle lies above the other. How can you explain to him the meaning of vertical angles, using the word vertex in your explanation? Sample answer: Two angles are vertical angles if they share the same vertex and their sides are opposite rays. It doesn’t matter how the angles are positioned. 1. Use a straightedge to draw lines CE and BD. Lesson 2-8 Proving Angle Relationships 113 43. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How do scissors illustrate supplementary angles? 4 Assess Include the following in your answer: • a description of the relationship among 1, 2, and 3, • an example of another way that you can tell the relationship between 1 and 3, and • an explanation of whether this relationship is the same for two angles complementary to the same angle. Open-Ended Assessment Writing Give students a list of theorems from this chapter. Have students choose a theorem and write a proof of it with their books closed. Assessment Options Standardized Test Practice Quiz (Lessons 2-7 and 2-8) is available on p. 120 of the Chapter 2 Resource Masters. 44. The measures of two complementary angles are in the ratio 4:1. What is the measure of the smaller angle? B A 15 B 18 C 24 D 36 45. ALGEBRA T is the set of all positive numbers n such that n 50 and n is an integer. What is the median of the members of set T? B A 4 B 16 C 20 D 25 Answers 43. Two angles that are supplementary to the same angle are congruent. Answers should include the following. • 1 and 2 are supplementary; 2 and 3 are supplementary. • 1 and 3 are vertical angles, and are therefore congruent. • If two angles are complementary to the same angle, then the angles are congruent. 46. Given: G is between F and H. H is between F and J. Prove: FG GJ FH HJ F G H X Y Mixed Review Write a two-column proof. (Lesson 2-7) 46–47. See margin. PROOF 46. Given: G is between F and H. J H is between G and J. H F G Prove: FG GJ FH HJ 47. Given: X is the midpoint of W Y . Prove: W WX YZ XZ X 48. PHOTOGRAPHY Film is fed through a camera by gears that catch the perforation in the film. The distance from the left edge of the film, A, to the right edge of the image, C, is the same as the distance from the left edge of the image, B, to the right edge of the film, D. Show that the two perforated strips are the same width. (Lesson 2-6) See p. 123G. Y Z A B C D J Proof: Statements (Reasons) 1.G is between F and H; H is between F and J. (Given) 2.FG GJ FJ, FH HJ FJ (Seg. Add. Post.) 3.FJ FH HJ (Sym. Prop.) 4.FG GJ FH HJ (Transitive Prop.) 47. Given: X is the midpoint of WY . Prove: WX YZ XZ W Maintain Your Skills Z Proof: Statements (Reasons) 1.X is the midpoint of WY . (Given) 2.WX XY (Def. of midpoint) 3.XY YZ XZ (Segment Add. Post.) 4.WX YZ XZ (Substitution) 114 Chapter 2 Reasoning and Proof For Exercises 49–55, refer to the figure at the right. I (Lesson 1-4) 50. PMQ QMN 52. POQ, QON, NOM, MOP P O Q Name two angles that have N as a vertex. ONM, MNR N M R Q bisects PMN, name two congruent angles. If M Name a point in the interior of LMQ. N or R L List all the angles that have O as the vertex. Does QML appear to be acute, obtuse, right, or straight? obtuse and NP 54. Name a pair of opposite rays. Sample answer: NR 55. List all the angles that have M N as a side. NML, NMP, NMO, RNM, ONM 49. 50. 51. 52. 53. 114 Chapter 2 Reasoning and Proof Answers (page 115) 9. 45 X N L O 135 A 10. 11. M Y B Z Study Guide and Review Vocabulary and Concept Check axiom (p. 89) biconditional (p. 81) compound statement (p. 67) conclusion (p. 75) conditional statement (p. 75) conjecture (p. 62) conjunction (p. 68) contrapositive (p. 77) converse (p. 77) counterexample (p. 63) deductive argument (p. 94) deductive reasoning (p. 82) disjunction (p. 68) formal proof (p. 95) hypothesis (p. 75) if-then statement (p. 75) inductive reasoning (p. 62) informal proof (p. 90) inverse (p. 77) Law of Detachment (p. 82) Law of Syllogism (p. 83) logically equivalent (p. 77) negation (p. 67) paragraph proof (p. 90) Vocabulary and Concept Check postulate (p. 89) proof (p. 90) related conditionals (p. 77) statement (p. 67) theorem (p. 90) truth table (p. 70) truth value (p. 67) two-column proof (p. 95) • This alphabetical list of vocabulary terms in Chapter 2 includes a page reference where each term was introduced. • Assessment A vocabulary test/review for Chapter 2 is available on p. 118 of the Chapter 2 Resource Masters. A complete list of postulates and theorems can be found on pages R1–R8. Exercises Choose the correct term to complete each sentence. 1. A (counterexample, conjecture ) is an educated guess based on known information. 2. The truth or falsity of a statement is called its (conclusion, truth value ). 3. Two or more statements can be joined to form a (conditional, compound ) statement. Lesson-by-Lesson Review 4. A conjunction is a compound statement formed by joining two or more statements using (or, and ). 5. The phrase immediately following the word if in a conditional statement is called the ( hypothesis , conclusion). For each lesson, • the main ideas are summarized, • additional examples review concepts, and • practice exercises are provided. 6. The (inverse, converse ) is formed by exchanging the hypothesis and the conclusion. 7. (Theorems, Postulates ) are accepted as true without proof. 8. A paragraph proof is a (an) ( informal proof , formal proof ). Vocabulary PuzzleMaker 2-1 Inductive Reasoning and Conjecture See pages 62–66. Example Concept Summary • Conjectures are based on observations and patterns. • Counterexamples can be used to show that a conjecture is false. Given that points P, Q, and R are collinear, determine whether the conjecture that Q is between P and R is true or false. If the conjecture is false, give a counterexample. In the figure, R is between P and Q. Since we can find P R a counterexample, the conjecture is false. Q Exercises Make a conjecture based on the given information. Draw a figure to illustrate your conjecture. See Example 2 on page 63. 9–11. See margin for figures. 9. A and B are supplementary. mA mB 180 10. X, Y, and Z are collinear and XY YZ. Y is the midpoint of XZ. 11. In quadrilateral LMNO, LM LO MN NO, and m L 90. LMNO is a square. www.geometryonline.com/vocabulary_review Chapter 2 Study Guide and Review 115 TM For more information about Foldables, see Teaching Mathematics with Foldables. Have students look through the chapter to make sure they have included notes and examples in their Foldables for each lesson of Chapter 2. Encourage students to refer to their Foldables while completing the Study Guide and Review and to use them in preparing for the Chapter Test. ELL The Vocabulary PuzzleMaker software improves students’ mathematics vocabulary using four puzzle formats— crossword, scramble, word search using a word list, and word search using clues. Students can work on a computer screen or from a printed handout. MindJogger Videoquizzes ELL MindJogger Videoquizzes provide an alternative review of concepts presented in this chapter. Students work in teams in a game show format to gain points for correct answers. The questions are presented in three rounds. Round 1 Concepts (5 questions) Round 2 Skills (4 questions) Round 3 Problem Solving (4 questions) Chapter 2 Study Guide and Review 115 1/16/2003 1:48 PM Study Guide and Review Answers 12. 1 0 and in a right triangle with right angle C, a 2 b 2 c 2. 13. In a right triangle with right angle C, a 2 b 2 c 2 or the sum of the measures of two supplementary angles is 180. 14. The sum of the measures of two supplementary angles is 180 and 1 0. 15. 1 0, and in a right triangle with right angle C, a2 b2 c2, or the sum of the measures of two supplementary angles is 180. 16. In a right triangle with right angle C, a2 b2 c2, or 1 0 or the sum of the measures of two supplementary angles is 180. 17. In a right triangle with right angle C, a2 b2 c2 and the sum of the measures of two supplementary angles is 180, and 1 0. 18. Converse: If an angle is obtuse, then it measures 120. False; the measure could be any value between 90 and 180. Inverse: If an angle measure does not equal 120, then it is not obtuse. False; the measure could be any value other than 120 between 90 and 180. Contrapositive: If an angle is not obtuse, then its measure does not equal 120; true. 19. Converse: If a month has 31 days, then it is March. False; July has 31 days. Inverse: If a month is not March, then it does not have 31 days. False; July has 31 days. Contrapositive: If a month does not have 31 days, then it is not March; true. 20. Converse: If a point lies on the y-axis, then its ordered pair has 0 for its x-coordinate; true. Inverse: If an ordered pair does not have 0 for its x-coordinate, then the point does not lie on the y-axis; true. Contrapositive: If a point does not lie on the y-axis, then its ordered pair does not have 0 for its x-coordinate; true. 116 Chapter 2 Reasoning and Proof T_Maria_Manko 115-121 GEO C2SGT- Chapter 2 X Study Guide and Review • Extra Practice, see pages xxx-xxx. • Mixed Problem Solving, see page xxx. 2-2 Logic See pages 67–74. Example Concept Summary • The negation of a statement has the opposite truth value of the original statement. • Venn diagrams and truth tables can be used to determine the truth values of statements. Use the following statements to write a compound statement for each conjunction. Then find its truth value. p: 15 5 q: The measure of a right angle equals 90. a. p and q 15 5, and the measure of a right angle equals 90. p and q is false because p is false and q is true. b. p q 15 5, or the measure of a right angle equals 90. p q is true because q is true. It does not matter that p is false. Exercises Use the following statements to write a compound statement for each conjunction. Then find its truth value. See Examples 1 and 2 on pages 68 and 69. p: 1 0 q: In a right triangle with right angle C, a2 b2 c 2. r: The sum of the measures of two supplementary angles is 180. 13. q or r true 14. r p false 12. p and q false 15. p (q r) false 16. q (p r) true 17. (q r) p false 12 – 17. See margin for statements. 2-3 Conditional Statements See pages 75–80. Example Concept Summary • Conditional statements are written in if-then form. • Form the converse, inverse, and contrapositive of an if-then statement by using negations and by exchanging the hypothesis and conclusion. Identify the hypothesis and conclusion of the statement The intersection of two planes is a line. Then write the statement in if-then form. Hypothesis: two planes intersect Conclusion: their intersection is a line If two planes intersect, then their intersection is a line. Exercises Write the converse, inverse, and contrapositive of each conditional statement. Determine whether each related conditional is true or false. If a statement is false, find a counterexample. See Example 4 on page 77. 18–20. See margin. 18. If an angle measure equals 120, then the angle is obtuse. 19. If the month is March, then it has 31 days. 20. If an ordered pair for a point has 0 for its x-coordinate, then the point lies on the y-axis. 116 Chapter 2 Reasoning and Proof Chapter 2 Study Guide and Review Study Guide and Review Determine the truth value of the following statement for each set of conditions. If the temperature is at most 0°C, then water freezes. See Example 3 on page 76. 21. The temperature is 10°C, and water freezes. true 22. The temperature is 15°C, and water freezes. true 23. The temperature is 2°C, and water does not freeze. false 24. The temperature is 30°C, and water does not freeze. true 2-4 Deductive Reasoning See pages 82–87. Example Concept Summary • The Law of Detachment and the Law of Syllogism can be used to determine the truth value of a compound statement. Use the Law of Syllogism to determine whether a valid conclusion can be reached from the following statements. (1) If a body in our solar system is the Sun, then it is a star. (2) Stars are in constant motion. p: a body in our solar system is the sun q: it is a star r: stars are in constant motion Statement (1): p → q Statement (2): q → r Since the given statements are true, use the Law of Syllogism to conclude p → r. That is, If a body in our solar system is the Sun, then it is in constant motion. Exercises Determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. See Example 1 on page 82. If two angles are adjacent, then they have a common vertex. 25. Given: 1 and 2 are adjacent angles. Valid; by definition, adjacent angles Conclusion: 1 and 2 have a common vertex. have a common vertex. 26. Given: 3 and 4 have a common vertex. Invalid; vertical angles also have Conclusion: 3 and 4 are adjacent angles. a common vertex. 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 follow, write invalid. See Example 3 on page 83. 27. (1) If a student attends North High School, then the student has an ID number. (2) Josh Michael attends North High School. (3) Josh Michael has an ID number. yes; Law of Detachment 28. (1) If a rectangle has four congruent sides, then it is a square. (2) A square has diagonals that are perpendicular. (3) A rectangle has diagonals that are perpendicular. invalid 29. (1) If you like pizza with everything, then you’ll like Cardo’s Pizza. yes; Law (2) If you like Cardo’s Pizza, then you are a pizza connoisseur. of Syllogism (3) If you like pizza with everything, then you are a pizza connoisseur. Chapter 2 Study Guide and Review 117 Chapter 2 Study Guide and Review 117 Study Guide and Review Chapter 2 Study Guide and Review 2-5 Postulates and Paragraph Proofs Answers 30. Never; the intersection of two lines is a point. 31. Always; if P is the midpoint of X Y , then X P PY . By definition of congruent segments, XP PY. 32. Sometimes; if M, X, and Y are collinear. 33. Sometimes; if the points are collinear. 34. Always; there is exactly one line through Q and R. The line lies in at least one plane. 35. Sometimes; if the right angles form a linear pair. 36. Always; the Reflexive Property states that 1 1. 37. Never; adjacent angles must share a common side, and vertical angles do not. 38. If M is the midpoint of AB , then See pages 89–93. Example Concept Summary • Use undefined terms, definitions, postulates, and theorems to prove that statements and conjectures are true. Determine whether the following statement is always, sometimes, or never true. Explain. Two points determine a line. According to a postulate relating to points and lines, two points determine a line. Thus, the statement is always true. Exercises Determine whether the following statements are always, sometimes, or never true. Explain. See Example 2 on page 90. 30–37. See margin. 30. The intersection of two lines can be a line. 31. If P is the midpoint of X Y , then XP PY. 32. If MX MY, then M is the midpoint of XY. 33. Three points determine a line. 34. Points Q and R lie in at least one plane. 35. If two angles are right angles, they are adjacent. 36. An angle is congruent to itself. 37. Vertical angles are adjacent. 38. PROOF Write a paragraph proof to prove that A Q if M is the midpoint of A B and Q is the midpoint 1 4 of A M , then AQ AB. See margin. 1 AM (AB). Since Q is the 2 1 midpoint of AM , AQ AM or 1 1 1 (AB) AB. 2 2 4 2 2-6 Algebraic Proof See pages 94–100. Example Concept Summary • The properties of equality used in algebra can be applied to the measures of segments and angles to verify and prove statements. 5 Given: 2x 6 3 x 3 Prove: x 9 Proof: Statements 5 1. 2x 6 3 x 3 1. Given 5 2. 3(2x 6) 3 3 x 3 3. 4. 5. 6. 7. 6x 18 9 5x 6x 18 5x 9 5x 5x x 18 9 x 18 18 9 18 x9 118 Chapter 2 Reasoning and Proof 118 Chapter 2 Reasoning and Proof Reasons 2. Multiplication Property 3. 4. 5. 6. 7. Distributive Property Subtraction Property Substitution Subtraction Property Substitution M B Study Guide and Review Chapter 2 Study Guide and Review Exercises State the property that justifies each statement. See Example 1 on page 94. 39. If 3(x 2) 6, then 3x 6 6. Dist. Prop. Answers 1 43. Given: 5 2 x 2 Prove: x 6 Proof: Statements (Reasons) 40. If 10x 20, then x 2. Div. Prop. 41. If AB 20 45, then AB 25. Subt. Prop. 42. If 3 CD and CD XY, then 3 XY. Transitive Prop. PROOF 1 2 Write a two-column proof. See Examples 2 and 4 on pages 95 and 96. 1.5 2 x (Given) 1 43. If 5 2 x, then x 6. 2 x 10 44. If x 1 , then x 4. 2 45. If AC = AB, AC = 4x 1, and AB 6x 13, then x 7. 2 46. If MN PQ and PQ RS, then MN RS. P x 10 Q Proof: Statements 1. 2. 3. 4. 5. x 10 2 1.x 1 (Given) T S R 1. 2. 3. 4. 5. Given Addition Property Substitution Segment Addition Postulate Substitution 3x 3 9.x 4 (Substitution) 45. Given: AC AB, AC 4x 1, AB 6x 13 Prove: x 7 A 1 2 51. If 2(XY) AB, then XY (AB). Div. or Mult. Prop. 6x 13 4x 1 52. If AB CD, then AB BC CD BC. Add. Prop. Chapter 2 Study Guide and Review 119 46. Given: MN PQ, PQ RS Prove: MN RS N P 12 3 8. (Div. Prop.) Exercises Justify each statement with a property of equality or a property of congruence. See Example 1 on page 102. 47. PS PS Reflexive Prop. 48. If XY OP, then OP XY. Symmetric Prop. 49. If AB 8 CD 8, then AB CD. Add. Prop. 50. If EF GH and GH LM, then EF LM. Transitive Prop. M x 10 2.2(x 1) 2 2 (Mult. Prop.) 3.2x 2 x 10 (Dist. Prop.) 4.2x 2 2 x 10 2 (Subt. Prop.) 5.2x x 12 (Substitution) 6.2x x x 12 x (Subt. Prop.) 7.3x 12 (Substitution) Reasons QT RT, TS TP QT TS RT TS QT TS RT TP QT TS QS, RT TP RP QS RP 44. Given: x 1 2 Prove: x 4 Proof: Statements (Reasons) Concept Summary • Use properties of equality and congruence to write proofs involving segments. Write a two-column proof. Given: QT RT, TS TP Prove: QS RP 1 2 5.6 x (Substitution) 6.x 6 (Sym. Prop.) 2-7 Proving Segment Relationships Example 4.2(3) 2 x (Mult. Prop) 43 – 46. See margin. See pages 101–106. 1 2.5 2 2 x 2 2 (Subt. Prop.) 1 3.3 x (Substitution) R Q S Proof: Statements (Reasons) 1.MN PQ, PQ RS (Given) 2.MN RS (Transitive Prop.) B C Proof: Statements (Reasons) 1.AC AB,AC 4x 1, AB 6x 13 (Given) 2.4x 1 6x 13 (Subst.) 3.4x 1 1 6x 13 1 (Subt. Prop.) 4.4x 6x 14 (Subst.) 5.4x 6x 6x 14 6x (Subt. Prop.) 6.2x 14 (Subst.) 2x 2 14 2 7. (Div. Prop.) 8.x 7 (Subst.) Chapter 2 Study Guide and Review 119 • Extra Practice, see pages 756–758. • Mixed Problem Solving, see page 783. Study Guide and Review PROOF Answers (page 121) 1. Sample answer: Formal is the two-column proof, informal can be paragraph proofs. 2. Sample answer: You can use a counterexample. 3. Sample answer: statements and reasons to justify statements 7. 3 2 and 3x 12 when x 4. 8. 3 2 or 3x 12 when x 4. 9. 3 2, or 3x 12 when x 4 and an equilateral triangle is also equiangular. 10. H: you eat an apple a day; C: the doctor will stay away; If you eat an apple a day, then the doctor will stay away. Converse: If the doctor stays away, then you eat an apple a day. Inverse: If you do not eat an apple a day, then the doctor will not stay away. Contrapositive: If the doctor does not stay away, then you do not eat an apple a day. 11. H: a stone is rolling; C: it gathers no moss; If a stone is rolling, then it gathers no moss. Converse: If a stone gathers no moss, then it is rolling. Inverse: If a stone is not rolling, then it gathers moss. Contrapositive: If a stone gathers moss, then it is not rolling. 16. Given: y 4x 9; x 2 Prove: y 17 Proof: Statements (Reasons) 1.y 4x 9; x 2 (Given) 2.y 4(2) 9 (Substitution) 3.y 8 9 (Substitution) 4.y 17 (Substitution) 17. Given: AM CN, MB ND A M B 53. Given: Prove: 54. Given: AB CD Prove: AC BD E A B C D C D A 53 – 54. See p. 123G. 2-8 Proving Angle Relationships See pages 107–114. Example Concept Summary • The properties of equality and congruence can be applied to angle relationships. Find the measure of each numbered angle. m1 55, since 1 is a vertical angle to the 55° angle. 2 and the 55° angle form a linear pair. 2 1 55° 55 + m2 180 Def. of supplementary m2 125 Subtract 55 from each side. Exercises Find the measure of each numbered angle. 55. m6 145 56. m7 23 57. m8 90 58. PROOF See Example 2 on page 108. 7 157° 6 35° 8 Copy and complete the proof. See Example 3 on page 109. Given: 1 and 2 form a linear pair. m2 2(m1) Prove: m1 60 Proof: Statements Reasons a. 1 and 2 form a linear pair. a. ? b. ? b. 1 and 2 are supplementary. c. ? m1 m2 180 d. m2 2(m1) e. ? f. ? m1 2(m1) 180 3(m1) 180 3(m1) 180 g. 3 3 h. ? m1 60 120 Chapter 2 Reasoning and Proof Prove: AB CD Chapter 2 Reasoning and Proof BC EC, CA CD BA DE B Paragraph Proof: We are given that AM CN, MB ND. By the D N C Addition Property, AM MB CN MB. By Substitution, AM MB CN ND. Using the Segment Addition Postulate, AB AM MB, and CD CN ND. Then, by Substitution AB CD. 120 Write a two-column proof. See Examples 1 and 2 on pages 102 and 103. 18. H: you are a hard-working person; C: you deserve a great vacation; If you are a hard-working person, then you deserve a great vacation. Given Supplement Theorem c. Definition of supplementary angles d. ? Given e. Substitution f. Substitution g. ? Division Property h. Substitution Practice Test Vocabulary and Concepts Assessment Options 1. Explain the difference between formal and informal proofs. 1–3. See margin. 2. Explain how you can prove that a conjecture is false. 3. Describe the parts of a two-column proof. Vocabulary Test A vocabulary test/review for Chapter 2 can be found on p. 118 of the Chapter 2 Resource Masters. Skills and Applications Chapter Tests There are six Chapter 2 Tests and an OpenEnded Assessment task available in the Chapter 2 Resource Masters. Determine whether each conjecture is true or false. Explain your answer and give a counterexample for any false conjecture. 4. Given: A B 5. Given: y is a real number 6. Given: 3a2 48 Conjecture: B A Conjecture: y 0 Conjecture: a 4 true; Symmetric Prop. false; y 2 false; a 4 Use the following statements to write a compound statement for each conjunction or disjunction. Then find its truth value. 7–9. See margin for statements. p: 3 2 q: 3x 12 when x 4. r: An equilateral triangle is also equiangular. 7. p and q false Form 9. p (q r) true 8. p or q true Identify the hypothesis and conclusion of each statement and write each statement in if-then form. Then write the converse, inverse, and contrapositive of each conditional. 10–11. See margin. 10. An apple a day keeps the doctor away. 11. A rolling stone gathers no moss. 12. 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. (1) Perpendicular lines intersect. (2) Lines m and n are perpendicular. (3) Lines m and n intersect. valid; Law of Detachment Find the measure of each numbered angle. 13. 1 22 14. 2 85 15. 3See 85margin. 16–17. 16.Write a two-column proof. If y 4x 9 and x 2, then y 17. D N Pages 105–106 107–108 109–110 111–112 113–114 115–116 ExamView® Pro B C 18. ADVERTISING Identify the hypothesis and conclusion of the following statement, then write it in if-then form. Hard working people deserve a great vacation. See margin. 19. STANDARDIZED TEST PRACTICE If two planes intersect, their intersection can be A I a line. II three noncollinear points. III two intersecting lines. A I only B II only C D I and II only III only Chapter 2 Practice Test www.geometryonline.com/chapter_test basic average average average average advanced Open-Ended Assessment Performance tasks for Chapter 2 can be found on p. 117 of the Chapter 2 Resource Masters. A sample scoring rubric for these tasks appears on p. A31. 17. Write a paragraph proof. Given: AM CN, MB ND Prove: AB CD M MC MC MC FR FR FR MC = multiple-choice questions FR = free-response questions 95˚ 2 3 1 73˚ A 1 2A 2B 2C 2D 3 Chapter 2 Tests Type Level 121 Portfolio Suggestion Introduction In a chapter of diverse material, highlight concepts that are important but may not be used often or regularly, so that they will remain fresh in students’ minds. Ask Students Search the chapter for items you found the most difficult. Record these items in your portfolio and write about how you were able to master the concepts. If you are still having difficulty with the concepts, write about the steps you could take to better your understanding of them. Use the networkable ExamView® Pro to: • Create multiple versions of tests. • Create modified tests for Inclusion students. • Edit existing questions and add your own questions. • Use built-in state curriculum correlations to create tests aligned with state standards. • Apply art to your tests from a program bank of artwork. Chapter 2 Practice Test 121 Standardized Test Practice These two pages contain practice questions in the various formats that can be found on the most frequently given standardized tests. 6. Which of the following is an example of inductive reasoning? (Lesson 2-1) C Part 1 Multiple Choice Record your answers on the answer sheet provided by your teacher or on a sheet of paper. A B C Practice 2Standardized Standardized Test Test Practice Student Record Sheet (Use with Sheet, pages 122–123 ofp. the Student Student Recording A1 Edition.) D Part 1 Multiple Choice B C D 4 A B C D 7 A B C D 2 A B C D 5 A B C D 8 A B C D 3 A B C D 6 A B C D B Carlos reads in his textbook that the measure of all right angles is 90. He conjectures that the measure of each right angle in a square equals 90. 1 7 1 7 2, 7, , 7 7 1 7, , 7, 7 2 7 1 7 2, , 7, 7 7 C Carlos measures the angles of several triangles and finds that their measures all add up to 180. He conjectures that the sum of the measures of the angles in any triangle is always 180. D Carlos knows that the sum of the measures of the angles in a square is always 360. He conjectures that if he draws a square, the sum of the measures of the angles will be 360. 7, 7, , 7 2 Select the best answer from the choices given and fill in the corresponding oval. A Carlos learns that the measures of all acute angles are less than 90. He conjectures that if he sees an acute angle, its measure will be less than 90. 1 1. Arrange the numbers 7, , 7, 7 2 in order 7 from least to greatest. (Prerequisite Skill) D A practice answer sheet for these two pages can be found on p. A1 of the Chapter 2 Resource Masters. 1 A 2. Points A and B lie on the line y 2x 3. Which of the following are coordinates of a point noncollinear with A and B? (Lesson 1-1)C Part 2 Short Response/Grid In Solve the problem and write your answer in the blank. For Questions 9 and 11, also enter your answer by writing each number or symbol in a box. Then fill in the corresponding oval for that number or symbol. 9 (grid in) 10 11 (grid in) 12 9 A (7, 11) B (4, 5) C (–2, –10) D (–5, –13) 11 . / . / . . 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 . / . / . . 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 3. Dana is measuring distance on a map. Which of the following tools should Dana use to make the most accurate measurement? (Lesson 1-2) A 7. Which of the following is the contrapositive of the statement If Rick buys hamburgers for lunch, then Denzel buys French fries and a large soda? (Lesson 2-2) A If Denzel does not buy French fries and a large soda, then Rick does not buy hamburgers for lunch. B If Rick does not buy hamburgers for lunch, then Denzel does not buy French fries and a large soda. C If Denzel buys French fries and a large soda, then Rick buys hamburgers for lunch. D If Rick buys hamburgers for lunch, then Denzel does not buy French fries and a large soda. Part 3 Extended Response Answers Record your answers for Questions 13–15 on the back of this paper. A centimeter ruler B protractor C yardstick D calculator 4. Point E is the midpoint of D F . If DE 8x 3 and EF 3x 7, what is x? (Lesson 1-3) B A Additional Practice See pp. 123–124 in the Chapter 2 Resource Masters for additional standardized test practice. 1 B 2 C 4 D E A 8. Which property could justify the first step in A B C 14x 6 8 solving 3 18? (Lesson 2-5) A A complementary angles A Division Property of Equality B congruent angles B Substitution Property of Equality C supplementary angles C Addition Property of Equality D vertical angles D Transitive Property of Equality 122 Chapter 2 Reasoning and Proof ExamView® Pro Special banks of standardized test questions similar to those on the SAT, ACT, TIMSS 8, NAEP 8, and state proficiency tests can be found on this CD-ROM. 122 Chapter 2 Reasoning and Proof 13 F 5. What is the relationship between ACF and DCF? (Lesson 1-6) D A Preparing for Standardized Tests For test-taking strategies and more practice, see pages 795– 810. Evaluating Extended Response Questions Part 2 Short Response/Grid In Record your answers on the answer sheet provided by your teacher or on a sheet of paper. Test-Taking Tip 9. Two cheerleaders stand at opposite corners of a football field. What is the shortest distance between them, to the nearest yard? (Lesson 1-3) Question 6 When answering a multiple-choice question, always read every answer choice and eliminate those you decide are definitely wrong. This way, you may deduce the correct answer. 131 yd 120 yd Part 3 Extended Response 10 20 30 40 50 40 30 20 10 1 53 3 yd Record your answers on a sheet of paper. Show your work. 10 20 30 40 50 40 30 20 10 10. Consider the conditional If I call in sick, then I will not get paid for the day. Based on the original conditional, what is the name of the conditional If I do not call in sick, then I will get paid for the day? (Lesson 2-2) inverse 11. Examine the following statements. p: Martina drank a cup of soy milk. q: A cup is 8 ounces. r: Eight ounces of soy milk contains 300 milligrams of calcium. Using the Law of Syllogism, how many milligrams of calcium did Martina get from drinking a cup of soy milk? (Lesson 2-4) 13. In any right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. From a single point in her yard, Marti measures and marks distances of 18 feet and 24 feet for two sides of her garden. Explain how Marti can ensure that the two sides of her garden form a right angle. (Lesson 1-3) See margin. Extended Response questions are graded by using a multilevel rubric that guides you in assessing a student’s knowledge of a particular concept. Goal: Find measures and prove an angle measure. Sample Scoring Rubric: The following rubric is a sample scoring device. You may wish to add more detail to this sample to meet your individual scoring needs. Score Criteria 4 A correct solution that is supported by well-developed, accurate explanations A generally correct solution, but may contain minor flaws in reasoning or computation A partially correct interpretation and/or solution to the problem A correct solution with no supporting evidence or explanation An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given 14. A farmer needs to make a 100-square-foot rectangular enclosure for her chickens. She wants to save money by purchasing the least amount of fencing possible to enclose the area. (Lesson 1-4) a–c. See margin. 3 a. What whole-number dimensions, to the nearest yard, will require the least amount of fencing? 1 b. Explain your procedure for finding the dimensions that will require the least amount of fencing. 0 2 300 12. In the following proof, what property justifies statement c? (Lesson 2-7) Segment Addition Given: AC MN Postulate Prove: AB BC MN B C A N M Proof: Statements C MN a. A b. AC MN c. AC AB BC d. AC BC MN c. Explain how the amount of fencing required to enclose the area changes as the dimensions change. Reasons a. Given b. Definition of congruent segments c. ? d. Substitution www.geometryonline.com/standardized_test 15. Given: 1 and 3 are vertical angles. m1 3x 5, m3 2x 8 Prove: m1 14 (Lesson 2-8) See p. 123G. 1 2 4 3 Chapter 2 Standardized Test Practice 123 Answers 13. Sample answer: Marti can measure a third distance c, the distance between the ends of the two sides, and make sure it satisfies the equation a 2 b 2 c 2. 14a. 10 yd by 10 yd 14b. Sample answer: Make a list of all possible whole-number lengths and widths that will form a 100-square-foot area. Then find the perimeter of each rectangle. Choose the length and width combination that has the smallest perimeter. 14c. As the length and width get closer to having the same measure as one another, the amount of fencing required decreases. Chapter 2 Standardized Test Practice 123 Pages 63–66, Lesson 2-1 1. Sample answer: After the news is over, it’s time for dinner. 2. Sometimes; the conjecture is true when E is between D and F; otherwise it is false. 3. Sample answer: When it is cloudy, it rains. Counterexample: It is often cloudy and it does not rain. 19. 30 20. 20 14. p q ( p q) r p q r T T T T T T T F T T T F T T T T F F T T F T T T T F T F T T F F T F T F F F F F 8 and an equilateral triangle has three 18. 64 congruent sides. 21. 22. y 8 4 m B(2, 1) –8 Additional Answers for Chapter 2 C (5, 10) –4 O 4 8x –4 23. 3 –8 4 A(–2, –11) 24. 25. A y P D R(6, 5) (–1, 7) C B 26. H I x O Q (6, –2) K J 27. P Q S R 28. A C B Pages 71–74, Lesson 2-2 11. p q p q 13. 123A 12. q r qr 19. 64 8 or an equilateral triangle has three congruent sides. 20. 64 8 and 0 0. 21. 0 0 and an obtuse angle measures greater than 90° and less than 180°. 22. An equilateral triangle has three congruent sides or 0 0. 23. An equilateral triangle has three congruent sides and an obtuse angle measures greater than 90° and less than 180°. 24. 64 8 and an obtuse angle measures greater than 90° and less than 180°. 25. An equilateral triangle has three congruent sides and 0 0. 26. 0 0 or 64 8 27. An obtuse angle measures greater than 90° and less than 180° or an equilateral triangle has three congruent sides. 28. 64 8 and an equilateral triangle has three congruent sides, or an obtuse angle measures greater than 90° and less than 180°. 29. An obtuse angle measures greater than 90° and less than 180°, or an equilateral triangle has three congruent sides and 0 0. 33. q r q and r 34. p q p or q T T T T T T T F F T F T F T F F T T F F F F F F p r p or r p q p and q T T T T T T T T T T T T F F T F T F T F F T T T F F F F F F T F T T F F F T T F T F F F F F F F p p r p r T T F F T F F F F T T T F F T F Chapter 2 Additional Answers 35. 36. 37. 39. 40. r q r 38. q p q p q F T T T T T F F F F T F F F p q r T T T F F F F T T F F T T T T F T F F F F T F F F T F F F T T T F F T F T F T T T T F F T T F F T F F F T T F T p q r T T T T T T F T F F T F T F F F F F r T T F T F F T F F F F F F T F T T F F T T T p r q r p (q r ) q r q r p (q r) F F F F F F T T T T T F T T F T T T T T F F F F F F T T F T T F T F T T T F Pages 78–80, Lesson 2-3 15. If you are in Colorado, then aspen trees cover high areas of the mountains. If you are in Florida, then cypress trees rise from the swamps. If you are in Vermont, then maple trees are prevalent. 16. H: 2x 6 10, C: x 2 17. H: you are a teenager; C: you are at least 13 years old 18. H: you have a driver’s license; C: you are at least 16 years old 19. H: three points lie on a line; C: the points are collinear 20. H: a man hasn’t discovered something he will die for; C: he isn’t fit to live 21. H: an angle measures between 0 and 90; C: the angle is acute 22. If you buy a 1-year fitness plan, then you get a free visit. 23. If you are a math teacher, then you love to solve problems. 24. If I think, then I am. 25. If two angles are adjacent, then they have a common side. 26. If two angles are vertical, then they are congruent. 27. If two triangles are equiangular, then they are equilateral. 40. Converse: If you live in Texas, then you live in Dallas. False; you could live in Austin. Inverse: If you don’t live in Dallas, then you don’t live in Texas. False; you could live in Austin. Contrapositive: If you don’t live in Texas, then you don’t live in Dallas; true. 41. Converse: If you are in good shape, then you exercise regularly; true. Inverse: If you do not exercise regularly, then you are not in good shape; true. Contrapositive: If you are not in good shape, then you do not exercise regularly. False; an ill person may exercise a lot, but still not be in good shape. 42. Converse: If the sum of two angles is 90, then they are complementary; true. Inverse: If two angles are not complementary, then their sum is not 90; true. Contrapositive: If the sum of two angles is not 90, then they are not complementary; true. 43. Converse: If a figure is a quadrilateral, then it is a rectangle; false, rhombus. Inverse: If a figure is not a rectangle, then it is not a quadrilateral; false, rhombus. Contrapositive: If a figure is not a quadrilateral, then it is not a rectangle; true. 44. Converse: If an angle has a measure of 90, then it is a right angle; true. Inverse: If an angle is not a right angle, then its measure is not 90; true. Contrapositive: If an angle does not have a measure of 90, then it is not a right angle; true. 45. Converse: If an angle has measure less than 90, then it is acute; true. Inverse: If an angle is not acute, then its measure is not less than 90; true. Contrapositive: If an angle’s measure is not less than 90, then it is not acute; true. 47. Sample answer: In Alaska, if there are more hours of daylight than darkness, then it is summer; true. In Alaska, if there are more hours of darkness than daylight, then it is winter; true. Page 80, Practice Quiz 1 1. W X 3. Y 4. p q r qr p (q r ) T T T T T T T F F T T F T F T T F F F T F T T T T F T F F F F F T F F F F F F F p q p p q T T F F T F F F F T T T F F T F Chapter 2 Additional Answers 123B Additional Answers for Chapter 2 p q Additional Answers for Chapter 2 5. Converse: If two angles have a C A common vertex, then the angles are adjacent. False; ABD is not D B adjacent to ABC. Inverse: If two angles are not A C adjacent, then they do not have B a common vertex. False, ABC E D and DBE have a common vertex and are not adjacent. Contrapositive: If two angles do not have a common vertex, then they are not adjacent; true. Pages 84–87, Lesson 2-4 33. Sample answer: Doctors and nurses use charts to assist in determining medications and their doses for patients. Answers should include the following. • Doctors need to note a patient’s symptoms to determine which medication to prescribe, then determine how much to prescribe based on weight, age, severity of the illness, and so on. • Doctors use what is known to be true about diseases and when symptoms appear, then deduce that the patient has a particular illness. 39. q r q r 40. p r p p r 41. 42. T T T T T F T T F F T F F F F T F F T T T F F F F F T T Animal p q r qr p (q r ) Arthropod T T T T T Butterfly T T F T T T F T T T T F F F F F T T T F F T F T F F F T T F F F F F F p q r T T T F F T T T F F F T T F T T T T T F F T F T F T T F F F F T F F F F F F T T T T F F F T F F q q r p (q r ) Pages 91–93, Lesson 2-5 6. Sometimes; if the planes have a common intersection, then their intersection is one line. 123C 10. Since P is the midpoint of Q R and ST , PQ PR 1 1 QR and PS PT ST by the definition of 2 2 midpoint. We are given QR S T so QR ST by the definition of congruent segments. By the Multiplication 1 1 Property, QR ST. So, by substitution, PQ PT. 2 2 16. Sometimes; the three points cannot be on the same line. 17. Always; if two points lie in a plane, then the entire line containing those points lies in that plane. 18. Never; the intersection of a line and a plane can be a point, but the intersection of two planes is a line. 19. Sometimes; the three points cannot be on the same line. 20. Always; one plane contains at least three points, so it must contain two. 21. Sometimes; and m could be skew, so they would not lie in the same plane. 22. Postulate 2.1; through any two points, there is exactly one line. 23. Postulate 2.5; if two points lie in a plane, then the entire line containing those points lies in that plane. 24. Postulate 2.2; through any three points not on the same line, there is exactly one plane. 25. Postulate 2.5; if two points lie in a plane, then the entire line containing those points lies in the plane. 26. Postulate 2.1; through any two points, there is exactly one line. 27. Postulate 2.2; through any three points not on the same line, there is exactly one plane. 38. Chapter 2 Additional Answers Pages 97–100, Lesson 2-6 10. Given: 25 7(y 3) 5y Prove: 2 y Proof: Statements (Reasons) 1. 25 7(y 3) 5y (Given) 2. 25 7y 21 5y (Dist. Prop.) 3. 25 2y 21 (Substitution) 4. 4 2y (Subt. Prop.) 5. 2 y (Div. Prop.) 11. Given: Rectangle ABCD, 10 A B AD 3, AB 10 3 3 Prove: AC BD D C 10 Proof: Statements (Reasons) 1. Rectangle ABCD, AD 3, AB 10 (Given) 2. Draw segments AC and DB. (Two points determine a line.) 3. ABC and BCD are right triangles. (Def. of rt. ) 4. AC 32 1 02, DB 32 1 02 (Pythag. Th.) 5. AC BD (Substitution) 12. Given: c 2 a2 b 2 Prove: a c 2 b2 Proof: Statements (Reasons) 1. c 2 a2 b2 (Given) 2. c 2 b2 a2 (Subt. Prop.) 3. a2 c 2 b2 (Reflexive Prop.) 4. a2 c 2 b2 (Square Root Prop.) 5. a c 2 b2 (Square Root Prop.) 1 7 26. Given: 4 a a 2 2 Prove: a 1 Proof: Statements (Reasons) 1 1. 4 a 2 1 2. 2 4 a 2 3 27. Given: 2y 2 8 13 4 Prove: y Proof: Statements (Reasons) 3 2 1. 2y 8 (Given) 3 2 2. 2 2y 2(8) (Mult. Prop.) 3. 4y 3 16 (Dist. Prop.) 4. 4y 13 (Subt. Prop.) 13 5. y (Div. Prop.) 4 1 28. Given: m 9 2 Prove: m 18 Proof: Statements (Reasons) 1 2 1. m 9 (Given) 1 2 2 3 3. 15 2x 3 (Dist. Prop.) 4. 15 2x 15 3 15 (Subt. Prop.) 5. 2x 12 (Substitution) 2x 2 8 a 7 2a (Dist. Prop.) 1 a 2a (Subt. Prop.) 1 1a (Add. Prop.) 1 a (Div. Prop.) a 1 (Symmetric Prop.) 2. 3 5 z 3(1) (Mult. Prop.) 12 2 6. (Div. Prop.) 7 a (Given) 2 7 2 a (Mult. Prop.) 2 2 3 1. 5 z 1 (Given) 2. 2 m 2(9) (Mult. Prop.) 7. x 6 (Substitution) 30. Given: XZ ZY, XZ 4x 1, and ZY 6x 13 Prove: x 7 X 4x 1 Z 6x 13 Proof: Y Statements (Reasons) 1. XZ ZY, XZ 4x 1, and ZY 6x 13 (Given) 2. 4x 1 6x 13 (Substitution) 3. 4x 1 4x 6x 13 4x (Subt. Prop.) 4. 1 2x 13 (Substitution) 5. 1 13 2x 13 13 (Add. Prop.) 6. 14 2x (Substitution) 14 2 2x 2 7. (Div. Prop.) 8. 7 x (Substitution) 9. x 7 (Symmetric Prop.) 31. Given: mACB mABC Prove: mXCA mYBA A Proof: X C B Statements (Reasons) 1. mACB mABC (Given) 2. mXCA mACB 180, mYBA mABC 180 (Def. of supp. ) 3. mXCA mACB mYBA mABC (Substitution) 4. mXCA mACB mYBA mACB (Substitution) 5. mXCA mYBA (Subt. Prop.) Y 3. m 18 (Substitution) Chapter 2 Additional Answers 123D Additional Answers for Chapter 2 3. 4. 5. 6. 7. 2 29. Given: 5 z 1 3 Prove: z 6 Proof: Statements (Reasons) Additional Answers for Chapter 2 Page 100, Practice Quiz 2 5. Given: 2(n 3) 5 3(n 1) Prove: n 2 Proof: Statements (Reasons) 1. 2(n 3) 5 3(n 1) (Given) 2. 2n 6 5 3n 3 (Dist. Prop.) 3. 2n 1 3n 3 (Substitution) 4. 2n 1 2n 3n 3 2n (Subt. Prop.) 5. 1 n 3 (Substitution) 6. 1 3 n 3 3 (Add. Prop.) 7. 2 n (Substitution) 8. n 2 (Symmetric Prop.) 20. Given: AB A C and PC Q B Prove: AP AQ Pages 103–106, Lesson 2-7 8. Given: A P CP A P P B DP C Prove: AB CD Proof: Statements (Reasons) 1. A P and BP D P (Given) CP 2. AP CP and BP DP (Def. of segs.) 3. AP PB AB (Seg. Add. Post.) 4. CP DP AB (Substitution) 5. CP PD CD (Seg. Add. Post.) 6. AB CD (Transitive Prop.) 7. AB (Def. of segs.) CD 9. Given: HI TU and H H J TV I Prove: IJ U V 22. T Proof: J Statements (Reasons) 1. H I TU and HJ T V (Given) 2. HI TU and HJ TV (Def. of segs.) 3. HI IJ HJ (Seg. Add. Post.) 4. TU IJ TV (Substitution) 5. TU UV TV (Seg. Add. Post.) 6. TU IJ TU UV (Substitution) 7. TU TU (Reflexive Prop.) 8. IJ UV (Subt. Prop.) 9. IJ U V (Def. of segs.) 19. Given: XY WZ and WZ AB Prove: XY AB Proof: Statements (Reasons) U D B 23. V 24. B X Proof: Y Statements (Reasons) 1. X Y WZ and WZ AB (Given) 2. XY WZ and WZ AB (Def. of segs.) 3. XY AB (Transitive Prop.) 4. XY AB (Def. of segs.) Chapter 2 Additional Answers Q P W 25. A 123E B C A 1. A B AC and PC Q B (Given) 2. AB AC, PC QB (Def. of segs.) 3. AB AQ QB, AC AP PC (Seg. Add. Post.) 4. AQ QB AP PC (Substitution) 5. AQ QB AP QB (Substitution) 6. QB QB (Reflexive Prop.) 7. AP AQ (Subt. Prop.) 8. AP AQ (Def. of segs.) Given: LM P N and P L X M XN M X N Prove: LX P X Proof: Statements (Reasons) 1. L M PN and X M XN (Given) 2. LM PN and XM XN (Def. of segs.) 3. LM LX XM, PN PX XN (Seg. Add. Post.) 4. LX XM PX XN (Substitution) 5. LX XN PX XN (Substitution) 6. XN XN (Reflexive Prop.) 7. LX PX (Subt. Prop.) 8. L X P X (Def. of segs.) Given: AB BC A B C Prove: AC 2BC Proof: Statements (Reasons) 1. AB BC (Given) 2. AC AB BC (Seg. Add. Post.) 3. AC BC BC (Substitution) 4. AC 2BC (Substitution) Given: A B A B Prove: AB AB Proof: Statements (Reasons) 1. A B (Given) 2. AB AB (Reflexive Prop.) 3. AB AB (Def. of segs.) Given: AB DE , C is the midpoint of BD . Prove: AC CE Z Proof: A B C D E Statements (Reasons) 1. A B DE , C is the midpoint of BD . (Given) 2. BC CD (Def. of midpoint) 3. AB DE (Def. of segs.) 4. AB BC CD DE (Add. Prop.) 5. AB BC AC, CD DE CE (Seg. Add. Post.) 6. AC CE (Substitution) 7. AC CE (Def. of segs.) 26. Given: A B and BC EF DE Prove: AC DF A C D B E F Proof: Statements (Reasons) 1. A B EF and B C D E (Given) 2. AB EF and BC DE (Def. of segs.) 3. AB BC DE EF (Add. Prop.) 4. AC AB BC, DF DE EF (Seg. Add. Post.) 5. AC DF (Substitution) 6. A C DF (Def. of segs.) 27. Sample answers: LN QO and L M M N RS S T Q P P O 1 2 3 4 1 2 1 Proof: Statements (Reasons) 1. ⊥ m (Given) 2. 1 and 2 are rt. . (⊥ lines intersect to form 4 rt. .) 3. 1 2 (All rt. are .) 36. Given: 1 2, 1 and 2 are supplementary. 1 2 Prove: 1 and 2 are rt. . Proof: Statements (Reasons) 1. 1 2, 1 and 2 are supplementary. (Given ) 2. m1 m2 180 (Def. of supplementary ) 3. m1 m2 (Def. of angle) 4. m1 m1 180 (Substitution) 5. 2(m1) 180 (Add. Prop.) 6. m1 90 (Div. Prop.) 7. m2 90 (Substitution (steps 3, 6)) 8. 1 and 2 are rt. . (Def. of rt. ) 37. Given: ABD CBD, ABD and D CBD form a linear pair. Prove: ABD and CBD are rt. . A m Proof: Statements (Reasons) 1. ⊥ m (Given) 2. 1 is a right angle. (Def. of ⊥) 3. m1 90 (Def. of rt. ) 4. 1 4 (Vert. are ) 5. m1 m4 (Def. of ) 6. m4 90 (Substitution) 7. 1 and 2 form a linear pair; 3 and 4 form a linear pair. (Def. of linear pair) 8. m1 m2 180, m4 m3 180 (Linear pairs are supplementary.) 9. 90 m2 180, 90 m3 180 (Substitution) 10. m2 90, m3 90 (Subt. Prop.) 11. 2, 3, 4 are rt. . (Def. of rt. (steps 6, 10)) m 2 3 4 B C Proof: Statements (Reasons) 1. ABD CBD, ABD and CBD form a linear pair. (Given) 2. ABD and CBD are supplementary. (Linear pairs are supplementary.) 3. ABD and CBD are rt. . (If are and suppl., they are rt. .) Chapter 2 Additional Answers 123F Additional Answers for Chapter 2 Pages 111–114, Lesson 2-8 25. Given: A Prove: A A Proof: A Statements (Reasons) 1. A is an angle. (Given) 2. mA mA (Reflexive Prop) 3. A A (Def. of angles) 26. Given: 1 2, 2 3 1 2 3 Prove: 1 3 Proof: Statements (Reasons) 1. 1 2, 2 3 (Given) 2. m1 m 2, m 2 m 3 (Def. of angles) 3. m1 m3 (Trans. Prop.) 4. 1 3 (Def. of angles) 33. Given: ⊥ m Prove: 2, 3, 4 are rt. 34. Given: 1 and 2 are rt. . Prove: 1 2 Proof: Statements (Reasons) 1. 1 and 2 are rt. . (Given) 2. m1 90, m2 90 (Def. of rt. ) 3. m1 m2 (Substitution) 4. 1 2 (Def. of angles) 35. Given: ⊥ m Prove: 1 2 38. Given: ABD YXZ Prove: CBD WXZ W Additional Answers for Chapter 2 Y Z B C Proof: Statements (Reasons) 1. ABD YXZ; ABD and CBD form a linear pair; YXZ and WXZ form a linear pair. (Given; from the figure) 2. mABD mCBD 180, mYXZ mWXZ 180 (Linear pairs are supplementary.) 3. mABD mCBD mYXZ mWXZ (Subst.) 4. mABD mYXZ (Def. of ) 5. mYXZ mCBD mYXZ mWXZ (Subst.) 6. mYXZ mYXZ (Reflexive Prop.) 7. mCBD mWXZ (Subt. Prop.) 8. CBD WXZ (Def. of ) 39. Given: mRSW mTSU Prove: mRST mWSU T R W U S Proof: Statements (Reasons) 1. mRSW mTSU (Given) 2. mRSW mRST mTSW, mTSU mTSW mWSU (Angle Addition Postulate) 3. mRST mTSW mTSW mWSU (Substitution) 4. mTSW mTSW (Reflexive Prop.) 5. mRST mWSU (Subt. Prop.) 48. Given: AC BD A B C D Prove: AB CD Proof: Statements (Reasons) 1. AC BD (Given) 2. AB BC AC, BC CD BD (Segment Addition Postulate) 3. BC BC (Reflexive Prop.) 4. AB BC BC CD (Substitution (2 and 3)) 5. AB CD (Subt. Prop.) 123G Chapter 2 Additional Answers E C D A X Page 115-120, Chapter 2 Study Guide and Review 53. Given: BC EC, CA CD B Prove: BA DE D Proof: A Statements (Reasons) 1. BC EC, CA CD (Given) 2. BC CA EC CA (Add. Prop.) 3. BC CA EC CD (Substitution) 4. BC CA BA, EC CD DE (Seg. Add. Post.) 5. BA DE (Substitution) 54. Given: AB CD A B C D Prove: AC BD Proof: Statements (Reasons) 1. AB CD (Given) 2. BC BC (Reflexive Prop.) 3. AB BC CD BC (Add. Prop.) 4. AB BC AC, CD BC BD (Seg. Add. Post.) 5. AC BD (Substitution) Page 123, Chapter 2 Standardized Test Practice 15. Given: 1 and 3 are vertical angles. m1 3x 5, m3 2x 8 Prove: m1 14 1 2 4 3 Proof: Statements (Reasons) a. 1 and 3 are vertical angles; m1 3x 5, m3 2x 8 (Given) b. 1 3 (Vert. are .) c. m1 m3 (Def. of ) d. 3x 5 2x 8 (Substitution) e. x 5 8 (Subt. Prop.) f. x 3 (Subt. Prop.) g. m1 3(3) 5 (Substitution) h. m1 14 (Substitution) Notes Additional Answers for Chapter 2 Chapter 2 Additional Answers 123H