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Notes Triangles Introduction In this unit students identify and classify triangles by various methods. They learn how to test for and prove triangle congruence and how to write coordinate proofs. Bisectors, medians, and altitudes of triangles are introduced. Students explore inequalities and triangles and learn to use indirect proof. You can use triangles and their properties to model and analyze many real-world situations. In this unit, you will learn about relationships in and among triangles, including congruence and similarity. Students apply their knowledge of ratios and proportions to similar figures and scale factors. They explore proportional parts of triangles and proportional relationships between similar triangles. Finally, students focus on right triangles. They use the Pythagorean Theorem and are introduced to trigonometric ratios. About the Photograph The large photograph is the Air Force Academy Chapel in Colorado Springs, Colorado. Its 17-spire structure is designed to resemble aircraft in synchronized flight. Have students use the Internet to find out more about this building. Chapter 4 Congruent Triangles Chapter 5 Assessment Options Relationships in Triangles Unit 2 Test Pages 413–414 of the Chapter 7 Resource Masters may be used as a test or review for Unit 2. This assessment contains both multiple-choice and short answer items. Chapter 6 Proportions and Similarity Chapter 7 Right Triangles and Trigonometry 174 Unit 2 Triangles ExamView® Pro This CD-ROM can be used to create additional unit tests and review worksheets. 174 Unit 2 Triangles An online, research-based, instructional, assessment, and intervention tool that provides specific feedback on student mastery of state and national standards, instant remediation, and a management system to track performance. For more information, contact mhdigitallearning.com. Real-Life Geometry Videos What’s Math Got to Do With It? Real-Life Geometry Videos engage students, showing them how math is used in everyday situations. Use Video 2 with this unit. Teaching Suggestions Have students study the USA TODAY Snapshot. • Ask them which region of the United States has the lowest percentage of children using the Internet. South • Have students speculate about why the Northeast has the greatest percentage of children using the Internet. • What percentage of children in the Midwest use the Internet? 32.1% Who Is Behind This Geometry Concept Anyway? USA TODAY Snapshots® Have you ever wondered who first developed some of the ideas you are learning in your geometry class? Today, many students use the Internet for learning and research. In this project, you will be using the Internet to research a topic in geometry. You will then prepare a portfolio or poster to display your findings. Where children use the Internet Percentage of children 3-17 using the Internet at home, by region: Northeast 35.5% Midwest West Lesson Page 4-6 218 5-1 241 6-6 325 7-1 347 32.1% 29.0% Log on to www.geometryonline.com/webquest. Begin your WebQuest by reading the Task. Continue working on your WebQuest as you study Unit 2. Additional USA TODAY Snapshots appearing in Unit 2: Chapter 4 Gross Domestic Product slides in 2001 (p. 206) Chapter 5 Sources of college information (p. 259) Chapter 6 Workplace manners declining (p. 296) Chapter 7 Bruins bring skills to Major League Soccer (p. 347) South 27.6% Source: Census Bureau By Sam Ward, USA TODAY Unit 2 Triangles 175 Internet Project Problem-Based Learning A WebQuest is an online project in which students do research on the Internet, gather data, and make presentations using word processing, graphing, page-making, or presentation software. In each chapter, students advance to the next step in their WebQuest. At the end of Chapter 7, the project culminates with a presentation of their findings. Teaching notes and sample answers are available in the WebQuest and Project Resources. Unit 2 Triangles 175 Congruent Triangles Chapter Overview and Pacing Year-long pacing: pages T20–T21. PACING (days) Regular Block LESSON OBJECTIVES Basic/ Average Advanced Basic/ Average Advanced 1 1 0.5 0.5 2 (with 4-2 Preview) 1 1 (with 4-2 Preview) 0.5 Congruent Triangles (pp. 192–198) • Name and label corresponding parts of congruent triangles. • Identify congruence transformations. 2 2 1 1 Proving Congruence—SSS, SAS (pp. 200–206) • Use the SSS Postulate to test for triangle congruence. • Use the SAS Postulate to test for triangle congruence. 2 2 1 1 Classifying Triangles (pp. 178–183) • Identify and classify triangles by angles. • Identify and classify triangles by sides. Angles of Triangles (pp. 184–191) Preview: Use a model to find the relationships among the measures of the interior angles of a triangle. • Apply the Angle Sum Theorem. • Apply the Exterior Angle Theorem. 2 (with 4-5 2 (with 4-5 1 (with 4-5 1 (with 4-5 Follow-Up) Follow-Up) Follow-Up) Follow-Up) Proving Congruence—ASA, AAS (pp. 207–215) • Use the ASA Postulate to test for triangle congruence. • Use the AAS Theorem to test for triangle congruence. Follow-Up: Use models to explore congruence in right triangles. Isosceles Triangles (pp. 216–221) • Use properties of isosceles triangles. • Use properties of equilateral triangles. 2 2 1 1 Triangles and Coordinate Proof (pp. 222–226) • Position and label triangles for use in coordinate proofs. • Write coordinate proofs. 2 2 1 1 Study Guide and Practice Test (pp. 227–231) Standardized Test Practice (pp. 232–233) 1 1 1 0.5 Chapter Assessment 1 1 0.5 0.5 15 14 8 7 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. 176A Chapter 4 Congruent Triangles Timesaving Tools ™ All-In-One Planner and Resource Center Chapter Resource Manager See pages T5 and T21. 183–184 185–186 187 188 189–190 191–192 193 194 195–196 197–198 199 200 201–202 203–204 205 206 207–208 209–210 211 212 213–214 215–216 217 218 240 219–220 221–222 223 224 240 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 4 RESOURCE MASTERS 1–2 239 239, 241 81–84 Materials 4-1 4-1 patty paper, protractor, grid paper GCC 23 4-2 4-2 (Preview: protractor, scissors) straightedge SC 7 4-3 4-3 straightedge 4-4 4-4 straightedge, compass, scissors 4-5 4-5 4-6 4-6 ruler, scissors, protractor, tracing paper 4-7 4-7 grid paper, straightedge 1–2 SC 8 10 straightedge, compass, scissors, patty paper, protractor, ruler (Follow-Up: ruler, protractor) 225–238, 242–244 *Key to Abbreviations: GCC Graphing Calculator and Computer Masters SC School-to-Career Masters Chapter 4 Congruent Triangles 176B Mathematical Connections and Background Continuity of Instruction Prior Knowledge In Chapter 1, students calculated distance between points on the coordinate plane using the Distance Formula. In Chapter 3, they learned that an intersection between parallel lines and a transversal creates a variety of congruent and supplementary angles. Classifying Triangles Triangles can be classified based on their angle measures. In an acute triangle, all of the angles are acute. In an obtuse triangle, one of the angles is obtuse. In a right triangle, one angle measures 90. When all of the angles of an acute triangle are congruent, it is called an equiangular triangle. Triangles can also be classified according to their number of congruent sides. No two sides of a scalene triangle are congruent. At least two sides of an isosceles triangle are congruent. All of the sides of an equilateral triangle are congruent. Equilateral triangles are a special kind of isosceles triangle. Angles of Triangles This Chapter In this chapter, students prove triangles congruent using SSS, SAS, ASA, and AAS. They also learn how to write two new types of proofs, the flow proof and the coordinate proof. Students classify triangles according to their angles or sides and apply the Angle Sum Theorem and the Exterior Angle Theorem. The special properties of isosceles and equilateral triangles are introduced, and students are expected to use those properties in proofs. Students also position and label triangles for use in coordinate proofs. The Angle Sum Theorem states that the sum of the measures of the interior angles of a triangle is always 180. This theorem can be applied to any triangle. It also leads to the Third Angle Theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. Each angle of a triangle has an exterior angle, which is formed by one side of the triangle and the extension of another side. The interior angles of the triangle not adjacent to a given exterior angle are called remote interior angles. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. This is the Exterior Angle Theorem. This lesson also introduces flow proofs. A flow proof organizes a series of statements in logical order. Arrows are used to indicate the order of the statements. Flow proofs can be written horizontally or vertically. Congruent Triangles Future Connections Students will need the knowledge gained from this chapter to master the skills taught later in this course. They must understand triangle congruence to be successful scholars of trigonometry, a precursor to calculus and all courses in higher math. 176C Chapter 4 Congruent Triangles Two triangles are congruent if and only if their corresponding parts are congruent. Certain transformations, including a slide, flip, and turn, do not affect congruence. These transformations are called congruence transformations. Congruence of triangles, like that of angles and segments, is reflexive, symmetric, and transitive. Proving Congruence—SSS, SAS Isosceles Triangles In this lesson you will construct a triangle in which three sides are congruent to the three sides of a given triangle. This activity demonstrates the SideSide-Side Postulate. Also written as SSS, it states that if the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. You will also construct a triangle in which two sides are congruent to two sides of a given triangle and the included angle is congruent to the included angle in the given triangle. This activity demonstrates the Side-Angle-Side Postulate, also written SAS. It states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Isosceles triangles have special terminology for their parts. The angle formed by the congruent sides is called the vertex angle. The two angles formed by the base and one of the congruent sides are called base angles. The congruent sides are called legs. Isosceles triangles also have special properties recognized in Isosceles Triangle Theorem and its converse: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. This theorem leads to corollaries about the angles of an equilateral triangle. The first states that a triangle is equilateral if and only if it is equiangular. The second states that each angle of an equilateral triangle measures 60°. Proving Congruence—ASA, AAS Triangles and Coordinate Proof The Angle-Side-Angle Postulate, written as ASA, works because the measures of two angles of a triangle and the side between them form a unique triangle. The postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. The Angle-Angle-Side, or AAS, Theorem follows from the ASA Postulate: If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. Right triangles have their own theorems to prove congruence. One of those is the LL Congruence Theorem, which is the SAS Postulate applied to right triangles. It states that if the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. The HA Theorem is based on the AAS Theorem: If the hypotenuse and acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the two triangles are congruent. The LA Theorem states that if one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent. The HL Postulate is based on SSA, a test that only works for right triangles. It states that if the hypotenuse and leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. The coordinate plane can be used in combination with algebra in a new method of proof called coordinate proof. Before beginning a coordinate proof, you will need to place the figure in the coordinate plane. It is important that you use coordinates that make computation as simple as possible. Using the origin as a vertex or center will help, and you should place at least one side of a polygon on an axis. If possible, keep the figure within the first quadrant. Once the triangle is placed, you can proceed with the proof. The Distance Formula, Slope Formula, and Midpoint Formula are often used in coordinate proof. Chapter 4 Congruent Triangles 176D and Assessment Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters ASSESSMENT INTERVENTION Type Student Edition Teacher Resources Ongoing Prerequisite Skills, pp. 177, 183, 191, 198, 206, 213, 221 Practice Quiz 1, p. 198 Practice Quiz 2, p. 221 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 1–2, 81–84 Quizzes, CRM pp. 239–240 Mid-Chapter Test, CRM p. 241 Study Guide and Intervention, CRM pp. 183–184, 189–190, 195–196, 201–202, 207–208, 213–214, 219–220 Mixed Review pp. 183, 191, 198, 206, 213, 221, 226 Cumulative Review, CRM p. 242 Error Analysis Find the Error, pp. 188, 203 Common Misconceptions, p. 178 Find the Error, TWE pp. 189, 203 Unlocking Misconceptions, TWE p. 202 Tips for New Teachers, TWE p. 210 Standardized Test Practice pp. 183, 191, 198, 206, 213, 217, 219, 221, 226, 231, 232, 233 TWE pp. 232–233 Standardized Test Practice, CRM pp. 243–244 Open-Ended Assessment Writing in Math, pp. 183, 191, 198, 205, 213, 221, 226 Open Ended, pp. 180, 188, 195, 203, 210, 219, 224 Standardized Test, p. 233 Modeling: TWE p. 198 Speaking: TWE pp. 183, 206, 221, 226 Writing: TWE pp. 191, 213 Open-Ended Assessment, CRM p. 237 Chapter Assessment Study Guide, pp. 227–230 Practice Test, p. 231 Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 225–230 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 231–236 Vocabulary Test/Review, CRM p. 238 For more information on Yearly ProgressPro, see p. 174. Geometry Lesson 4-1 4-2 4-3 4-4 4-5 4-6 4-7 Yearly ProgressPro Skill Lesson Classifying Triangles Angles of Triangles Congruent Triangles Proving Congruence—SSS, SAS Proving Congruence—ASA, AAS Isosceles Triangles Triangles and Coordinate Proof GeomPASS: Tutorial Plus, Lesson 10 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. 176E Chapter 4 Congruent Triangles Technology/Internet Reading and Writing in Mathematics Glencoe Geometry provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition Additional Resources • Foldables Study Organizer, p. 177 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 180, 188, 195, 203, 210, 219, 224) • Reading Mathematics, p. 199 • Writing in Math questions in every lesson, pp. 183, 191, 198, 205, 213, 221, 226 • Reading Study Tip, pp. 186, 207 • WebQuest, p. 216 Teacher Wraparound Edition • Foldables Study Organizer, pp. 177, 227 • Study Notebook suggestions, pp. 181, 184, 188, 194, 199, 203, 210, 214, 219, 224 • Modeling activities, p. 198 • Speaking activities, pp. 183, 206, 221, 226 • Writing activities, pp. 191, 213 • ELL Resources, pp. 176, 182, 190, 197, 199, 205, 212, 220, 225, 227 • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 4 Resource Masters, pp. vii-viii) • Proof Builder helps students learn and understand theorems and postulates from the chapter. (Chapter 4 Resource Masters, pp. ix–x) • Reading to Learn Mathematics master for each lesson (Chapter 4 Resource Masters, pp. 187, 193, 199, 205, 211, 217, 223) • 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. Lesson 4-1 Lesson 4-2 Lesson 4-7 Building on Prior Knowledge Using Manipulatives Flexible Groups If possible, pair each English Language Learner with a bilingual student. Give each pair of students posterboard to cut into several right triangles in different sizes, each with the right angle labeled. Have students measure the acute angles with a protractor. Ask them to explain in their own words why Corollaries 4.1 and 4.2 are true. Understanding how to best place a triangle on the coordinate plane can be a difficult concept for students. Divide the class into small groups. Give each group a coordinate plane made from posterboard, and several different triangles (isosceles, equilateral, right triangles, scalene, etc.). Have each group find as many ways as possible to place each triangle on the coordinate plane. Then have each group name the coordinates of the vertices of the triangles. Students often get confused as to what scale to use when measuring with a protractor. Ask students to label the angles as acute or obtuse before they begin to measure. Then they can determine which measure is reasonable for the angle they are measuring. Chapter 4 Congruent Triangles 176F Congruent Triangles Notes Have students read over the list of objectives and make a list of any words with which they are not familiar. 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. • Lesson 4-1 Classify triangles. • Lesson 4-2 Apply the Angle Sum Theorem and the Exterior Angle Theorem. • Lesson 4-3 Identify corresponding parts of congruent triangles. • Lessons 4-4 and 4-5 Test for triangle congruence using SSS, SAS, ASA, and AAS. • Lesson 4-6 Use properties of isosceles and equilateral triangles. • Lesson 4-7 Write coordinate proofs. Key Vocabulary • • • • • exterior angle (p. 186) flow proof (p. 187) corollary (p. 188) congruent triangles (p. 192) coordinate proof (p. 222) Triangles are found everywhere you look. Triangles with the same size and shape can even be found on the tail of a whale. You will learn more about orca whales in Lesson 4-4. Lesson 4-1 4-2 Preview 4-2 4-3 4-4 4-5 4-5 Follow-Up 4-6 4-7 NCTM Standards Local Objectives 3, 6, 8, 9, 10 3, 6, 10 3, 6, 8, 9, 10 3, 6, 8, 9, 10 3, 6, 7, 8, 9, 10 3, 6, 7, 8, 9, 10 3, 7, 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 176 Chapter 4 Congruent Triangles 176 Chapter 4 Congruent Triangles Daniel J. Cox/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 4 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 4 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 4. For Lesson 4-1 Solve Equations Solve each equation. (For review, see pages 737 and 738.) 1 1 1. 2x 18 5 ⫺6 2. 3m 16 12 9 2 23 3. 4y 12 16 1 4. 10 8 3z ⫺ 3 1 3 2 5. 6 2a 2 6. b 9 15 ⫺36 2 4 3 For Lessons 4-2, 4-4, and 4-5 Congruent Angles Name the indicated angles or pairs of angles if p 㛳 q and m 㛳 ᐉ. (For review, see Lesson 3-1.) 5 6 9 1 7. angles congruent to ⬔8 ⬔2, ⬔12, ⬔15, ⬔6, ⬔9, ⬔3, ⬔13 7 8 2 10 11 3 13 8. angles congruent to ⬔13 ⬔2, ⬔12, ⬔15, ⬔6, ⬔9, ⬔3, ⬔8 ⬔12, ⬔15 9. angles supplementary to ⬔1 ⬔6, ⬔9, ⬔3, ⬔13, ⬔2, ⬔8, ⬔7, ⬔10 10. angles supplementary to ⬔12 ⬔4, ⬔16, ⬔11, ⬔14, ⬔5, ⬔1, For Lessons 4-3 and 4-7 m 4 12 15 16 14 p ᐉ q This section provides a review of the basic concepts needed before beginning Chapter 4. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pages 1–2, 81–84. 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. For Lesson Prerequisite Skill 4-2 Angles Formed by Parallel Lines and a Transversal, p. 183 Properties of Congruence, p. 191 Distance Formula, p. 198 Bisectors of Segments and Angles, p. 206 Classification of triangles by sides, p. 213 Finding Midpoints, p. 221 Distance Formula 4-3 Find the distance between each pair of points. Round to the nearest tenth. (For review, see Lesson 1-3.) 11. (6, 8), (4, 3) 11.2 12. (15, 12), (6, 18) 21.8 13. (11, 8), (3, 4) 14.6 14. (10, 4), (8, 7) 21.1 4-4 4-5 4-6 Triangles Make this Foldable to help you organize your notes. Begin with two 4-7 sheets of grid paper and one sheet of construction paper. Fold and Cut Staple and Label Staple the edge to form a booklet. Then label each page with a lesson number and title. Stack the grid paper on the construction paper. Fold diagonally as shown and cut off the excess. Congruent Triangles Reading and Writing As you read and study the chapter, use your journal for sketches and examples of terms associated with triangles and sample proofs. Chapter 4 Congruent Triangles 177 TM For more information about Foldables, see Teaching Mathematics with Foldables. Writer’s Journal Use this Foldable for student writing about triangles. After students make their Foldable journal, have them label the pages to correspond to the seven lessons in this chapter. Students use their Foldable to take notes, define terms, record concepts, and write examples. Writer’s journals can also be used by students to record the direction and progress of learning, describe positive and negative experiences during learning, write about personal associations and experiences called to mind during learning, and to list examples of ways in which new knowledge has or will be used in their daily lives. Chapter 4 Congruent Triangles 177 Lesson Notes Classifying Triangles • Identify and classify triangles by angles. 1 Focus 5-Minute Check Transparency 4-1 Use as a quiz or review of Chapter 3. Mathematical Background notes are available for this lesson on p. 176C. are triangles important in construction? Ask students: • Why does the shape of a triangle offer good support for a roof? As gravity and the weight of roof material act on the two top legs of the triangle, the third leg prevents them from collapsing or spreading apart. Triangular supports within the truss offer even more support. • How would you classify the angles of a triangular truss with angle measures 96°, 42°, and 42°? obtuse, acute, acute • Identify and classify triangles by sides. Vocabulary • • • • • • • acute triangle obtuse triangle right triangle equiangular triangle scalene triangle isosceles triangle equilateral triangle are triangles important in construction? Many structures use triangular shapes as braces for construction. The roof sections of houses are made of triangular trusses that support the roof and the house. CLASSIFY TRIANGLES BY ANGLES Recall that a triangle is a three-sided polygon. Triangle ABC, written ABC, has parts that are named using the letters A, B, and C. B B, B C, and C A. • The sides of ABC are A • The vertices are A, B, and C. • The angles are ABC or B, BCA or C, A C and BAC or A. There are two ways to classify triangles. One way is by their angles. All triangles have at least two acute angles, but the third angle is used to classify the triangle. Study Tip Common Misconceptions These classifications are distinct groups. For example, a triangle cannot be right and acute. Classifying Triangles by Angles In an acute triangle , all of the angles are acute. In an obtuse triangle , one angle is obtuse. 67˚ 42˚ 13˚ 37˚ 76˚ all angle measures 90 In a right triangle , one angle is right. 142˚ 25˚ one angle measure 90 90˚ 48˚ one angle measure 90 An acute triangle with all angles congruent is an equiangular triangle . Example 1 Classify Triangles by Angles ARCHITECTURE The roof of this house is made up of three different triangles. Use a protractor to classify DFH, DFG, and HFG as acute, equiangular, obtuse, or right. DFH has all angles with measures less than 90, so it is an acute triangle. DFG and HFG both have one angle with measure equal to 90. Both of these are right triangles. 178 Chapter 4 Congruent Triangles (t)Martin Jones/CORBIS, (b)David Scott/Index Stock Resource Manager Workbook and Reproducible Masters Chapter 4 Resource Masters • Study Guide and Intervention, pp. 183–184 • Skills Practice, p. 185 • Practice, p. 186 • Reading to Learn Mathematics, p. 187 • Enrichment, p. 188 Prerequisite Skills Workbook, pp. 1–2 Teaching Geometry With Manipulatives Masters, pp. 1, 16, 69 Transparencies 5-Minute Check Transparency 4-1 Answer Key Transparencies Technology Interactive Chalkboard F D G H CLASSIFY TRIANGLES BY SIDES Triangles can also be classified according to the number of congruent sides they have. To indicate that sides of a triangle are congruent, an equal number of hash marks are drawn on the corresponding sides. Classifying Triangles by Sides No two sides of a scalene triangle are congruent. At least two sides of an isosceles triangle are congruent. All of the sides of an equilateral triangle are congruent. 2 Teach CLASSIFY TRIANGLES BY ANGLES In-Class Example Power Point® 1 ARCHITECTURE The triangular truss below is modeled for steel construction. Use a protractor to classify JMN, JKO, and OLN as acute, equiangular, obtuse or right. An equilateral triangle is a special kind of isosceles triangle. M L 1. Yes, all edges of the paper are an equal length. Equilateral Triangles Model Analyze 2–3. See students’ work. • Align three pieces of patty paper as indicated. Draw a dot at X. • Fold the patty paper through X and Y and through X and Z. 1. Is XYZ equilateral? Explain. J 2. Use three pieces of patty paper to make a triangle that is isosceles, but not equilateral. X Y K Z Example 2 Classify Triangles by Sides CLASSIFY TRIANGLES BY SIDES A In-Class Examples B C Power Point® Teaching Tip Explain to students that all equilateral triangles are isosceles, but not all isosceles triangles are equilateral. D E Example 3 Find Missing Values 2 Identify the indicated ALGEBRA Find x and the measure of each side of equilateral triangle RST if RS x 9, ST 2x, and RT 3x 9. Since RST is equilateral, RS ST. R triangles in the figure if V VX UX . U S x+9 x 9 2x Substitution 9 x Subtract x from each side. V 2x 3x – 9 Y T T Next, substitute to find the length of each side. RS x 9 ST 2x RT 3x 9 9 9 or 18 2(9) or 18 3(9) 9 or 18 For RST, x 9, and the measure of each side is 18. www.geometryonline.com/extra_examples N JMN is obtuse. JKO is right. OLN is equiangular. 3. Use three pieces of patty paper to make a scalene triangle. Identify the indicated type of triangle in the figure. a. isosceles triangles b. scalene triangles Isosceles triangles have Scalene triangles have at least two sides no congruent sides. congruent. So, ABD AEB, AED, ACB, and EBD are isosceles. ACD, BCE, and DCE are scalene. O Lesson 4-1 Classifying Triangles 179 Geometry Activity Materials: patty paper, pencil • Have students use a protractor to measure the angles of XYZ, and ask them to consider whether an equilateral triangle could be obtuse or right. • Students can fold a corner of the patty paper to form a right triangle. They can also use a ruler and protractor to draw and label different combinations of triangles, such as an acute scalene triangle, an obtuse isosceles triangle, and so on. U W Z X a. isosceles triangles UTX, UVX b. scalene triangles VYX, ZTX, VZU, YTU, VWX, ZUX, YXU 3 ALGEBRA Find d and the measure of each side of equilateral triangle KLM if KL d 2, LM 12 d, and KM 4d 13. d 5 and the measure of each side is 7. Lesson 4-1 Classifying Triangles 179 In-Class Example Example 4 Use the Distance Formula Power Point® 4 COORDINATE GEOMETRY Find the measures of the sides of RST. Classify the triangle by sides. Look Back COORDINATE GEOMETRY Find the measures of the sides of DEC. Classify the triangle by sides. Use the Distance Formula to find the lengths of each side. To review the Distance Formula, see Lesson 1-3. 2)2 (3 2 )2 EC (5 Study Tip y S (4, 4) y D ED (5 3)2 (3 9 )2 49 1 64 36 50 100 E C O DC (3 2 )2 (9 2)2 x 1 49 O 50 x Since E C and D C have the same length, DEC is isosceles. T (8,–1) R (–1, –3) RS 74 ; ST 41 ; RT 85 ; RST is scalene. Concept Check 1. Explain how a triangle can be classified in two ways. 1–2. See margin. 2. OPEN ENDED Determine whether each of the following statements is always, sometimes, or never true. Explain. 3. Always; equiangular triangles have three acute angles. 3. Equiangular triangles are also acute. 4. Right triangles are acute. Never; right triangles have one right angle and acute Answers 1. Triangles are classified by sides and angles. For example, a triangle can have a right angle and have no two sides congruent. 2. Sample answer: Draw a triangle that is isosceles and right. triangles have all acute angles. Guided Practice GUIDED PRACTICE KEY Exercises Examples 5–8 9, 10 11 12 1 3 4 2 Use a protractor to classify each triangle as acute, equiangular, obtuse, or right. obtuse 6. equiangular 5. 7. Identify the obtuse triangles if MJK KLM, mMJK 126, and mJNM 52. J K N M 8. Identify the right triangles if , G H DF , and G I EF . IJ 㛳 GH E MJK, KLM, JKN, LMN G D H L 9. ALGEBRA Find x, JM, MN, and JN if JMN is an isosceles triangle with JM M N . J J R 4x 3x – 9 x–2 F 10. ALGEBRA Find x, QR, RS, and QS if QRS is an equilateral triangle. M 2x – 5 GHD, GHJ, IJF, EIG I N x 4, JM 3, MN 3, JN 2 180 Chapter 4 Congruent Triangles 1 2 Q 2x + 1 6x – 1 Differentiated Instruction Naturalist Certain forms of algae are triangular in structure. Three-sided leaves are said to have a triangular shape. Some wings of birds and insects are triangular. Blue spruce trees grow in a triangular shape. Cats have triangular ears. Students can use these examples, find more throughout the chapter, or come up with their own ideas, and classify triangles found in nature. 180 Chapter 4 Congruent Triangles S x , QR 2, RS 2, QS 2 11. Find the measures of the sides of TWZ with vertices at T(2, 6), W(4, 5), and Z(3, 0). Classify the triangle. TW 125 , WZ 74 , TZ 61; scalene Application 12. QUILTING The star-shaped composite quilting square is made up of four different triangles. Use a ruler to classify the four triangles by sides. 8 scalene triangles (green), 8 isosceles triangles in the middle (blue), 4 isosceles triangles around the middle (yellow) and 4 isosceles at the corners of the square (purple) Study Notebook ★ indicates increased difficulty Practice and Apply For Exercises See Examples 13–18 19, 21–25 26–29 30, 31 32–37, 40, 41 1 1, 2 3 2 4 Use a protractor to classify each triangle as acute, equiangular, obtuse, or right. right 14. acute 15. acute 13. 16. Extra Practice 17. obtuse 18. right About the Exercises… 19. ASTRONOMY On May 5, 2002, Venus, Saturn, and Mars were aligned in a triangular formation. Use a protractor or ruler to classify the triangle formed by sides and angles. equilateral, equiangular Organization by Objective • Classify Triangles by Angles: 13–18, 19, 21 • Classify Triangles by Sides: 19, 21, 22–39 Mars Saturn Venus 20. RESEARCH Use the Internet or other resource to find out how astronomers can predict planetary alignment. See students’ work. 21. ARCHITECTURE The restored and decorated Victorian houses in San Francisco are called the “Painted Ladies.” Use a protractor to classify the triangles indicated in the photo by sides and angles. isosceles, acute 22. AGB, AGC, DGB, DGC The Painted Ladies are located in Alamo Square. The area is one of 11 designated historic districts in San Francisco. Source: www.sfvisitor.org 27. x 5, MN 9, MP 9, NP 9 28. x 8, QR 14, RS 14, QS 14 Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 4. • include any other item(s) that they find helpful in mastering the skills in this lesson. obtuse See page 760. Architecture 3 Practice/Apply Identify the indicated type of triangles in the figure if A BD DC CA BC AD B and . 22. right 23. obtuse BAC, CDB 24. scalene AGB, AGC, 25. isosceles ABD, ACD, DGB, DGC B A G D BAC, CDB ALGEBRA Find x and the measure of each side of the triangle. C 26. GHJ is isosceles, with HG JG , GH x 7, GJ 3x 5, and HJ x 1. x 6, GH 13, GJ 13, HJ 5 27. MPN is equilateral with MN 3x 6, MP x 4, and NP 2x 1. 28. QRS is equilateral. QR is two less than two times a number, RS is six more than the number, and QS is ten less than three times the number. LJ. JL is five less than two times a number. JK is 29. JKL is isosceles with KJ three more than the number. KL is one less than the number. Find the measure of each side. x 8, JL 11, JK 11, KL 7 Odd/Even Assignments Exercises 13–18, 22–29, and 32–41 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Exercise 20 requires the Internet or other research materials. Assignment Guide Basic: 13–37 odd, 42–57 Average: 13–41 odd, 42–57 Advanced: 14–40 even, 42–52 (optional: 53–57) Lesson 4-1 Classifying Triangles 181 Joseph Sohm/Stock Boston Interactive Chalkboard PowerPoint® Presentations 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 Lesson 4-1 Classifying Triangles 181 NAME ______________________________________________ DATE ★ 30. CRYSTAL The top of the crystal bowl shown is ____________ PERIOD _____ Study Guide andIntervention Intervention, 4-1 Study Guide and circular. The diameter at the top of the bowl is MN MN. P is the midpoint of M N , and O P . If MN 24 and OP 12, determine whether MPO and NPO are equilateral. p. 183 (shown) Classifying Triangles and p. 184 Classify Triangles by Angles One way to classify a triangle is by the measures of its angles. • If one of the angles of a triangle is an obtuse angle, then the triangle is an obtuse triangle. • If one of the angles of a triangle is a right angle, then the triangle is a right triangle. a. Lesson 4-1 • If all three angles of an acute triangle are congruent, then the triangle is an equiangular triangle. Classify each triangle. A B 60 E 120 35 D 25 F The triangle has one angle that is obtuse. It is an obtuse triangle. c. G 90 H 60 30 J The triangle has one right angle. It is a right triangle. Exercises Classify each triangle as acute, equiangular, obtuse, or right. 1. K 2. N 30 67 L 23 65 65 Lexington to Nashville, 265 miles from Cairo to Lexington, and 144 miles from Cairo to Nashville. Q 60 60 R obtuse 60 S equiangular 5. W T 6. B 60 45 50 X V acute Gl 3. O P M right 4. U 30 120 90 45 90 28 F Y right 92 D obtuse NAME ______________________________________________ DATE /M G Hill 183 ____________ Gl PERIOD G _____ Skills Practice, 4-1 Practice (Average) p. 185 and Practice, 186 (shown) Classifyingp. Triangles Use a protractor to classify each triangle as acute, equiangular, obtuse, or right. 1. 2. 3. obtuse acute B E 5. obtuse ABC, CDE BED, BDC 6. scalene A D C 7. isosceles ABC, CDE 32. AB 29 , BC 4, AC 29 ; isosceles Find the measures of the sides of ABC and classify each triangle by its sides. 32. A(5, 4), B(3, 1), C(7, 1) 33. A(4, 1), B(5, 6), C(3, 7) 34. A(7, 9), B(7, 1), C(4, 1) 35. A(3, 1), B(2, 1), C(2, 3) 33. AB 106 , BC 233 , AC 65; scalene 36. A(0, 5), B(53, 2), C(0,1) 37. A(9, 0), B5, 63 , C(1, 0) ABD, BED, BDC ALGEBRA Find x and the measure of each side of the triangle. 8. FGH is equilateral with FG x 5, GH 3x 9, and FH 2x 2. x 7, FG 12, GH 12, FH 12 9. LMN is isosceles, L is the vertex angle, LM 3x 2, LN 2x 1, and MN 5x 2. x 3, LM 7, LN 7, MN 13 Find the measures of the sides of KPL and classify each triangle by its sides. 10. K(3, 2) P(2, 1), L(2, 3) KP 26 , PL 42 , LK 26 ; isosceles 11. K(5, 3), P(3, 4), L(1, 1) KP 53 , PL 5, LK 213 ; scalene ★ ★ 34. AB 10, BC 11, AC 221 ; scalene right Identify the indicated type of triangles if A B A D B D D C , BE E D , AB ⊥B C , and ED ⊥D C . 4. right 38. PROOF Write a two-column proof to prove that EQL is equiangular. See p. 233A. 35. AB 29 , BC 4, AC 29 ; isosceles E M Q N P L R U I 37. AB 124 , 40. COORDINATE GEOMETRY BC 124 , AC 8; ★ Show that S is the midpoint of isosceles R T and U is the midpoint of T V . ★ 41. COORDINATE GEOMETRY Show that ADC is isosceles. See p. 233A. See p. 233A. KP 210 , PL 52 , LK 52 ; isosceles T (4, 14) y 13. DESIGN Diana entered the design at the right in a logo contest sponsored by a wildlife environmental group. Use a protractor. How many right angles are there? 5 NAME ______________________________________________ DATE /M G Hill 186 y D 12 S (7, 8) ____________ Gl PERIOD G _____ Reading 4-1 Readingto to Learn Learn Mathematics ELL Mathematics, p. 187 Classifying Triangles Pre-Activity 39. PROOF Write a paragraph proof to prove that RPM is an obtuse triangle if mNPM 33. See p. 233A. 33˚ 36. AB 84 , BC 84, AC 6; isosceles 12. K(2, 6), P(4, 0), L(3, 1) Gl O 31. MAPS The total distance from Nashville, Lexington Tennessee, to Cairo, Illinois, to Lexington, Kentucky, and back to Nashville, Tennessee, is 593 miles. The distance from Cairo to Cairo Lexington is 81 more miles than the distance from Lexington to Nashville. The Nashville distance from Cairo to Nashville is 40 miles less than the distance from Nashville to Lexington. Classify the triangle formed by its sides. Scalene; it is 184 miles from C All three angles are congruent, so all three angles have measure 60°. The triangle is an equiangular triangle. b. N P No, MO NO 288 • If all three of the angles of a triangle are acute angles, then the triangle is an acute triangle. Example M U (0, 8) R (10, 2) Why are triangles important in construction? 8 4 Read the introduction to Lesson 4-1 at the top of page 178 in your textbook. • Why are triangles used for braces in construction rather than other shapes? ( a2 , b) 4 V (4, 2) O 4 8 (0, 0) A x O C (a, 0) x Sample answer: Triangles lie in a plane and are rigid shapes. • Why do you think that isosceles triangles are used more often than scalene triangles in construction? Sample answer: Isosceles 42. CRITICAL THINKING K L is a segment representing one side of isosceles right LM triangle KLM, with K(2, 6), and L(4, 2). KLM is a right angle, and K L . Describe how to find the coordinates of vertex M and name these coordinates. triangles are symmetrical. Reading the Lesson 1. Supply the correct numbers to complete each sentence. a. In an obtuse triangle, there are 2 acute angle(s), 0 right angle(s), and Use the Distance Formula and Slope Formula; (0, 0) or (8, 4). 1 obtuse angle(s). b. In an acute triangle, there are 3 acute angle(s), 0 right angle(s), and 0 obtuse angle(s). c. In a right triangle, there are 2 acute angle(s), 182 Chapter 4 Congruent Triangles 1 right angle(s), and 0 obtuse angle(s). 2. Determine whether each statement is always, sometimes, or never true. a. A right triangle is scalene. sometimes NAME ______________________________________________ DATE b. An obtuse triangle is isosceles. sometimes 4-1 Enrichment Enrichment, c. An equilateral triangle is a right triangle. never d. An equilateral triangle is isosceles. always ____________ PERIOD _____ p. 188 e. An acute triangle is isosceles. sometimes f. A scalene triangle is obtuse. sometimes Reading Mathematics 3. Describe each triangle by as many of the following words as apply: acute, obtuse, right, scalene, isosceles, or equilateral. a. b. 70 80 30 acute, scalene c. 135 obtuse, isosceles 4 3 5 right, scalene Helping You Remember 4. A good way to remember a new mathematical term is to relate it to a nonmathematical definition of the same word. How is the use of the word acute, when used to describe acute pain, related to the use of the word acute when used to describe an acute angle or an acute triangle? Sample answer: Both are related to the meaning of acute as sharp. An acute pain is a sharp pain, and an acute angle can be thought of as an angle with a sharp point. In an acute triangle all of the angles are acute. 182 Chapter 4 Congruent Triangles When you read geometry, you may need to draw a diagram to make the text easier to understand. Example Consider three points, A, B, and C on a coordinate grid. The y-coordinates of A and B are the same. The x-coordinate of B is greater than the x-coordinate of A. Both coordinates of C are greater than the corresponding coordinates of B. Is triangle ABC acute, right, or obtuse? To answer this question, first draw a sample triangle that fits the description. Side AB must be a horizontal segment because the y-coordinates are the same. Point C must be located to the right and up from point B. From the diagram you can see that triangle ABC must be obtuse. y Q A B O x 43. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. Why are triangles important in construction? 4 Assess Include the following in your answer: • describe how to classify triangles, and • if one type of triangle is used more often in architecture than other types. Standardized Test Practice Open-Ended Assessment 44. Classify ABC with vertices A(1, 1), B(1, 3), and C(3, 1). C A scalene acute B equilateral C isosceles acute D isosceles right 45. ALGEBRA Find the value of y if the mean of x, y, 15, and 35 is 25 and the mean of x, 15, and 35 is 27. B A 18 B 19 C 31 D 36 Maintain Your Skills Mixed Review Graph each line. Construct a perpendicular segment through the given point. Then find the distance from the point to the line. (Lesson 3-6) 46. y x 2, (2, 2) 18 47. x y 2, (3, 3) 46–48. See margin for graphs. Find x so that p 㛳 q . (Lesson 3-5) 49. 15 50. 45 p q p 110˚ q 8 48. y 7, (6, 2) 9 Getting Ready for Lesson 4-2 51. 44 Prerequisite Skill Students will learn about angles of triangles in Lesson 4-2. They will use angle relationships with the Angle Sum Theorem and the Exterior Angle Theorem to find angle measures. Use Exercises 53–57 to determine your students’ familiarity with angles formed by parallel lines and a transversal. 57˚ p (3x – 50)˚ (4x + 10)˚ (2x – 5)˚ q (3x – 9)˚ For this proof, the reasons in the right column are not in the proper order. Reorder the reasons to properly match the statements in the left column. (Lesson 2-6) 52. Given: 3x 4 x 10 Order should be: 1. Given Prove: x 3 Proof: Statements a. 3x 4 x 10 b. 2x 4 10 c. 2x 6 d. x 3 Getting Ready for the Next Lesson 53. any three: 2 and 11, 3 and 6, 4 and 7, 3 and 12, 7 and 10, 8 and 11 54. 1 and 4, 1 and 10, 5 and 2, 5 and 8, 9 and 6, 9 and 12 Speaking Ask students to call out the classifications of selected triangles from the book or on the board by both angles and sides. Provide angle measures, side measures, congruency tick-marks, or right angle symbols for figures on the board. Label the triangles, and encourage students to use proper terminology to refer to the triangle, its angles, and its segments. 2. Subtraction Property 3. Addition Property 4. Division Property 46. Reasons 1. Subtraction Property 2. Division Property 3. Given 4. Addition Property y yx2 BC AC RQ PR PQ PREREQUISITE SKILL In the figure, A B 㛳 , 㛳 , and 㛳 . Name the indicated angles or pairs of angles. x O (2, –2) (To review angles formed by parallel lines and a transversal, see Lessons 3-1 and 3-2.) 53. 54. 55. 56. 57. three pairs of alternate interior angles six pairs of corresponding angles all angles congruent to 3 6, 9, and 12 all angles congruent to 7 1, 4, and 10 all angles congruent to 11 2, 5, and 8 www.geometryonline.com/self_check_quiz R 47. 9 C 8 6 Q 5 10 11 7 4 3 2 B y A (3, 3) 12 1 xy2 P x O Lesson 4-1 Classifying Triangles 183 Answers 43. Sample answer: Triangles are used in construction as structural support. Answers should include the following. • Triangles can be classified by sides and angles. If the measure of each angle is less than 90, the triangle is acute. If the measure of one angle is greater than 90, the triangle is obtuse. If one angle equals 90°, the triangle is right. If each angle has the same measure, the triangle is equiangular. If no two sides are congruent, the triangle is scalene. If at least two sides are congruent, it is isosceles. If all of the sides are congruent, the triangle is equilateral. • Isosceles triangles seem to be used more often in architecture and construction. 48. y y7 x O (6, –2) Lesson 4-1 Classifying Triangles 183 Geometry Activity A Preview of Lesson 4-2 Getting Started A Preview of Lesson 4-2 Angles of Triangles There are special relationships among the angles of a triangle. scissors Step 2 Step 3 Teach • Advise students to label the obtuse angle B when they are first working through Activity 1. They can also repeat Activity 1 using one of the acute angles as angle B to further verify concepts. • As an extension, students can repeat Activity 1 using an acute triangle or a right triangle. They can also cut along the segments DF, FE, and DE and arrange angles A, B, and C over their congruent counterparts in DEF to see that the angles have equal measures. Assess In Exercises 1–12 students determine angle measures of the triangles used in this activity, find relationships, and make conjectures that will lead them to the Angle Sum Theorem and the Exterior Angle Theorem. Step 4 Step 5 Step 6 E C Analyze the Model Describe the relationship between each pair. 1. A and DFA congruent 2. B and DFE congruent 3. C and EFC congruent 4. What is the sum of the measures of DFA, DFE, and EFC? 180 5. What is the sum of the measures of A, B, and C? 180 6. Make a conjecture about the sum of the measures of the angles of any triangle. The sum of the measures of the angles of any triangle is 180. In the figure at the right, 4 is called an exterior angle of the triangle. 1 and 2 are the remote interior angles of 4. Step 1 Step 2 Step 3 Step 4 2 1 Find the relationship among the interior and exterior angles of a triangle. Trace ABC from Activity 1 onto a piece of paper. Label the vertices. Extend A C to draw an exterior angle at C. Tear A and B off the triangle from Activity 1. Place A and B over the exterior angle. 4 3 E D F C A Materials protractor Step 1 Find the relationship among the measures B of the interior angles of a triangle. D Draw an obtuse triangle and cut it out. Label the vertices A, B, and C. F A Find the midpoint of A B by matching A to B. Label this point D. Find the midpoint of B C by matching B to C. Label this point E. Draw D E . Fold ABC along DE . Label the point where B touches A C as F. Draw D F and F E . Measure each angle. Analyze the Model 7. mA mB is the measure of the exterior angle at C. 7. 8. 9. 10. 11. 12. Make a conjecture about the relationship of A, B, and the exterior angle at C. Repeat the steps for the exterior angles of A and B. See students’ work. Is your conjecture true for all exterior angles of a triangle? yes Repeat Activity 2 with an acute triangle. 10–11. See students’ work. Repeat Activity 2 with a right triangle. Make a conjecture about the measure of an exterior angle and the sum of the measures of its remote interior angles. The measure of an exterior angle is equal to the sum of measures of 184 Chapter 4 Congruent Triangles the two remote interior angles. Resource Manager Study Notebook Ask students to summarize what they have learned about the relationships among the measures of the interior and exterior angles of triangles. 184 Chapter 4 Congruent Triangles B Objective Find the relationships among the measures of the interior and exterior angles of a triangle. Teaching Geometry with Manipulatives Glencoe Mathematics Classroom Manipulative Kit • p. 70 (student recording sheet) • p. 16 (protractor) • protractor • scissors Lesson Notes Angles of Triangles • Apply the Angle Sum Theorem. 1 Focus • Apply the Exterior Angle Theorem. Vocabulary • • • • exterior angle remote interior angles flow proof corollary are the angles of triangles used to make kites? 5-Minute Check Transparency 4-2 Use as a quiz or review of Lesson 4-1. The Drachen Foundation coordinates the annual Miniature Kite Contest. This kite won second place in the Most Beautiful Kite category in 2001. The overall dimensions are 10.5 centimeters by 9.5 centimeters. The wings of the beetle are triangular. Mathematical Background notes are available for this lesson on p. 176C. ANGLE SUM THEOREM If the measures of two of the angles of a triangle are known, how can the measure of the third angle be determined? The Angle Sum Theorem explains that the sum of the measures of the angles of any triangle is always 180. Theorem 4.1 Angle Sum Theorem The sum of the X measures of the angles of a triangle is 180. Example: mW mX mY 180 W Proof Study Tip Angle Sum Theorem Look Back Given: ABC Recall that sometimes extra lines have to be drawn to complete a proof. These are called auxiliary lines. Prove: mC m2 mB 180 X A Y 1 2 3 C Proof: Statements 1. ABC CB 2. Draw XY through A parallel to . 3. 1 and CAY form a linear pair. 4. 1 and CAY are supplementary. 5. 6. 7. 8. 9. 10. Y m1 mCAY 180 mCAY m2 m3 m1 m2 m3 180 1 C, 3 B m1 mC, m3 mB mC m2 mB 180 are the angles of triangles used to make kites? Ask students: • Assuming that the wings are equal in size and the angle between the two wings is 90°, what type of triangle is formed if you draw a line to connect one wing-tip to the other wing-tip? right isosceles triangle • Are the wings of a real beetle perfectly triangular in shape? no B Reasons 1. Given 2. Parallel Postulate 3. Def. of a linear pair 4. If 2 form a linear pair, they are supplementary. 5. Def. of suppl. 6. Angle Addition Postulate 7. Substitution 8. Alt. Int. Theorem 9. Def. of 10. Substitution Lesson 4-2 Angles of Triangles 185 Courtesy The Drachen Foundation Resource Manager Workbook and Reproducible Masters Chapter 4 Resource Masters • Study Guide and Intervention, pp. 189–190 • Skills Practice, p. 191 • Practice, p. 192 • Reading to Learn Mathematics, p. 193 • Enrichment, p. 194 • Assessment, p. 239 Graphing Calculator and Computer Masters, p. 23 Prerequisite Skills Workbook, pp. 81–84 Teaching Geometry With Manipulatives Masters, p. 71 Transparencies 5-Minute Check Transparency 4-2 Answer Key Transparencies Technology Interactive Chalkboard Lesson x-x Lesson Title 185 If we know the measures of two angles of a triangle, we can find the measure of the third. 2 Teach Example 1 Interior Angles ANGLE SUM THEOREM In-Class Example Find the missing angle measures. Find m1 first because the measures of two angles of the triangle are known. Power Point® m1 110 180 measures. m1 70 79 1 74 28˚ m1 28 82 180 Angle Sum Theorem 1 Find the missing angle 43 82˚ 1 2 Simplify. 3 1 and 2 are congruent vertical angles. So m2 70. 2 3 m3 68 70 180 m3 138 180 m1 63; m2 63; m3 38 m3 42 68˚ Subtract 110 from each side. Angle Sum Theorem Simplify. Subtract 138 from each side. Therefore, m1 70, m2 70, and m3 42. The Angle Sum Theorem leads to a useful theorem about the angles in two triangles. Theorem 4.2 Third Angle Theorem If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. C A B D F C A E B D F E 3 remote interior angles Example: If A F and C D, then B E. You will prove this theorem in Exercise 44. EXTERIOR ANGLE THEOREM Study Tip Reading Math Remote means far away and interior means inside. The remote interior angles are the interior angles farthest from the exterior angle. Each angle of a triangle has an exterior angle. An exterior angle is formed by one side of a triangle and the extension of another side. The interior angles of the triangle not adjacent to a given exterior angle are called remote interior angles of the exterior angle. exterior angle 2 1 Theorem 4.3 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. Example: mYZP mX mY Y X Z P 186 Chapter 4 Congruent Triangles Differentiated Instruction Visual/Spatial Tell students that the Angle Sum Theorem and Exterior Angle Theorem are both based on the idea that a straight angle measures 180°. Show that if they cut the angles of any triangle and place them right next to one another, they form a straight line. This visually demonstrates why the sum of the interior angles of a triangle measures 180°. 186 Chapter 4 Congruent Triangles We will use a flow proof to prove this theorem. A flow proof organizes a series of statements in logical order, starting with the given statements. Each statement is written in a box with the reason verifying the statement written below the box. Arrows are used to indicate how the statements relate to each other. Proof Exterior Angle Theorem EXTERIOR ANGLE THEOREM In-Class Example 2 Find the measure of each numbered angle in the figure. C Write a flow proof of the Exterior Angle Theorem. Given: ABC Prove: mCBD mA mC 5 Flow Proof: D ABC CBD and ABC form a linear pair. Given Definition of linear pair B A If 2 s form a linear pair, they are supplementary. mA mABC C 180 mCBD mABC = 180 Definition of supplementary 2 1 29 In Chapter 3, students used angle relationships to find angle measures. In this lesson, students will apply their knowledge of vertical angles, supplementary angles, and complementary angles along with the Angle Sum Theorem and the Exterior Angle Theorem to find angle measures in figures. Substitution Property mA mC mCBD Subtraction Property Example 2 Exterior Angles 128 41 64 Building on Prior Knowledge mA mABC mC mCBD mABC Find the measure of each numbered angle in the figure. Exterior Angle Theorem m1 50 78 38 4 3 32 m1 70, m2 110, m3 46, m4 102, and m5 37. CBD and ABC are supplementary. Angle Sum Theorem Power Point® 3 2 50˚ Simplify. 1 78˚ 120˚ 4 5 56˚ m1 m2 180 If 2 form a linear pair, they are suppl. 128 m2 180 Substitution m2 52 Subtract 128 from each side. m2 m3 120 Exterior Angle Theorem 52 m3 120 Substitution m3 68 120 m4 180 m4 60 m5 m4 56 Subtract 52 from each side. If 2 form a linear pair, they are suppl. Subtract 120 from each side. Exterior Angle Theorem 60 56 Substitution 116 Simplify. Therefore, m1 128, m2 52, m3 68, m4 60, and m5 116. www.geometryonline.com/extra_examples Lesson 4-2 Angles of Triangles 187 Lesson 4-2 Angles of Triangles 187 In-Class Example A statement that can be easily proved using a theorem is often called a corollary of that theorem. A corollary, just like a theorem, can be used as a reason in a proof. Power Point® 3 GARDENING The flower bed Corollaries shown is in the shape of a right triangle. Find mA if mC is 20. B 4.1 C 4.2 The acute angles of a right triangle are complementary. G There can be at most one right or obtuse angle in a triangle. P K acute ❀ ❀ ❀ ❀ ❀ ❀ ❀ 20 ✿ ✿ ✿ ✿ ✿ ✿ ✿ ❀ ❀ ❀ ❀ ❀ ❀ ✿ ✿ ✿ ✿ ❀ ❀ ❀ ✿ 142˚ H A M J Q L R acute Example: mG mJ 90 mA 70 You will prove Corollaries 4.1 and 4.2 in Exercises 42 and 43. Example 3 Right Angles 3 Practice/Apply SKI JUMPING Ski jumper Simon Ammann of Switzerland forms a right triangle with his skis and his line of sight. Find m2 if m1 is 27. Use Corollary 4.1 to write an equation. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 4. • include any other item(s) that they find helpful in mastering the skills in this lesson. Answer 1. Sample answer: 2 and 3 are the remote interior angles of exterior 1. 1 m1 m2 90 27 m2 90 Substitution m2 63 Subtract 27 from each side. Concept Check 2. Najee; the sum of the measures of the remote interior angles is equal to the measure of the corresponding exterior angle. Guided Practice GUIDED PRACTICE KEY Exercises Examples 3–4 5–7 8–10 1 2 3 2 1 2 3 1. OPEN ENDED Draw a triangle. Label one exterior angle and its remote interior angles. See margin. 2. FIND THE ERROR Najee and Kara are discussing the Exterior Angle Theorem. 2 1 3 4 Najee Kara mΩ1 + mΩ2 = mΩ4 mΩ1 + mΩ2 + mΩ4 = 180 Who is correct? Explain your reasoning. Find the missing angle measure. 43 3. Toledo 85˚ Pittsburgh 62˚ 52˚ Cincinnati Find each measure. 5. m1 55 6. m2 33 7. m3 147 188 Chapter 4 Congruent Triangles Adam Pretty/Getty Images 188 Chapter 4 Congruent Triangles 99 4. 32˚ 23˚ 1 2 3 22˚ 19˚ Find each measure. 8. m1 65 9. m2 25 D 1 25˚ 65˚ E Application FIND THE ERROR Explain to students that Kara’s equation could only be true in the special case when m3 m4 90. 2 G F 10. SKI JUMPING American ski jumper Eric Bergoust forms a right angle with his skis. If m2 70, find m1. 20 2 About the Exercises… 1 Organization by Objective • Angle Sum Theorem: 11–17 • Exterior Angle Theorem: 18–38 ★ indicates increased difficulty Practice and Apply For Exercises See Examples 11–17 18–31 32–35 36–38 1 2 3 2 Find the missing angle measures. 11. 93 40˚ Odd/Even Assignments Exercises 11–38 are structured so that students practice the same concepts whether they are assigned odd or even problems. 70.5, 70.5 12. 47˚ Assignment Guide Extra Practice See page 761. 39˚ 65, 65 13. 63 14. 50˚ 27˚ Find each measure. 15. m1 76 16. m2 76 17. m3 49 55˚ 47˚ 1 2 57˚ 3 Find each measure if m4 m5. 19. m2 53 18. m1 64 20. m3 116 21. m4 32 22. m5 32 23. m6 44 24. m7 89 Find each measure. 25. m1 123 26. m2 28 27. m3 14 Basic: 11–35 odd, 41, 43, 45, 46–64 Average: 11–45 odd, 46–64 Advanced: 12–44 even, 45–58 (optional: 59–64) 95˚ 7 63˚ 2 69˚ 47˚ 1 5 6 136˚ 3 4 109˚ 1 2 33˚ 3 24˚ Lesson 4-2 Angles of Triangles 189 Doug Pensinger/Getty Images Lesson 4-2 Angles of Triangles 189 NAME ______________________________________________ DATE SPEED SKATING For Exercises 28–31, use the following information. Speed skater Catriona Lemay Doan of Canada forms at least two sets of triangles and exterior angles as she skates. Use the measures of given angles to find each measure. ____________ PERIOD _____ Study Guide andIntervention Intervention, 4-2 Study Guide and p. 189 Angles(shown) of Triangles and p. 190 Angle Sum Theorem If the measures of two angles of a triangle are known, the measure of the third angle can always be found. Angle Sum Theorem The sum of the measures of the angles of a triangle is 180. In the figure at the right, mA mB mC 180. B A Example 1 28. 29. 30. 31. Find the missing angle measures. S B 35 90 T mR mS mT 180 25 35 mT 180 60 mT 180 mT 120 A 58 C 1 2 Angle Sum Theorem Substitution Add. Subtract 60 from each side. D 3 108 E m1 mA mB m1 58 90 m1 148 m1 180 180 180 32 Angle Sum Theorem Substitution Add. Subtract 148 from each side. m2 32 m3 m2 mE m3 32 108 m3 140 m3 Vertical angles are congruent. 180 180 180 40 Lesson 4-2 25 R C Example 2 Find mT. m1 m2 m3 m4 1 54 53 137 103 126˚ 2 73˚ 3 43˚ 34˚ 4 Online Research Data Update Use the Internet or other resource to find the world record in speed skating. Visit www.geometryonline.com/data_update to learn more. Angle Sum Theorem Substitution Add. Subtract 140 from each side. Exercises Find the measure of each numbered angle. 1. m1 28 M 62 90 1 P 60 W U 1 30 T 4. M 3 66 1 58 m1 30, m2 60 R 1 2 30 Q 6. A 50 Catriona Lemay Doan is the first Canadian to win a Gold medal in the same event in two consecutive Olympic games. m1 8 20 152 1 G D NAME ______________________________________________ DATE /M G Hill 189 Gl m1 56, m2 56, m3 74 N 2 O S Speed Skating R P T 60 W Find each measure if mDGF 53 and mAGC 40. 33. m2 50 32. m1 37 34. m3 50 35. m4 40 1 2 5. m1 120 S 30 Q m1 30, m2 60 3. V 2. N ____________ Gl PERIOD G _____ Skills Practice, p. 191 and 4-2 Practice (Average) Practice, p. 192 (shown) Angles of Triangles 18 72 40 ★ 36. m1 53 ★ 37. m2 129 ★ 38. m3 153 55 Find the measure of each angle. 3 58 3. m1 97 1 2 B HOUSING For Exercises 36–38, use the following information. The two braces for the roof of a house form triangles. Find each measure. 85 2. 4 3 G Source: www.catrionalemaydoan. com ? A F 1 2 Find the missing angle measures. 1. D C 101˚ 128˚ 3 26˚ 103˚ 2 1 35 4. m2 83 39 5. m3 62 Find the measure of each angle. ★ 5 2 6. m1 104 3 1 7. m4 45 70 36 118 6 4 65 68 39. flow proof Given: FGI IGH F H GI Prove: F H 82 8. m3 65 9. m2 79 10. m5 73 11. m6 147 Find the measure of each angle if BAD and BDC are right angles and mABC 84. B For Exercises 39–44, write the specified type of proof. 39–44. See PROOF Given: ABCD is a quadrilateral. Prove: mDAB mB mBCD mD 360 A G 1 pp. 233A–233B. ★ 40. two-column B 64 C 12. m1 26 2 A D 13. m2 32 14. CONSTRUCTION The diagram shows an example of the Pratt Truss used in bridge construction. Use the diagram to find m1. 145 F NAME ______________________________________________ DATE /M G Hill 192 Gl ELL Mathematics, p. 193 Angles of Triangles How are the angles of triangles used to make kites? The frame of the simplest kind of kite divides the kite into four triangles. Describe these four triangles and how they are related to each other. Sample answer: There are two pairs of right triangles that have the same size and shape. 42. flow proof of Corollary 4.1 44. two-column proof of Theorem 4.2 m1 48, m2 60, m3 72 Reading the Lesson 1. Refer to the figure. 1 E A a. Name the three interior angles of the triangle. (Use three letters to name each angle.) BAC, ABC, BCA D B 39 b. Name three exterior angles of the triangle. (Use three letters to name each angle.) EAB, DBC, FCA c. Name the remote interior angles of EAB. ABC, BCA A 23 C F d. Find the measure of each angle without using a protractor. ii. ABC 118 iii. ACF 157 190 Chapter 4 Congruent Triangles Jed Jacobsohn/Getty Images iv. EAB 141 2. Indicate whether each statement is true or false. If the statement is false, replace the underlined word or number with a word or number that will make the statement true. a. The acute angles of a right triangle are supplementary. false; complementary b. The sum of the measures of the angles of any triangle is 100. false; 180 c. A triangle can have at most one right angle or acute angle. false; obtuse d. If two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent. true e. The measure of an exterior angle of a triangle is equal to the difference of the measures of the two remote interior angles. false; sum NAME ______________________________________________ DATE 4-2 Enrichment Enrichment, ____________ PERIOD _____ p. 194 Finding Angle Measures in Triangles You can use algebra to solve problems involving triangles. f. If the measures of two angles of a triangle are 62 and 93, then the measure of the third angle is 35. false; 25 Example In triangle ABC, mA, is twice mB, and mC is 8 more than mB. What is the measure of each angle? g. An exterior angle of a triangle forms a linear pair with an interior angle of the triangle. true Write and solve an equation. Let x mB. Helping You Remember 3. Many students remember mathematical ideas and facts more easily if they see them demonstrated visually rather than having them stated in words. Describe a visual way to demonstrate the Angle Sum Theorem. Sample answer: Cut off the angles of a triangle and place them side-by-side on one side of a line so that their vertices meet at a common point. The result will show three angles whose measures add up to 180. 190 D H and BC are opposite 45. CRITICAL THINKING BA rays. The measures of 1, 2, and 3 are in a 4:5:6 ratio. Find the measure of each angle. Read the introduction to Lesson 4-2 at the top of page 185 in your textbook. i. DBC 62 I 41. two-column proof of Theorem 4.3 43. paragraph proof of Corollary 4.2 ____________ Gl PERIOD G _____ Reading 4-2 Readingto to Learn Learn Mathematics Pre-Activity C 1 55 Chapter 4 Congruent Triangles mA mB mC 180 2x x (x 8) 180 4x 8 180 4x 172 x 43 So, m A 2(43) or 86, mB 43, and mC 43 8 or 51. Solve each problem. 1. In triangle DEF, mE is three times mD, and mF is 9 less than mE. What is the measure of each angle? 2. In triangle RST, mT is 5 more than mR, and mS is 10 less than mT. What is the measure of each angle? 2 B 3 C 46. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How are the angles of triangles used to make kites? 4 Assess Include the following in your answer: • if two angles of two triangles are congruent, how you can find the measure of the third angle, and • if one angle measures 90, describe the other two angles. Standardized Test Practice 47. In the triangle, what is the measure of Z? A A 18 B 24 C 72 D X 2a˚ 90 a˚ 2 Y Z 48. ALGEBRA The measure of the second angle of a triangle is three times the measure of the first, and the measure of the third angle is 25 more than the measure of the first. Find the measure of each angle. B A 25, 85, 70 B 31, 93, 56 C 39, 87, 54 D 42, 54, 84 Identify the indicated type of triangle if AD EB EC AC B C , , bisects B D , and mAED 125. (Lesson 4-1) 49. scalene triangles AED 50. obtuse triangles BEC, AED 51. isosceles triangles BEC B C E A 125˚ D 53. 20 units Find the distance between each pair of parallel lines. (Lesson 3-6) 52. y x 6, y x 10 128 units 53. y 2x 3, y 2x 7 54. 4x y 20, 4x y 3 17 units 55. 2x 3y 9, 2x 3y 6 117 units 13 13 Find x, y, and z in each figure. (Lesson 3-2) 56. z˚ 57. 58. (5z 2)˚ 3x˚ 42˚ (4x 6)˚ (2y 8)˚ Getting Ready for the Next Lesson 142˚ x 34, y 15, z 142 68˚ y˚ x 112, y 28, z 22 48˚ z˚ x 16, y 90, z 42 PREREQUISITE SKILL Name the property of congruence that justifies each statement. (To review properties of congruence, see Lessons 2-5 and 2-6.) 59. 1 1 and AB AB . reflexive 60. If A B X Y, then X Y A B. symmetric 61. If 1 2, then 2 1. symmetric 62. If 2 3 and 3 4, then 2 4. transitive 63. If P Q X Y and X Y H K, then P Q H K. transitive 64. If A B C D, C D P Q, and P Q X Y, then A B X Y. transitive www.geometryonline.com/self_check_quiz Prerequisite Skill Students will learn about congruent triangles in Lesson 4-3. They will use properties of congruent segments and angles to identify corresponding parts of congruent triangles and to prove congruency between a triangle and its transformed image. Use Exercises 59–64 to determine your students’ familiarity with properties of congruence for segments and angles. Assessment Options x˚ 4y˚ Writing Draw an acute triangle with two angles that measure 44° and 56°, an obtuse triangle with angles 110° and 40°, and an isosceles triangle sitting on a line with two angles measuring 75°. Ask students to use the theorems in this lesson to find the missing angle measures in each triangle and then write a paragraph summarizing how they found the measures. Getting Ready for Lesson 4-3 Maintain Your Skills Mixed Review Open-Ended Assessment Lesson 4-2 Angles of Triangles Quiz (Lessons 4-1 and 4-2) is available on p. 239 of the Chapter 4 Resource Masters. Answer 191 46. Sample answer: The shape of a kite is symmetric. If triangles are used on one side of the kite, congruent triangles are used on the opposite side. The wings of this kite are made from congruent right triangles. Answers should include the following. • By the Third Angle Theorem, if two angles of two congruent triangles are congruent, then the third angles of each triangle are congruent. • If one angle measures 90, the other two angles are both acute. Lesson 4-2 Angles of Triangles 191 Lesson Notes Congruent Triangles • Name and label corresponding parts of congruent triangles. 1 Focus 5-Minute Check Transparency 4-3 Use as a quiz or review of Lesson 4-2. • Identify congruence transformations. Vocabulary • congruent triangles • congruence transformations Mathematical Background notes are available for this lesson on p. 176C. are triangles used in bridges? Ask students: • What types of triangles do you notice in the construction of the bridge? acute triangles • What do you notice about the size and shape of the triangles? The triangles appear to be the same size and shape. are triangles used in bridges? In 1930, construction started on the West End Bridge in Pittsburgh, Pennsylvania. The arch of the bridge is trussed, not solid. Steel rods are arranged in a triangular web that lends structure and stability to the bridge. CORRESPONDING PARTS OF CONGRUENT TRIANGLES Triangles that are the same size and shape are congruent triangles . Each triangle has three angles and three sides. If all six of the corresponding parts of two triangles are congruent, then the triangles are congruent. B F C G A E ← ← If ABC is congruent to EFG, the vertices of the two triangles correspond in the same order as the letters naming the triangles. ← ← ← ← ABC EFG This correspondence of vertices can be used to name the corresponding congruent sides and angles of the two triangles. A E B F C G EF AB B C FG A C EG The corresponding sides and angles can be determined from any congruence statement by following the order of the letters. Definition of Congruent Triangles (CPCTC) Study Tip Congruent Parts In congruent triangles, congruent sides are opposite congruent angles. 192 Two triangles are congruent if and only if their corresponding parts are congruent. CPCTC stands for corresponding parts of congruent triangles are congruent. “If and only if” is used to show that both the conditional and its converse are true. Chapter 4 Congruent Triangles Aaron Haupt Resource Manager Workbook and Reproducible Masters Chapter 4 Resource Masters • Study Guide and Intervention, pp. 195–196 • Skills Practice, p. 197 • Practice, p. 198 • Reading to Learn Mathematics, p. 199 • Enrichment, p. 200 School-to-Career Masters, p. 7 Teaching Geometry With Manipulatives Masters, pp. 72, 73 Transparencies 5-Minute Check Transparency 4-3 Answer Key Transparencies Technology Interactive Chalkboard Example 1 Corresponding Congruent Parts FURNITURE DESIGN The seat and legs of this stool form two triangles. Suppose the measures in inches are QR 12, RS 23, QS 24, RT 12, TV 24, and RV 23. a. Name the corresponding congruent angles and sides. Q T QRS TRV S V QR T R R S RV Q T R 2 Teach CORRESPONDING PARTS OF CONGRUENT TRIANGLES In-Class Example S Q S TV V Teaching Tip Students can use tick marks on sides and angles to help visually organize the corresponding parts of congruent triangles. b. Name the congruent triangles. QRS TRV 1 ARCHITECTURE A tower roof Like congruence of segments and angles, congruence of triangles is reflexive, symmetric, and transitive. is composed of congruent triangles all converging toward a point at the top. Properties of Triangle Congruence Theorem 4.4 Power Point® I Congruence of triangles is reflexive, symmetric, and transitive. Reflexive JKL JKL K Symmetric Transitive If JKL PQR, then PQR JKL. If JKL PQR, and PQR XYZ, then JKL XYZ K L J K L J Q K L J Q R Y L P J R P H Z Theorem 4.4 (Transitive) Given: ABC DEF DEF GHI Prove: ABC GHI A Proof: Statements 1. ABC DEF b. Name the congruent triangles. HIJ LIK I F C G D a. Name the corresponding congruent angles and sides of HIJ and LIK. HJI LKI; ILK IHJ; HIJ LIK; HI LI ; H J LK ; JI KI H E B L K X You will prove the symmetric and reflexive parts of Theorem 4.4 in Exercises 33 and 35, respectively. Proof J Reasons 1. Given 2. A D, B E, C F 2. CPCTC AB DE EF AC DF , B C , 3. DEF GHI 3. Given 4. D G, E H, F I 4. CPCTC DE GH HI, D GI , E F F 5. A G, B H, C I 5. Congruence of angles is transitive. 6. AB GH BC HI, A GI , C 6. Congruence of segments is transitive. 7. ABC GHI 7. Def. of s www.geometryonline.com/extra_examples Lesson 4-3 Congruent Triangles 193 Private Collection/Bridgeman Art Library Differentiated Instruction Auditory/Musical Explain to students that congruency can be appealing to both the eyes and the ears. Point out that if students use beats to model two congruent equilateral triangles, they could use three equally-spaced drum beats for the first and then repeat the exact same rhythm for the second. An isosceles beat could consist of two quick beats and one slow beat or vice versa. Tell students that often in music, a “congruent” rhythm is used throughout a song. A popular example is the song, “Louie, Louie.” Lesson 4-3 Congruent Triangles 193 IDENTIFY CONGRUENCE TRANSFORMATIONS In-Class Example Power Point® Teaching Tip In Chapter 9, students will be introduced to the formal names for the slide, flip, and turn transformations. Study Tip Naming Congruent Triangles IDENTIFY CONGRUENCE TRANSFORMATIONS In the figures below, ABC is congruent to DEF. If you slide DEF up and to the right, DEF is still congruent to ABC. E' B There are six ways to name each pair of congruent triangles. E slide A D' D F' C F The congruency does not change whether you turn DEF or flip DEF. ABC is still congruent to DEF. 2 COORDINATE GEOMETRY S y E D' Study Tip T R O E D Transformations R x T S a. Verify that RST RST. RS RS 34 ST ST 17 TR TR 17 b. Name the congruence transformation for RST and RST. turn 3 Practice/Apply Not all of the transformations preserve congruence. Only transformations that do not change the size or shape of the triangle are congruence transformations. F D F' F F' Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 4. • include an example from each theorem introduced in this lesson. • include any other item(s) that they find helpful in mastering the skills in this lesson. 194 Chapter 4 Congruent Triangles E' E' If you slide, flip, or turn a triangle, the size and shape do not change. These three transformations are called congruence transformations . Example 2 Transformations in the Coordinate Plane COORDINATE GEOMETRY The vertices of CDE are C(5, 7), D(8, 6), and E(3, 3). The vertices of CDE are C(5, 7), D(8, 6), and E(3, 3). D a. Verify that CDE CDE. Use the Distance Formula to find the length of each side in the triangles. –8 [8 (5)]2 (6 7)2 DC 9 1 or 10 DE [8 (3)]2 (6 3)2 25 9 or 34 Study Notebook flip D' turn CE [5 (3)]2 (7 3)2 4 16 or 20 C 8 y C' D' 4 E –4 E' O 4 8x DC (8 5 )2 (6 7)2 9 1 or 10 DE (8 3 )2 (6 3)2 25 9 or 34 2 (7 CE (5 3) 3)2 4 16 or 20 By the definition of congruence, D DC DE DE CE C , , and C E . Use a protractor to measure the angles of the triangles. You will find that the measures are the same. DC DE CE In conclusion, because DC , D E , and C E , D D, C C, and E E, DCE DCE. b. Name the congruence transformation for CDE and CDE. CDE is a flip of CDE. 194 Chapter 4 Congruent Triangles Concept Check Guided Practice GUIDED PRACTICE KEY Exercises Examples 3–6, 8 7 1 2 1. Explain how slides, flips, and turns preserve congruence. See margin. 2. OPEN ENDED Draw a pair of congruent triangles and label the congruent sides and angles. See margin. Identify the congruent triangles in each figure. D AFC DFB 4. H HJT TKH 3. A K F C 5. W S, X T, Z J, W X ST, XZ TJ, W Z SJ B J T 5. If WXZ STJ, name the congruent angles and congruent sides. B 6. QUILTING In the quilt design, assume that angles and segments that appear to be congruent are congruent. Indicate which triangles are congruent. See margin. 8. GARDENING This garden lattice will be covered with morning glories in the summer. Wesley wants to save two triangular areas for artwork. If GHJ KLP, name the corresponding congruent angles and sides. A G E 7. The coordinates of the vertices of QRT and QRT are Q(4, 3), Q(4, 3), R(4, 2), R(4, 2), T(1, 2), and T(1, 2). Verify that QRT QRT. Then name the congruence transformation. See margin. Application M L F C K D H G L J K 9–22, 27–35 23–26 1 Answers Identify the congruent triangles in each figure. 9. CFH JKL 10. K S RSV TSV L F 2 1. The sides and the angles of the triangle are not affected by a congruence transformation, so congruence is preserved. 2. Sample answer: J Extra Practice See page 761. V H C 11. R WPZ QVS P T 12. F EFH GHF E Z S W G Q V 13– 16. See margin. Basic: 9–19 odd, 23, 27–33, 36–51 Average: 9–35 odd, 36–51 Advanced: 10–36 even, 37–48 (optional: 49–51) All: Quiz 1 (1–5) P Practice and Apply See Examples Odd/Even Assignments Exercises 9–35 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide H ★ indicates increased difficulty For Exercises Organization by Objective • Corresponding Parts of Congruent Triangles: 9–22, 27–35 • Identify Congruence Transformations: 23–26 N J G K, H L, J P, G H KL, H J LP, J KP G About the Exercises… H Name the congruent angles and sides for each pair of congruent triangles. 13. TUV XYZ 14. CDG RSW 15. BCF DGH 16. ADG HKL Lesson 4-3 Congruent Triangles 195 6. BME, ANG, DKH, CLF; EMJ, GNJ, HKJ, FLJ; BAJ, ADJ, DCJ, CBJ; BCD, ADC, CBA, DAB; BLJ, AMJ, JND, JKC, BMJ, ANJ, JKD, JLC 7. QR 5, QR 5, RT 3, RT 3, QT 34, and QT 34. Use a protractor to confirm that the corresponding angles are congruent; flip. 13. T X, U Y, V Z, TU XY , UV YZ, TV XZ 14. C R, D S, G W, CD RS , DG SW , CG R W 15. B D, C G, F H, BC DG , CF GH , BF DH 16. A H, D K, G L, AD HK , DG KL, AG HL Lesson 4-3 Congruent Triangles 195 Answers 22. Flip; PQ 2, PQ 2, QV 4, QV 4, PV 20, and PV 20. Use a protractor to confirm that the corresponding angles are congruent. 23. Flip; MN 8, MN 8, NP 2, NP 2, MP 68, and MP 68. Use a protractor to confirm that the corresponding angles are congruent. 24. Slide; GH 10, GH 10, HF 8, HF 8, GF 10, and GF 10. Use a protractor to confirm that the corresponding angles are congruent. 25. Turn; JK 40, JK 40, KL 29, KL 29, JL 17, and JL 17. Use a protractor to confirm that the corresponding angles are congruent. 26. False; A X, B Y, and C Z but the corresponding sides are not congruent. Y B X Z A C 27. D 18. s 1–4, s 5–12, s 13–20 19. s 1, 5, 6, and 11, s 3, 8, 10, and 12, s 2, 4, 7, and 9 Assume that segments and angles that appear to be congruent in the numbered triangles are congruent. Indicate which triangles are congruent. 17. 18. 19. 5 9 2 1 21. MOSAICS 1 9 8 6 10 10 7 3 11 8 4 12 2 12 7 3 8 6 11 4 10 14 13 17 18 16 15 19 20 9 5 S V B U F E T A C D The picture at the left is the center of a Roman mosaic. Because the Verify that each of the following preserves congruence and name the congruence transformation. 22–25. See margin for verification. 22. PQV PQV flip 23. MNP MNP flip P y Q' Q y M N P' P V Mosaics A mosaic is composed of glass, marble, or ceramic pieces often arranged in a pattern. The pieces, or tesserae, are set in cement. Mosaics are used to decorate walls, floors, and gardens. x O V' P' M' x O ★ ★ 24. GHF GHF slide y H N' 25. JKL JKL turn y J H' L L' F F' K' x O F x O G' K J' C 12 H J S Determine whether each statement is true or false. Draw an example or counterexample for each. 26. Two triangles with corresponding congruent angles are congruent. false 27. Two triangles with angles and sides congruent are congruent. true 26–27. See margin for drawings. 6 10 12 R G ★ 6 10 Q 31. K 36 80 D E L 36 Chapter 4 Congruent Triangles North Carolina Museum of Art, Raleigh. Gift of Mr. & Mrs. Gordon Hanes 64 80 196 28. UMBRELLAS Umbrellas usually have eight congruent triangular sections with ribs of equal length. Are the statements JAD IAE and JAD EAI both correct? Explain. Both statements are correct because the spokes are the same length, EA IA, and AE AI. 64 F 35. Given: DEF Prove: DEF DEF Proof: DEF Given E D F DE DE, EF EF, DF DF Congruence of segments is reflexive. D D, E E, F F Congruence of is reflexive. DEF DEF Def. of s 196 1 2 in another triangle. What else do you need to know to conclude that the four triangles are congruent? G 29. J 7 ★ four triangles connect to a square, they have at least one side congruent to a side A B 6 21. We need to know ★ 20. All of the small triangles in the figure at the right are congruent. Name three larger congruent that all of the corretriangles. UFS, TDV, ACB sponding angles are congruent and that the other corresponding sides are congruent. Source: www.dimosaico.com E 5 4 3 Chapter 4 Congruent Triangles F D B A J I E C G ALGEBRA For Exercises 29 and 30, use the following information. QRS GHJ, RS 12, QR 10, QS 6, and HJ 2x 4. 29. Draw and label a figure to show the congruent triangles. See margin. 30. Find x. 8 NAME ______________________________________________ DATE p. 195 (shown) Congruent Triangles and p. 196 Corresponding Parts of Congruent Triangles X R, Y S, Z T, XY RS, YZ ST, XZ RT RS XY, ST YZ, RT XZ 2. K ? 34. a. Given b. Given c. Congruence of segments is symmetric. d. Given e. Def. of lines f. Given g. Def. of lines h. All right are . i. Given j. Alt. int. are . k. Given l. Alt. int. are . m. Def. of s XYZ RST ? D 3. K L J M C ABC DCB G L 5. B K JKM LMK 6. R D U E J A E J; F K; G L; EF JK ; EG JL ; FG KL ? B A Gl C T A D; ABC DCB; ACB DBC; AB DC ; AC DB ; B C CB R T; RSU TSU; RUS TUS; RU TU ; RS TS ; U S SU NAME ______________________________________________ DATE /M G Hill 195 Skills Practice, 4-3 Practice (Average) ? S Lesson 4-3 L C 4. F Y RST XYZ T Name the corresponding congruent angles and sides for the congruent triangles. Z R X, S Y, T Z, R Z B 33. PROOF The statements below can be used to prove that congruence of triangles is symmetric. Use the statements to construct a correct flow proof. Provide the reasons for each statement. See p. 233B. T Y S X Exercises 1. A X T A Identify the congruent triangles in each figure. ABC JKL S R C Example If XYZ RST, name the pairs of congruent angles and congruent sides. X R, Y S, Z T XY RS , XZ RT , YZ ST J R S B Triangles that have the same size and same shape are congruent triangles. Two triangles are congruent if and only if all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are congruent. In the figure, ABC RST. ALGEBRA For Exercises 31 and 32, use the following information. JKL DEF, mJ 36, mE 64, and mF 3x 52. 31. Draw and label a figure to show the congruent triangles. See margin. 28 32. Find x. 3 Given: RST XYZ Prove: XYZ RST Flow Proof: ____________ PERIOD _____ Study Guide andIntervention Intervention, 4-3 Study Guide and ____________ Gl PERIOD G _____ p. 197 and Practice, 198 (shown) Congruent p. Triangles Identify the congruent triangles in each figure. 34. PROOF 1. Copy the flow proof and provide the reasons for each statement. 2. B CD CB DC Given: AB , A D , A D , BC CD BC AB , A D 㛳 , A B 㛳 Prove: ACD CAB B C 4 S LMN QPN 3. GKP LMN 2 G L, K M, P N; GK LM , KP MN , GP LN 1 4. ANC RBV D A R, N B, C V; AN RB , NC BV , AC RV Verify that each of the following transformations preserves congruence, and name the congruence transformation. 5. PST PST a. ? b. ? Q L D Name the congruent angles and sides for each pair of congruent triangles. Flow Proof: AD CB P N C ABC DRS 3 A AB CD M R A AC CA c. ? AD DC d. ? AB BC AD BC f. ? i. ? AB CD k. ? O B is a rt. . 1 4 2 3 e. ? g. ? j. ? l. ? L N O x T T P y M S S D is a rt. . 6. LMN LMN y x L P N M D B h. ? PS 13 , P S 13 , LM 22 , LM 22 , , ST 5 , PT 10 , ST 5 MN 29 , MN 29 , , P P , PT 10 LN 7, LN 7, L L, S S, T T; flip M M , N N; flip QUILTING For Exercises 7 and 8, refer to the quilt design. A C D E G F 7. Indicate the triangles that appear to be congruent. ABI EBF, CBD HBG ACD CAB 8. Name the congruent angles and congruent sides of a pair of congruent triangles. m. ? I H Sample answer: A E, ABI EBF, I F; EB , B I BF , AI EF AB NAME ______________________________________________ DATE /M G Hill 198 Gl ★ 35. B ____________ Gl PERIOD G _____ Reading 4-3 Readingto to Learn Learn Mathematics Write a flow proof to prove Congruence of triangles is reflexive. (Theorem 4.4) See margin. PROOF Mathematics, p. 199 Congruent Triangles Pre-Activity ELL Why are triangles used in bridges? Read the introduction to Lesson 4-3 at the top of page 192 in your textbook. 36. CRITICAL THINKING RST is isosceles with RS = RT, M, N, and P are midpoints of their sides, S MPS, MP and NP . What else do you need to know to prove that SMP TNP? SMP TNP, MPS NPT S R diagonal braces make the structure stronger and prevent it from being deformed when it has to withstand a heavy load. M N Reading the Lesson 1. If RST UWV, complete each pair of congruent parts. P T R U S W V U W S R U RT T V T S WV 2. Identify the congruent triangles in each diagram. Lesson 4-3 Congruent Triangles 197 a. B ABC ADC b. PQS RQS Q C A S D P c. M NAME ______________________________________________ DATE 4-3 Enrichment Enrichment, Q ____________ PERIOD _____ p. 200 T V N O S P MNO QPO U RTV USV 3. Determine whether each statement says that congruence of triangles is reflexive, symmetric, or transitive. Transformations in The Coordinate Plane The following statement tells one way to map preimage points to image points in the coordinate plane. a. If the first of two triangles is congruent to the second triangle, then the second triangle is congruent to the first. symmetric (x, y) → (x 6, y 3) y (x, y) → (x 6, y 3) This can be read, “The point with coordinates (x, y) is mapped to the point with coordinates ( x 6, y 3).” With this transformation, for example, (3, 5) is mapped to (3 6, 5 3) or (9, 2). The figure shows how the triangle ABC is mapped to triangle XYZ. R d. R B b. If there are three triangles for which the first is congruent to the second and the second is congruent to the third, then the first triangle is congruent to the third. transitive Y c. Every triangle is congruent to itself. reflexive A O C x Helping You Remember X Z 1. Does the transformation above appear to be a congruence transformation? Explain your answer. Yes; the transformation slides the figure to the lower right without changing its size or shape. 4. A good way to remember something is to explain it to someone else. Your classmate Ben is having trouble writing congruence statements for triangles because he thinks he has to match up three pairs of sides and three pairs of angles. How can you help him understand how to write correct congruence statements more easily? Sample answer: Write the three vertices of one triangle in any order. Then write the corresponding vertices of the second triangle in the same order. If the angles are written in the correct correspondence, the sides will automatically be in the correct correspondence also. Draw the transformation image for each figure. Then tell whether the Lesson 4-3 Congruent Triangles 197 Lesson 4-3 www.geometryonline.com/self_check_quiz In the bridge shown in the photograph in your textbook, diagonal braces were used to divide squares into two isosceles right triangles. Why do you think these braces are used on the bridge? Sample answer: The 37. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. Why are triangles used in bridges? 4 Assess Include the following in your answer: • whether the shape of the triangle matters, and • whether the triangles appear congruent. Open-Ended Assessment Modeling Ask students to name examples of how congruent triangles are modeled in objects or in nature. Then they can name the corresponding congruent angles and sides and determine the congruence transformations applied to the triangles. The umbrella for Exercise 28 on p. 196 is an example of rotated triangles, and students can name corresponding parts. Standardized Test Practice 38. Determine which statement is true given ABC XYZ. B A 39. ALGEBRA A Assessment Options Practice Quiz 1 The quiz provides students with a brief review of the concepts and skills in Lessons 4-1 through 4-3. Lesson numbers are given to the right of the exercises or instruction lines so students can review concepts not yet mastered. Answers 37. Sample answer: Triangles are used in bridge design for structure and support. Answers should include the following. • The shape of the triangle does not matter. • Some of the triangles used in the bridge supports seem to be congruent. 198 Chapter 4 Congruent Triangles B A C XZ A B YZ C D cannot be determined D 185 Find the length of DF if D(5, 4) and F(3, 7). D 5 B 13 57 C Maintain Your Skills Mixed Review Find x. (Lesson 4-2) 40. 75 41. 58 42. 75 x˚ x˚ 40˚ Getting Ready for Lesson 4-4 Prerequisite Skill Students will learn about proving congruence using SSS and SAS in Lesson 4-4. They will use the Distance Formula to find side lengths of triangles in a coordinate plane. Use Exercises 49–51 to determine your students’ familiarity with the Distance Formula. B C ZX x˚ 43. x 3, BC 10, CD 10, BD 5 x˚ 30˚ 115˚ 100˚ 42˚ Find x and the measure of each side of the triangle. (Lesson 4-1) 43. BCD is isosceles with BC CD , BC 2x 4, BD x 2, and CD 10. 44. Triangle HKT is equilateral with HK x 7 and HT 4x 8. x 5; HK 12, HT 12, KT 12 Write an equation in slope-intercept form for the line that satisfies the given conditions. (Lesson 3-4) 45. y 3x 3 3 3 2 45. contains (0, 3) and (4, 3) 46. m , y-intercept 8 y x 8 4 4 47. parallel to y 4x 1; 48. m 4, contains (3, 2) y 4x 10 contains (3, 1) y 4x 11 Getting Ready for the Next Lesson PREREQUISITE SKILL Find the distance between each pair of points. (To review the Distance Formula, see Lesson 1-4.) 49. (1, 7), (1, 6) 5 50. (8, 2), (4, 2) 32 P ractice Quiz 1 51. (3, 5), (5, 2) Lessons 4-1 through 4-3 1. Identify the isosceles triangles in the figure, if F H and D G are congruent perpendicular bisectors. (Lesson 4-1) DFJ, GJF, HJG, DJH, ALGEBRA ABC is equilateral with AB 2x, BC 4x 7, and AC x 3.5. (Lesson 4-1) 2. Find x. x 3.5 3. Find the measure of each side. F D m1 60, m2 110, m3 49 (Lesson 4-2) J H AB BC AC 7 4. Find the measure of each numbered angle. 1 3 50˚ 5. M J, N K, P L; M N JK, N P KL, M P JL 2 21˚ 70˚ 5. If MNP JKL, name the corresponding congruent angles and sides. (Lesson 4-3) 198 Chapter 4 Congruent Triangles 13 G Reading Mathematics Making Concept Maps Getting Started When studying a chapter, it is wise to record the main topics and vocabulary you encounter. In this chapter, some of the new vocabulary words were triangle, acute triangle, obtuse triangle, right triangle, equiangular triangle, scalene triangle, isosceles triangle, and equilateral triangle. The triangles are all related by the size of the angles or the number of congruent sides. Advise students to create their own version of this concept map and place it in their study notebooks. Encourage them to use colored pencils or highlighters to group related items. A graphic organizer called a concept map is a convenient way to show these relationships. A concept map is shown below for the different types of triangles. The main ideas are in boxes. Any information that describes how to move from one box to the next is placed along the arrows. Teach Making Concept Maps Students should discern from the concept map that if all the angle measures of a triangle are equal, then the triangle must also be acute, equilateral, and isosceles. Similarly, if all sides of a triangle are congruent, then the triangle must also be acute, equiangular, and isosceles. Classifying Triangles Classify by angle measure. Classify by the number of congruent sides. Angles Sides Measure of one angle is 90. Measure of one angle is greater than 90. Measures of all angles are less than 90. At least 2 sides congruent No sides congruent Right Obtuse Acute Isosceles Scalene 3 congruent angles 3 sides congruent Equiangular Equilateral Assess Study Notebook Ask students to summarize what they have learned about using concept maps to review chapter material and enhance their knowledge of chapter concepts. Reading to Learn 1. Describe how to use the concept map to classify triangles by their side lengths. See margin. 2. In ABC, mA 48, mB 41, and mC 91. Use the concept map to classify ABC. obtuse 3. Identify the type of triangle that is linked to both classifications. equiangular or equilateral Reading Mathematics Making Concept Maps 199 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. Answer 1. Sample answer: If side lengths are given, determine the number of congruent sides and name the triangle. Some isosceles triangles are equilateral triangles. Reading Mathematics Making Concept Maps 199 Lesson Notes Proving Congruence—SSS, SAS • Use the SSS Postulate to test for triangle congruence. 1 Focus 5-Minute Check Transparency 4-4 Use as a quiz or review of Lesson 4-3. • Use the SAS Postulate to test for triangle congruence. do land surveyors use congruent triangles? Vocabulary • included angle Land surveyors mark and establish property boundaries. To check a measurement, they mark out a right triangle and then mark a second triangle that is congruent to the first. Mathematical Background notes are available for this lesson on p. 176D. SSS POSTULATE Is it always necessary to show that all of the corresponding parts of two triangles are congruent to prove that the triangles are congruent? In this lesson, we will explore two other methods to prove that triangles are congruent. do land surveyors use congruent triangles? Ask students: • What does it mean for two triangles to be congruent? All three corresponding sides and all three corresponding angles are congruent. • Would two congruent triangles have the same perimeter? Explain. Yes, the three corresponding sides are congruent so the sum of the measures of the sides of the triangles would be equal. Congruent Triangles Using Sides 1 Draw a triangle and label the vertices X, Y, and Z. 2 Use a straightedge to draw any line and select a point R. Use a compass to construct R S on such that R S X Z . 3 Using R as the center, draw an arc with radius equal to XY. Y Z X 4 R Using S as the center, draw an arc with radius equal to YZ. 5 S Let T be the point of intersection of the two arcs. Draw R T and S T to form RST. R 6 S Cut out RST and place it over XYZ. How does RST compare to XYZ? RST XYZ T R S R S If the corresponding sides of two triangles are congruent, then the triangles are congruent. This is the Side-Side-Side Postulate, and is written as SSS. 200 Chapter 4 Congruent Triangles Paul Conklin/PhotoEdit Resource Manager Workbook and Reproducible Masters Chapter 4 Resource Masters • Study Guide and Intervention, pp. 201–202 • Skills Practice, p. 203 • Practice, p. 204 • Reading to Learn Mathematics, p. 205 • Enrichment, p. 206 • Assessment, pp. 239, 241 Prerequisite Skills Workbook, pp. 1–2 Teaching Geometry With Manipulatives Masters, pp. 8, 75 Transparencies 5-Minute Check Transparency 4-4 Answer Key Transparencies Technology Interactive Chalkboard Postulate 4.1 Side-Side-Side Congruence If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. Abbreviation: SSS 2 Teach Z SSS POSTULATE B A In-Class Examples C ABC ZXY Y Teaching Tip Explain to students that when naming congruent triangles, it is customary to use triangle names in the same order as their congruent parts. BYA CYA uses appropriate order to signify the corresponding sides and angles that are congruent in the two triangles. It would be incorrect to write BAY CYA. Example 1 Use SSS in Proofs B MARINE BIOLOGY The tail of an orca whale can be viewed as two triangles that share a common side. Write a two-column proof to prove that BYA CYA if A B A C and Y C Y. B C Y A AC BY CY Given: AB ; Prove: BYA CYA Proof: Statements Reasons 1. A AC CY B ; B Y AY 2. A Y 3. BYA CYA 1. Given 2. Reflexive Property 3. SSS 1 ENTOMOLOGY The wings of one type of moth form two triangles. Write a two-column proof to prove that FEG FH , FE HI, HIG if EI and G is the midpoint of I and F H . both E Example 2 SSS on the Coordinate Plane COORDINATE GEOMETRY Determine whether RTZ JKL for R(2, 5), Z(1, 1), T(5, 2), L(3, 0), K(7, 1), and J(4, 4). Explain. Use the Distance Formula to show that the corresponding sides are congruent. y F R I J T K G Z L O x E (2 5 )2 (5 2)2 RT 9 9 or 18 1)2 16 1 or 17 1 16 or 17 KL [7 (3)]2 (1 0)2 16 1 or 17 RZ (2 (5 1)2 1)2 H Statements (Reasons) 1. FE HI; G is the midpoint of EI; G is the midpoint of F H. (Given) 2. FG HG ; EG I G. (Midpoint Theorem) 3. FEG HIG. (SSS) JK [4 (7)]2 (4 1)2 9 9 or 18 TZ (5 (2 1)2 Power Point® X JL [4 (3)]2 (4 0)2 1 16 or 17 COORDINATE GEOMETRY Determine whether WDV 2 MLP. Explain. RT JK, TZ KL, and RZ JL. By definition of congruent segments, all corresponding segments are congruent. Therefore, RTZ JKL by SSS. y SAS POSTULATE Suppose you are given the measures of two sides and the angle they form, called the included angle . These conditions describe a unique triangle. Two triangles in which corresponding sides and the included pairs of angles are congruent provide another way to show that triangles are congruent. www.geometryonline.com/extra_examples D O P x V W Lesson 4-4 Proving Congruence—SSS, SAS 201 Jeffrey Rich/Pictor International/PictureQuest L M Differentiated Instruction Logical/Mathematical Students can use a systematic approach to write the proofs for problems and examples in this lesson. Have students start by looking for possible methods of proof using SSS or SAS. Then they should examine the problem to determine how much necessary information is given and how they can find any other information that they need for the proof. Finally, they can draw on prior knowledge of midpoints, distances, angle relationships, and so on, to extract any other necessary information and compile the facts for the final proof. WD ML, DV LP, and VW PM. By definition of congruent segments, all corresponding segments are congruent. Therefore, WDV MLP by SSS. Lesson 4-4 Proving Congruence—SSS, SAS 201 SAS POSTULATE Postulate 4.2 In-Class Examples Side-Angle-Side Congruence If two sides and Power Point® B 3 Write a proof for the following. F Abbreviation: SAS R D the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. A S E C ABC FDE You can also construct congruent triangles given two sides and the included angle. Q T Q || T S Given: R Q R TS Prove: QRT STR Proof: Statements (Reasons) 1. RQ || TS , RQ TS (Given) 2. QRT STR (Alt. int. are .) 3. RT TR (Reflexive Property) 4. QRT STR (SAS) Congruent Triangles using Two Sides and the Included Angle 1 Draw a triangle and label its vertices A, B, and C. Select a point K on line m . Use a compass to construct K L on m such that KL B C . 2 a. AB that JK . Draw JL to complete JKL. J m B 5 K C L m m K K L Cut out JKL and place it over ABC. How does JKL compare to ABC? JKL ABC SAS Study Tip b. 4 Construct JK such A 4 Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. Construct an angle congruent to B as a side using KL of the angle and point K as the vertex. 3 SSS Example 3 Use SAS in Proofs Flow Proofs Write a flow proof. Flow proofs can be written vertically or horizontally. Given: X is the midpoint of BD . X is the midpoint of A C . D A Prove: DXC BXA X C Flow Proof: B X is the midpoint of DB. DX BX Given Midpoint Theorem X is the midpoint of AC. CX AX DXC BXA Given Midpoint Theorem SAS DXC BXA Vertical s are . 202 Chapter 4 Congruent Triangles Unlocking Misconceptions Figures Point out that figures will not always be marked and that it is up to students to draw on their knowledge of geometric concepts to prove congruence. Stress the importance of using only information that is given and not forming any assumptions about two figures just because they appear to be congruent. 202 Chapter 4 Congruent Triangles L Example 4 Identify Congruent Triangles 3 Practice/Apply Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. a. b. Study Notebook Each pair of corresponding sides are congruent. The triangles are congruent by the SSS Postulate. Concept Check The triangles have three pairs of corresponding angles congruent. This does not match the SSS Postulate or the SAS Postulate. It is not possible to prove the triangles congruent. 1. OPEN ENDED Draw a triangle and label the vertices. Name two sides and the included angle. See margin. Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 4. • include a simple example of a proof using SSS and a proof using SAS. • include any other item(s) that they find helpful in mastering the skills in this lesson. 2. FIND THE ERROR Carmelita and Jonathan are trying to determine whether ABC is congruent to DEF. A Jonathan Congruence cannot be determined. Carmelita πABC πDEF by SAS D 2 78˚ C 1.5 B 2 F 48˚ 1.5 E Who is correct and why? Jonathan; the measure of DEF is needed to use SAS. Guided Practice GUIDED PRACTICE KEY Exercises Examples 3–4 5–6 7–8 9 2 3 4 1 Determine whether EFG MNP given the coordinates of the vertices. Explain. 3. E(4, 3), F(2, 1), G(2, 3), M(4, 3), N(2, 1), P(2, 3) 3–4. See margin. 4. E(2, 2), F(4, 6), G(3, 1), M(2, 2), N(4, 6), P(3, 1) 5. Write a flow proof. B E and B Given: D C bisect each other. Prove: DGB EGC 6. Write a two-column proof. Given: KM 㛳 JL , K M JL Prove: JKM MLJ K D G E J M About the Exercises… C Exercise 5 Organization by Objective • SSS Postulate: 10–13, 20–21, 28–29 • SAS Postulate: 14–19 L Exercise 6 5 – 6. See p. 233B. Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. SAS 8. SSS 7. Lesson 4-4 Proving Congruence—SSS, SAS 203 Answer 1. Sample answer: In QRS, R is the included angle of the sides QR and R S . R Q S FIND THE ERROR Even though the figures look congruent and E appears to have the same measure as B, this cannot be assumed based on the information given in the figures. Carmelita’s answer would have been correct if it could be shown that mE 78. 3. EG 2, MP 2, FG 4, NP 4, EF 20 , and MN 20. The corresponding sides have the same measure and are congruent. EFG MNP by SSS. 4. EG 10, FG 26, EF 68, MP 2, NP 26, and MN 20. The corresponding sides are not congruent, so the triangles are not congruent. Odd/Even Assignments Exercises 10–27 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide Basic: 11–25 odd, 28–47 Average: 11–27 odd, 28–47 Advanced: 10–26 even, 28–43 (optional: 44–47) Lesson 4-4 Proving Congruence—SSS, SAS 203 Application Answers 9. Given: T is the midpoint of 苶 SQ 苶. 苶R 苶⬵苶 QR 苶 S Prove: 䉭SRT ⬵ 䉭QRT R S T Q Proof: Statements (Reasons) 1.T is the midpoint of 苶 SQ 苶. (Given) 2.S 苶T苶 ⬵ 苶TQ 苶 (Def. of midpoint) 3.S 苶R 苶⬵苶 QR 苶 (Given) 4.R 苶T苶 ⬵ 苶 RT苶 (Reflexive Prop.) 5.䉭SRT ⬵ 䉭QRT (SSS) 10. JK ⫽兹苶 20, KL ⫽ 兹61 苶, JL ⫽ 兹53 苶, FG ⫽ 兹苶 20, GH ⫽ 兹61 苶, and FH ⫽ 兹53 苶. Each pair of corresponding sides have the same measure so they are congruent. 䉭JKL ⬵ 䉭FGH by SSS. 11. JK ⫽ 兹10 苶, KL ⫽ 兹10 苶, 20, FG ⫽ 兹苶2, GH ⫽ 兹苶 50, JL ⫽ 兹苶 and FH ⫽ 6. The corresponding sides are not congruent so 䉭JKL is not congruent to 䉭FGH. 12. JK ⫽ 兹苶 50, KL ⫽ 兹苶 13, JL ⫽ 5, FG ⫽ 兹苶8, GH ⫽ 兹苶 13, and FH ⫽ 5. The corresponding sides are not congruent so 䉭JKL is not congruent to 䉭FGH. 13. JK ⫽ 兹苶 10, KL ⫽ 兹苶 10, JL ⫽ 兹苶 20, FG ⫽ 兹苶 10, GH ⫽ 兹苶 10, and FH ⫽ 兹苶 20. Each pair of corresponding sides have the same measure so they are congruent. 䉭JKL ⬵ 䉭FGH by SSS. 9. PRECISION FLIGHT The United States Navy Flight Demonstration Squadron, the Blue Angels, fly in a formation that can be viewed as two triangles with a common side. Write a two-column proof to prove that 䉭SRT ⬵ 䉭QRT if T is the midpoint QR of 苶 SQ 苶 and S 苶R 苶⬵苶 苶. See margin. R T S Q ★ indicates increased difficulty Practice and Apply For Exercises See Examples 10–13 14–19 20–21, 28–29 22–27 2 3 1 4 Extra Practice See page 761. Determine whether 䉭JKL ⬵ 䉭FGH given the coordinates of the vertices. Explain. 10. J(⫺3, 2), K(⫺7, 4), L(⫺1, 9), F(2, 3), G(4, 7), H(9, 1) 11. J(⫺1, 1), K(⫺2, ⫺2), L(⫺5, ⫺1), F(2, ⫺1), G(3, ⫺2), H(2, 5) 12. J(⫺1, ⫺1), K(0, 6), L(2, 3), F(3, 1), G(5, 3), H(8, 1) 13. J(3, 9), K(4, 6), L(1, 5), F(1, 7), G(2, 4), H(⫺1, 3) 10–13. See margin. Write a flow proof. 14–19. See p. 233C. 14. Given: 苶 AE FC BC 15. Given: 苶 RQ TQ WQ 苶⬵苶 苶, A 苶B 苶⬵苶 苶, 苶⬵苶 苶⬵Y 苶Q 苶⬵苶 苶 B苶 E⬵苶 B苶 F ⬔RQY ⬵ ⬔WQT 苶 Prove: 䉭AFB ⬵ 䉭CEB Prove: 䉭QWT ⬵ 䉭QYR R B Y Q C F E A W Write a two-column proof. 16. Given: 䉭CDE is isosceles. G is the midpoint of 苶 CE 苶. Prove: 䉭CDG ⬵ 䉭EDG D T 17. Given: 䉭MRN ⬵ 䉭QRP ⬔MNP ⬵ ⬔QPN Prove: 䉭MNP ⬵ 䉭QPN M Q R C G N E 18. Given: 苶 AC GC 苶⬵苶 苶 EC 苶 苶 bisects A 苶G 苶. Prove: 䉭GEC ⬵ 䉭AEC P ★ 19. Given: 䉭GHJ ⬵ 䉭LKJ Prove: 䉭GHL ⬵ 䉭LKG H K A J E C G G 204 Chapter 4 Congruent Triangles Elaine Thompson/AP/Wide World Photos 29. Sample answer: The properties of congruent triangles help land surveyors double check measurements. Answers should include the following. • If each pair of corresponding angles and sides are congruent, the triangles are congruent by definition. If two pairs of corresponding sides and the included angle are congruent, the triangles are congruent by SAS. If each pair of corresponding sides are congruent, the triangles are congruent by SSS. • Sample answer: Architects also use congruent triangles when designing buildings. 204 Chapter 4 Congruent Triangles L 20–21. See p. 233C. 20. CATS A cat’s ear is triangular in shape. Write a two-column proof to PN prove RST PNM if RS , MP RT , S N, and T M. 21. GEESE This photograph shows a flock of geese flying in formation. Write a two-column proof to prove that EFG HFG, if EF HF and G is the midpoint of E H . NAME ______________________________________________ DATE ____________ PERIOD _____ Study Guide andIntervention Intervention, 4-4 Study Guide and p. 201 andSAS p. 202 Proving(shown) Congruence—SSS, SSS Postulate You know that two triangles are congruent if corresponding sides are congruent and corresponding angles are congruent. The Side-Side-Side (SSS) Postulate lets you show that two triangles are congruent if you know only that the sides of one triangle are congruent to the sides of the second triangle. If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. SSS Postulate P R Example Write a two-column proof. Given: A B DB and C is the midpoint of A D . Prove: ABC DBC E S N B A M G F T H Statements Reasons B DB 1. A 1. Given 2. C is the midpoint of A D . 2. Given D C DC 3. AC 3. Definition of midpoint BC 4. BC 4. Reflexive Property of 5. ABC DBC 5. SSS Postulate Exercises Write a two-column proof. Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. SSS 23. not possible 22. not possible Z X T U R S XY , AC XZ , B C YZ Given: AB Prove: ABC XYZ UT , RT US Given: RS Prove: RST UTS Statements Statements Reasons Reasons 1.R S UT 1. Given US RT 2.S T TS 2. Refl. Prop. 3.RST UTS 3. SSS Post. NAME ______________________________________________ DATE /M G Hill 201 ____________ Gl PERIOD G _____ Skills Practice, 4-4 Practice (Average) SSS or SAS 25. 2. Y C 1.A B XY 1.Given XZ AC C B YZ 2.ABC XYZ 2. SSS Post. Gl 24. B A Lesson 4-4 1. p. 203 and Practice, p. 204 (shown) Proving Congruence—SSS, SAS Determine whether DEF PQR given the coordinates of the vertices. Explain. 1. D(6, 1), E(1, 2), F(1, 4), P(0, 5), Q(7, 6), R(5, 0) DE 52 , PQ 52 , EF 210 , QR 210 , DF 52 , PR 52 . DEF PQR by SSS since corresponding sides have the same measure and are congruent. 2. D(7, 3), E(4, 1), F(2, 5), P(2, 2), Q(5, 4), R(0, 5) DE 13 , PQ 13, EF 25 , QR 26 , DF 29 , PR 13 . Corresponding sides are not congruent, so DEF is not congruent to PQR. 3. Write a flow proof. Given: R S TS V is the midpoint of RT . Prove: RSV TSV Baseball The infield is a square 90 feet on each side. A B SV SV Reflexive Property RS TS Given RSV TSV V is the midpoint of RT. Given SSS RV VT Definition of midpoint Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. 4. F 5. not possible 6. SAS or SSS SSS 7. INDIRECT MEASUREMENT To measure the width of a sinkhole on his property, Harmon marked off congruent triangles as shown in the diagram. How does he know that the lengths AB and AB are equal? C NAME ______________________________________________ DATE /M G Hill 204 Gl Answer the question that was posed at the beginning of the lesson. See margin. How do land surveyors use congruent triangles? Include the following in your answer: • description of three methods to prove triangles congruent, and • another example of a career that uses properties of congruent triangles. Mathematics, p. 205 Proving Congruence—SSS, SAS Pre-Activity Lesson 4-4 Proving Congruence—SSS, SAS 205 (tl)G.K. & Vikki Hart/PhotoDisc, (tr)Chase Swift/CORBIS, (b)Index Stock NAME ______________________________________________ DATE Read the introduction to Lesson 4-4 at the top of page 200 in your textbook. Why do you think that land surveyors would use congruent right triangles rather than other congruent triangles to establish property boundaries? Sample answer: Land is usually divided into rectangular lots, so their boundaries meet at right angles. Reading the Lesson 1. Refer to the figure. N a. Name the sides of LMN for which L is the included angle. M L , L N M ____________ PERIOD _____ p. 206 L N , NM L M , M N 2. Determine whether you have enough information to prove that the two triangles in each figure are congruent. If so, write a congruence statement and name the congruence postulate that you would use. If not, write not possible. b. B E D D C Congruent Parts of Regular Polygonal Regions Congruent figures are figures that have exactly the same size and shape. There are many ways to divide regular polygonal regions into congruent parts. Three ways to divide an equilateral triangular region are shown. You can verify that the parts are congruent by tracing one part, then rotating, sliding, or reflecting that part on top of the other parts. L c. Name the sides of LMN for which M is the included angle. a. A 4-4 Enrichment Enrichment, ELL How do land surveyors use congruent triangles? b. Name the sides of LMN for which N is the included angle. www.geometryonline.com/self_check_quiz B C ____________ Gl PERIOD G _____ Reading 4-4 Readingto to Learn Learn Mathematics 29. WRITING IN MATH A Since ACB and ACB are vertical angles, they are A B AC and BC B C. So congruent. In the figure, AC ABC ABC by SAS. By CPCTC, the lengths AB and AB are equal. E D Source: www.mlb.com S T Proof: 26–27. See pp. 233C–233D. 28. CRITICAL THINKING Devise a plan and write a two-column proof for the following. E F B, A E F C, See p. 233D. Given: D A DB DB E , C F Prove: ABD CDB R V ABD CBD; SAS c. E H and D G bisect each other. not possible d. R G E U F D F G H DEF GHF; SAS S Lesson 4-4 BASEBALL For Exercises 26 and 27, use the following information. A baseball diamond is a square with four right angles and all sides congruent. ★ 26. Write a two-column proof to prove that the distance from first base to third base is the same as the distance from home plate to second base. ★ 27. Write a two-column proof to prove that the angle formed by second base, home plate, and third base is the same as the angle formed by second base, home plate, and first base. T RSU TSU; SSS Helping You Remember 1. Divide each square into four congruent parts. Use three different ways. Sample answers are shown. 3. Find three words that explain what it means to say that two triangles are congruent and that can help you recall the meaning of the SSS Postulate. Sample answer: Congruent triangles are triangles that are the same size and shape, and the SSS Postulate ensures that two triangles with three corresponding sides congruent will be the same size and shape. Lesson 4-4 Proving Congruence—SSS, SAS 205 4 Assess Standardized Test Practice Open-Ended Assessment Speaking Ask students to explain in their own words how they can use SSS and SAS to prove triangle congruence. Then students can explain how they would approach proving triangle congruence for some of the exercises in Lesson 4-4. 30. Which of the following statements about the figure is true? C A 90 a b B a b 90 b˚ a˚ C a b 90 D a b 31. Classify the triangle with the measures of the angles in the ratio 3:6:7. B A isosceles B acute C obtuse D right Maintain Your Skills Mixed Review Identify the congruent triangles in each figure. 32. B E Getting Ready for Lesson 4-5 33. (Lesson 4-3) X 34. M N L P C W Z Y WXZ YXZ Prerequisite Skill Students will learn about proving congruence with ASA and AAS in Lesson 4-5. They will apply the Angle Sum Theorem and concepts of angle and segment bisection toward proving triangle congruence. Use Exercises 44–47 to determine your students’ familiarity with bisectors of segments and angles. A D Find each measure if P QR Q . (Lesson 4-2) 35. m2 78 37. m5 68 39. m1 59 LMP NPM ACB DCE P Q 1 36. m3 102 38. m4 22 40. m6 34 56˚ 43˚ T For Exercises 41–43, use the graphic at the right. (Lesson 3-3) 41. Find the rate of change from first quarter to the second quarter. 1 42. Find the rate of change from the second quarter to the third quarter. 1.4 43. Compare the rate of change from the first quarter to the second, and the second quarter to the third. Which had the greater rate of change? Assessment Options Quiz (Lessons 4-3 and 4-4) is available on p. 239 of the Chapter 4 Resource Masters. Mid-Chapter Test (Lessons 4-1 through 4-4) is available on p. 241 of the Chapter 4 Resource Masters. There is a steeper rate of decline from the second quarter to the third. 4 3 5 78˚ 2 6 R USA TODAY Snapshots® GDP slides in 2001 Gross domestic product in private industries, which generate 88% of GDP, slowed to 4.1% in 2000 from 4.8% in 1999. 1.3% Percentage changes for the first three quarters of 2001: 0.3% 0% –1.1% First quarter Second quarter Third quarter Source: The Bureau of Economic Analysis By Shannon Reilly and Suzy Parker, USA TODAY Getting Ready for the Next Lesson and AE are angle bisectors PREREQUISITE SKILL BD and segment bisectors. Name the indicated segments and angles. B 44. 45. 46. 47. a segment congruent to EC BE an angle congruent to ABD CBD an angle congruent to BDC BDA a segment congruent to A D CD 206 Chapter 4 Congruent Triangles 206 Chapter 4 Congruent Triangles E X (To review bisectors of segments and angles, see Lessons 1-5 and 1-6.) A D C Proving Congruence—ASA, AAS Lesson Notes • Use the ASA Postulate to test for triangle congruence. 1 Focus • Use the AAS Theorem to test for triangle congruence. Vocabulary are congruent triangles used in construction? • included side 5-Minute Check Transparency 4-5 Use as a quiz or review of Lesson 4-4. The Bank of China Tower in Hong Kong has triangular trusses for structural support. These trusses form congruent triangles. In this lesson, we will explore two additional methods of proving triangles congruent. Mathematical Background notes are available for this lesson on p. 176D. ASA POSTULATE Suppose you were given the measures of two angles of a triangle and the side between them, the included side . Do these measures form a unique triangle? Congruent Triangles Using Two Angles and Included Side 1 Draw a triangle and label its vertices A, B, and C. 2 Draw any line 3 Construct an angle m and select a point L. Construct L K such that L K C B . congruent to C at L as a side of using LK the angle. 4 Construct an angle congruent to B at as a side K using LK of the angle. Label the point where the new sides of the angles meet J. J A C B m L K m m L L K 5 Cut out JKL and place it over ABC. How does JKL compare to ABC? are congruent triangles used in construction? Ask students: • How do the congruent triangles in the trusses contribute to the appearance of the structure? Sample answer: They make the structure visually appealing. • How would using congruent triangles make the structure easier to assemble? The triangles could be manufactured in bulk and construction workers could place any triangle in any location. K JKL ABC This construction leads to the Angle-Side-Angle Postulate, written as ASA. Study Tip Postulate 4.3 Reading Math Angle-Side-Angle Congruence If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. The included side refers to the side that each of the angles share. C04-174C T C W H G R RTW CGH Abbreviation: ASA Lesson 4-5 Proving Congruence—ASA, AAS 207 Sylvain Grandadam/Photo Researchers Resource Manager Workbook and Reproducible Masters Chapter 4 Resource Masters • Study Guide and Intervention, pp. 207–208 • Skills Practice, p. 209 • Practice, p. 210 • Reading to Learn Mathematics, p. 211 • Enrichment, p. 212 Teaching Geometry With Manipulatives Masters, pp. 8, 16, 17, 77 Transparencies 5-Minute Check Transparency 4-5 Real-World Transparency 4 Answer Key Transparencies Technology GeomPASS: Tutorial Plus, Lesson 10 Interactive Chalkboard Multimedia Applications: Virtual Activities Lesson x-x Lesson Title 207 Example 1 Use ASA in Proofs 2 Teach D Proof: W E because alternate interior angles are congruent. By the Midpoint Theorem, WL EL. Since vertical angles are congruent, WLR ELD. WRL EDL by ASA. P CP CP Since CP bisects BCR and BPR, BCP RCP and BPC RPC. by the Reflexive Property. By ASA, BCP RCP. Given: L is the midpoint . of WE R || E D W Prove: WRL EDL E B Proof: Power Point® 1 Write a paragraph proof. W BCP RCP Prove: In-Class Examples L R Write a paragraph proof. Given: CP bisects BCR and BPR. ASA POSTULATE R C AAS THEOREM Suppose you are given the measures of two angles and a nonincluded side. Is this information sufficient to prove two triangles congruent? Angle-Angle-Side Congruence Model 1. Draw a triangle on a piece of patty paper. Label the vertices A, B, and C. 2. Copy AB , B, and C on another piece of patty paper and cut them out. B B A A C B B A 3. Assemble them to form a triangle in which the side is not the included side of the angles. B B C C B C Analyze 1. They are congruent. 1. Place the original ABC over the assembled figure. How do the two triangles compare? 2. Make a conjecture about two triangles with two angles and the nonincluded side of one triangle congruent to two angles and the nonincluded side of the other triangle. The triangles are congruent. This activity leads to the Angle-Angle-Side Theorem, written as AAS. Theorem 4.5 Angle-Angle-Side Congruence If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. J Abbreviation: AAS Example: JKL CAB Proof K A L C B Theorem 4.5 T Given: M S, J R, M P S T M R Prove: JMP RST Proof: Statements 1. M S, J R, M P S T 2. P T 3. JMP RST J Reasons 1. Given 2. Third Angle Theorem 3. ASA 208 Chapter 4 Congruent Triangles Geometry Activity Materials: patty paper, straightedge, scissors • When students are copying B and C, tell them to extend the sides so that they are longer than the sides in the original triangle. Explain that these sides represent unknown side lengths. • In Step 2, make sure students copy C and not A. 208 Chapter 4 Congruent Triangles S P Study Tip Overlapping Triangles When triangles overlap, it is a good idea to draw each triangle separately and label the congruent parts. Example 2 Use AAS in Proofs AAS THEOREM B Write a flow proof. Given: EAD EBC BC AD A A E BE Prove: Flow Proof: In-Class Examples D C Power Point® 2 Write a proof. E Given: NKL NJM KL JM Prove: LN MN EAD EBC Given AD BC ADE BCE AE BE Given AAS CPCTC J K M L E E Reflexive Property N Proof: Statements (Reasons) 1. N N (Reflex. Prop. of ) 2. NKL NJM (Given) 3. KL JM (Given) 4. JNM KNL (AAS) 5. LN MN (CPCTC) You have learned several methods for proving triangle congruence. The Concept Summary lists ways to help you determine which method to use. Methods to Prove Triangle Congruence Definition of Congruent Triangles All corresponding parts of one triangle are congruent to the corresponding parts of the other triangle. SSS The three sides of one triangle must be congruent to the three sides of the other triangle. SAS Two sides and the included angle of one triangle must be congruent to two sides and the included angle of the other triangle. ASA Two angles and the included side of one triangle must be congruent to two angles and the included side of the other triangle. AAS Two angles and a nonincluded side of one triangle must be congruent to two angles and side of the other triangle. 3 STANCES When Ms. Gomez puts her hands on her hips, she forms two triangles with her upper body and arms. Suppose her arm lengths AB and DE measure 9 inches, and AC and EF measure 11 inches. Also suppose that you are C DF . given that B Determine whether ABC EDF. Justify your answer. Example 3 Determine if Triangles Are Congruent Architect About 28% of architects are self-employed. Architects design a variety of buildings including offices, retail spaces, and schools. ARCHITECTURE This glass chapel was designed by Frank Lloyd Wright’s son, Lloyd Wright. TV Suppose the redwood supports, T U and , measure 3 feet, TY 1.6 feet, and mU and mV are 31. Determine whether TYU TYV. Justify your answer. V Y D B U Explore We are given three measurements of each triangle. We need to determine whether the two triangles are congruent. Plan Since mU mV, U V. TV TY Likewise, TU TV so TU , and TY TY so T Y . Check each possibility using the five methods you know. Solve We are given information about side-side-angle (SSA). This is not a method to prove two triangles congruent. Online Research For information about a career as an architect, visit: www.geometryonline. com/careers T A E C F With B C DF, you could use SSS to prove ABC EDF. (continued on the next page) www.geometryonline.com/extra_examples Lesson 4-5 Proving Congruence—ASA, AAS 209 (l)Dennis MacDonald/PhotoEdit, (r)Michael Newman/PhotoEdit Differentiated Instruction Intrapersonal Ask students to study the proofs for the examples in this lesson and note the properties that recur, such as the reflexive properties of angles and segments, bisectors, midpoints, and so on. Students can start a list of helpful tools and things to watch for when they are working proofs and include recurring properties, theorems, formulas and methods that they can refer to in later lessons. They can also look at the order of the steps in paragraph proofs, flow proofs, and two-column proofs for similarities and differences. Lesson 4-5 Proving Congruence—ASA, AAS 209 Intervention A student may ask about proving congruence with AAA. Explain that while congruent triangles do share three congruent angles, AAA is not a possible tool for proving congruence of triangles because two triangles with three corresponding congruent angles can be similar but not congruent. Provide students with an example of two different-sized similar triangles. Examine Use a compass, protractor, and ruler to draw a triangle with the given measurements. 1.6 cm For simplicity of measurement, we will use centimeters instead of feet, so the measurements of the construction and those 31° of the support beams will be proportional. 3.0 cm • Draw a segment 3.0 centimeters long. • At one end, draw an angle of 31°. Extend the line longer than 3.0 centimeters. • At the other end of the segment, draw an arc with a radius of 1.6 centimeters such that it intersects the line. New Notice that there are two possible segments that could determine the triangle. Since the given measurements do not lead to a unique triangle, we cannot show that the triangles are congruent. Concept Check 1–2. See margin. 3 Practice/Apply Third Angle Theorem. Postulates are accepted as true without proof. Guided Practice GUIDED PRACTICE KEY Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 4. • include any other item(s) that they find helpful in mastering the skills in this lesson. 1. Find a counterexample to show why AAA (Angle-Angle-Angle) cannot be used to prove triangle congruence. 2. OPEN ENDED Draw a triangle and label the vertices. Name two angles and the included side. 3. Explain why AAS is a theorem, not a postulate. AAS can be proven using the Exercises Examples 4, 6 5, 7 8 1 2 3 Write a flow proof. 4–5. See p. 233D. KJ, G HJ 4. Given: GH 㛳 K 㛳 Prove: GJK JGH G YZ 5. Given: XW 㛳 , X Z Prove: WXY YZW X H K Y W J Z Write a paragraph proof. 6–7. See p. 233D. 6. Given: QS bisects RST; R T. 7. Given: E K, DGH DHG KH Prove: QRS QTS EG Prove: EGD KHD R D Q Answers S T 1. Two triangles can have corresponding congruent angles without corresponding congruent sides. A D, B E, and C F. However, AB DE, so ABC DEF. Application E 8. PARACHUTES Suppose S T and M L each measure R and M 7 feet, S K each measure 5.5 feet, and mT mL 49. Determine whether SRT MKL. Justify your answer. See margin. D A C C A 210 210 Chapter 4 Congruent Triangles F 2. Sample answer: In ABC, A B is the included side of A and B. B Chapter 4 Congruent Triangles H K R 8. This cannot be determined. The information given cannot be used with any of the triangle congruence postulates, theorems or the definition of congruent triangles. By construction, two different triangles can be shown with the given information. Therefore, it cannot be determined if SRT MKL. 5.5 49 7 5.5 K M S T B E G L ★ indicates increased difficulty Practice and Apply For Exercises See Examples 9, 11, 14, 15–18 10, 12, 13, 19, 20 21–28 2 1 F Organization by Objective • ASA Postulate: 10, 12, 13, 19, 20 • AAS Theorem: 9, 11, 14–18, 21–28 J E D 3 K Odd/Even Assignments Exercises 9–28 are structured so that students practice the same concepts whether they are assigned odd or even problems. G Extra Practice See page 762. About the Exercises… Write a flow proof. 9–14. See pp. 233D– 233E. 9. Given: EF GH 10. Given: DE DK GH 㛳 , E F 㛳 JK , bisects JE . K K H Prove: EGD JGK Prove: E K H G E 12. Given: EJ 㛳 EF GH FK KH , JG 㛳 , Prove: EJG FKH 11. Given: V S, TV QS Prove: V SR R T J S 1 K Assignment Guide Basic: 9–17 odd, 21–29 odd, 30–41 Average: 9–29 odd, 30–41 Advanced: 10–28 even, 29–38 (optional: 39–41) 2 R V E Q PQ 13. Given: MN , M Q 2 3 Prove: MLP QLN F G H 14. Given: Z is the midpoint of CT . Y㛳 E C T Prove: Y Z E Z L E Answers C 15. Given: NOM POR, NM ⊥ MR , P R ⊥ MR , NM PR Prove: MO OR Z 1 M 2 N 3 4 P T Q Y Write a paragraph proof. 15–18. See margin. 15. Given: NOM POR, 16. Given: DL bisects B N , N M M R XLN XDB R M R, N M P R Prove: L DB P N Prove: M O O R B N M D X O L R F T B D X L C J R X H G E N SY 18. Given: TX 㛳 TXY TSY Prove: TSY YXT 17. Given: F J, E H E GH C F HJ Prove: E O Proof: Since N M ⊥M R and PR ⊥ MR , M and R are right angles. M R because all right angles are congruent. We know that NOM POR and NM PR . By AAS, NMO PRO. M O OR by CPCTC. 16. Given: DL bisects B N . XLN XDB Prove: LN DB P M P N S Y Lesson 4-5 Proving Congruence—ASA, AAS 17. Given: F J, E H F EC GH H G Prove: EF HJ C E Proof: We are given that J F J, E H, and EC GH . By the Reflexive Property, CG CG . Segment addition results in EG EC CG and CH CG GH. By the definition of congruence, EC GH and CG CG. Substitute to find EG CH. By AAS, EFG HJC. By CPCTC, EF HJ. 211 N Proof: Since D L bisects B N , BX XN . XLN XDB . LXN DXB because vertical angles are congruent. LXN DXB by AAS. LN DB by CPCTC. 18. Given: TX || S Y X T TXY TSY Prove: TSY YXT Proof: Since TX || S Y , S Y YTX TYS by Alternate Interior Angles Theorem. TY TY by the Reflexive Property. Given TXY TSY, TSY YXT by AAS. Lesson 4-5 Proving Congruence—ASA, AAS 211 NAME ______________________________________________ DATE Write a two-column proof. 19–20. See pp. 233E–233F. ★ 20. Given: BMI KMT ★ 19. Given: MYT NYT PT MTY NTY IP Prove: RYM RYN Prove: IPK TPB ____________ PERIOD _____ Study Guide andIntervention Intervention, 4-5 Study Guide and p. 207 andAAS p. 208 Proving(shown) Congruence—ASA, ASA Postulate The Angle-Side-Angle (ASA) Postulate lets you show that two triangles are congruent. If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. ASA Postulate M Example Find the missing congruent parts so that the triangles can be proved congruent by the ASA Postulate. Then write the triangle congruence. a. B B I P E A C D M F Two pairs of corresponding angles are congruent, A D and C F. If the and D F are congruent, then ABC DEF by the ASA Postulate. included sides AC b. S R X R T W T Y T Y XY . If S X, then RST YXW by the ASA Postulate. R Y and SR What corresponding parts must be congruent in order to prove that the triangles are congruent by the ASA Postulate? Write the triangle congruence statement. 1. 2. C D W A Z W Y WY ; XYW ZYW; WXY WZY 5. B GARDENING For Exercises 21 and 22, use the following information. Beth is planning a garden. She wants the triangular sections, CFD and HFG, to be congruent. F is the midpoint of DG , and DG 16 feet. 21–22. See p. 233E. B Y A DC BC ; CDE CBA 4. A 3. X B E E D C ABE CBD; ABE CBD 6. V R B D T C BD DB ; ADB CBD; ABD CDB A S T VT ; RST UVT C E Lesson 4-5 D S ACB E; ABC CDE NAME ______________________________________________ DATE /M G Hill 207 ____________ Gl PERIOD G _____ Skills Practice, 4-5 Practice (Average) p. 209 and Practice, p. 210 (shown) Proving Congruence—ASA, AAS 1. Write a flow proof. T . Given: S is the midpoint of Q Q R || T U Prove: QSR TSU R U Sample proof: S is the midpoint of QT. QS TS Def.of midpoint Given QR || TU Q T Alt. Int. are . Given QSR TSU ASA QSR TSU Vertical are . Kites 2. Write a paragraph proof. Given: D F bisects DEF. GE Prove: D G FG D The largest kite ever flown was 210 feet long and 72 feet wide. E G F Proof: Since it is given that GE bisects DEF, DEG FEG by the definition of an angle bisector. It is given that D F. By the Reflexive Property, GE GE . So DEG FEG by AAS. Therefore DG FG by CPCTC. ARCHITECTURE For Exercises 3 and 4, use the following information. An architect used the window design in the diagram when remodeling an art studio. AB and C B each measure 3 feet. Source: Guinness Book of World Records B A D C 3. Suppose D is the midpoint of A C . Determine whether ABD CBD. Justify your answer. CD Since D is the midpoint of A C , AD by the definition of midpoint. A B CB by the definition of congruent segments. By the Reflexive Property, BD BD . So ABD CBD by SSS. 4. Suppose A C. Determine whether ABD CBD. Justify your answer. We are given A B CB and A C. BD BD by the Reflexive Property. Since SSA cannot be used to prove that triangles are congruent, we cannot say whether ABD CBD. NAME ______________________________________________ DATE /M G Hill 210 Gl ELL Mathematics, p. 211 Proving Congruence—ASA, AAS How are congruent triangles used in construction? Read the introduction to Lesson 4-5 at the top of page 207 in your textbook. Which of the triangles in the photograph in your textbook appear to be congruent? Sample answer: The four right triangles are congruent to each other. The two obtuse isosceles triangles are congruent to each other. Reading the Lesson 1. Explain in your own words the difference between how the ASA Postulate and the AAS Theorem are used to prove that two triangles are congruent. Sample answer: In ASA, you use two pairs of congruent angles and the included congruent sides. In AAS, you use two pairs of congruent angles and a pair of nonincluded congruent sides. B, D, E, G, H 2. Which of the following conditions are sufficient to prove that two triangles are congruent? A. Two sides of one triangle are congruent to two sides of the other triangle. B. The three sides of one triangles are congruent to the three sides of the other triangle. C. The three angles of one triangle are congruent to the three angles of the other triangle. D. All six corresponding parts of two triangles are congruent. E. Two angles and the included side of one triangle are congruent to two sides and the included angle of the other triangle. F. Two sides and a nonincluded angle of one triangle are congruent to two sides and a nonincluded angle of the other triangle. G. Two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the other triangle. H. Two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. I. Two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of the other triangle. 3. Determine whether you have enough information to prove that the two triangles in each figure are congruent. If so, write a congruence statement and name the congruence postulate or theorem that you would use. If not, write not possible. AEB DEC; AAS E a. b. T is the midpoint of R U . U S T A B C RST UVT; ASA D R V Helping You Remember 4. A good way to remember mathematical ideas is to summarize them in a general statement. If you want to prove triangles congruent by using three pairs of corresponding parts, what is a good way to remember which combinations of parts will work? Sample answer: At least one pair of corresponding parts must be sides. If you use two pairs of sides and one pair of angles, the angles must be the included angles. If you use two pairs of angles and one pair of sides, then the sides must both be included by the angles or must both be corresponding nonincluded sides. 212 F D H K Chapter 4 Congruent Triangles J 23. If N is the midpoint of JL and K M JL , determine whether JKN LKN. Justify your answer. M L M and NJM NLM, determine whether 24. If J JNM LNM. Justify your answer. Complete each congruence statement and the postulate or theorem that applies. M R V and 2 5, then 25. If I INM ? by ? . VNR, AAS or ASA M R V and I R M V, then 26. If I M IRN ? by ? . VMN, ASA or AAS V and R M bisect each other, then 27. If I RVN ? by ? . MIN, SAS 28. If MIR RVM and 1 6, then MRV ? by ? . RMI, AAS or ASA 212 M I 4 R 3 1 2 N 5 6 7 8 V See margin. Chapter 4 Congruent Triangles Courtesy Peter Lynn Kites NAME ______________________________________________ DATE 4-5 Enrichment Enrichment, ____________ PERIOD _____ p. 212 Congruent Triangles in the Coordinate Plane If you know the coordinates of the vertices of two triangles in the coordinate plane, you can often decide whether the two triangles are congruent. There may be more than one way to do this. 1. Consider ABD and CDB whose vertices have coordinates A(0, 0), B(2, 5), C(9, 5), and D(7, 0). Briefly describe how you can use what you know about congruent triangles and the coordinate plane to show that ABD CDB. You may wish to make a sketch to help get you started. Sample answer: Show that the slopes of A B and C D are BC equal and that the slopes of A D and are equal. Conclude B C D and B C A D . Use the angle relationships for that A parallel lines and a transversal and the fact that BD is a common side for the triangles to conclude that ABD CDB by ASA. L N 29. CRITICAL THINKING Aiko wants to estimate the distance between herself and a duck. She adjusts the visor of her cap so that it is in line with her line of sight to the duck. She keeps her neck stiff and turns her body to establish a line of sight to a point on the ground. Then she paces out the distance to the new point. Is the distance from the duck the same as the distance she just paced out? Explain your reasoning. ____________ Gl PERIOD G _____ Reading 4-5 Readingto to Learn Learn Mathematics Pre-Activity G KITES For Exercises 23 and 24, use the following information. Austin is building a kite. Suppose JL is 2 feet, JM is 2.7 feet, and the measure of NJM is 68. 23–24. See p. 233F. T S Q C 21. Suppose C D and G H each measure 4 feet and the measure of CFD is 29. Determine whether CFD HFG. Justify your answer. H, and C H D G. 22. Suppose F is the midpoint of C Determine whether CFD HFG. Justify your answer. U Gl K N Exercises Answer 29. Since Aiko is perpendicular to the ground, two right angles are formed and right angles are congruent. The angles of sight are the same and her height is the same for each triangle. The triangles are congruent by ASA. By CPCTC, the distances are the same. The method is valid. 30. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How are congruent triangles used in construction? 4 Assess Include the following in your answer: • explain how to determine whether the triangles are congruent, and • why it is important that triangles used for structural support are congruent. Standardized Test Practice 31. In ABC, AD and D C are angle bisectors and mB 76. What is the measure of ADC? D A 26 B 52 C 76 D 128 Open-Ended Assessment Writing Have students practice writing different versions of proofs for each example. For Example 1, students can write a flow proof and a two-column proof. B D A C 32. ALGEBRA For a positive integer x, 1 percent of x percent of 10,000 equals A A x. B 10x. C 100x. D Getting Ready for Lesson 4-6 1000x. Prerequisite Skill Students will learn about isosceles triangles in Lesson 4-6. They will use congruence postulates and theorems when writing proofs. Use Exercises 39–41 to determine your students’ familiarity with the classification of triangles by sides. Maintain Your Skills Mixed Review Write a flow proof. (Lesson 4-4) 33–34. See p. 233F. 33. Given: BA DE BE 34. Given: XZ WY , D A Prove: BEA DAE X Z bisects W Y . Prove: WZX YZX B D W C Z A E Answers Y X 35–36. See margin. 30. Sample answer: The triangular trusses support the structure. Answers should include the following. • To determine whether two triangles are congruent, information is needed about consecutive side-angle-side, side-side-side, angle-sideangle, angle-angle-side, or about each angle and each side. • Triangles that are congruent will support weight better because the pressure will be evenly divided. 35. Turn; RS 2, RS 2, ST 1, ST 1, RT 1, RT 1. Use a protractor to confirm that the corresponding angles are congruent. 36. Flip; MP 2, MP 2, MN 3, MN 3, NP 13, NP 13. Use a protractor to confirm that the corresponding angles are congruent. Verify that each of the following preserves congruence and name the congruence transformation. (Lesson 4-3) y y 35. 36. T S O S' T' x M Getting Ready for the Next Lesson P P' M' x O R' 37. If people are happy, then they rarely correct their faults. N' N R Write each statement in if-then form. (Lesson 2-3) 37. Happy people rarely correct their faults. 38. A champion is afraid of losing. If a person is a champion, then he or she is afraid of losing. PREREQUISITE SKILL Classify each triangle according to its sides. (To review classification by sides, see Lesson 4-1.) 39. 40. equilateral 41. isosceles isosceles www.geometryonline.com/self_check_quiz Lesson 4-5 Proving Congruence—ASA, AAS 213 Teacher to Teacher Karyn S. Cummins, Franklin Central High School Indianapolis, IN After proofs are introduced I write up several simple proofs on card stock. I then cut the statements and reasons apart, give them to the students and have them reconstruct them. Lesson 4-5 Proving Congruence—ASA, AAS 213 Geometry Activity A Follow-Up of Lesson 4-5 Getting Started Objective Explore congruence in right triangles. Materials ruler protractor Teach • Explain that right triangles are unique and typically have special relationships. Students will want to check for these relationships when they are working on proofs. • Remind students that the right triangle theorems do not work for acute or obtuse triangles, but only for right triangles. Assess Exercises 1–3 guide students through SAS, ASA, and AAS, and introduce LL, HA, and LA. Exercises 4–6 demonstrate that SSA works with right triangles and forms the HL Postulate. Exercises 7–11 use the right triangle congruence theorems and postulate in proofs. A Follow-Up of Lesson 4-5 Congruence in Right Triangles In Lessons 4-4 and 4-5, you learned theorems and postulates to prove triangles congruent. Do these theorems and postulates apply to right triangles? Activity 1 Triangle Congruence Model Study each pair of right triangles. a. b. c. Analyze 1. Is each pair of triangles congruent? If so, which congruence theorem or postulate applies? yes; a. SAS, b. ASA, c. AAS 2. Rewrite the congruence rules from Exercise 1 using leg, (L), or hypotenuse, (H), to replace side. Omit the A for any right angle since we know that all right triangles contain a right angle and all right angles are congruent. a. LL, b. LA, c. HA 3. Make a conjecture If you know that the corresponding legs of two right None; two pairs of triangles are congruent, what other information do you need to declare legs congruent is the triangles congruent? Explain. sufficient for prov- ing right triangles congruent. In Lesson 4-5, you learned that SSA is not a valid test for determining triangle congruence. Can SSA be used to prove right triangles congruent? Activity 2 SSA and Right Triangles Make a Model How many right triangles exist that have a hypotenuse of 10 centimeters and a leg of 7 centimeters? Draw X Y so that XY 7 centimeters. Use a protractor to draw a ray from Y that is perpendicular to X Y . Open your compass to a width of 10 centimeters. Place the point at X and draw a long arc to intersect the ray. Label the intersection Z and draw X Z to complete XYZ. Z 10 cm Study Notebook Ask students to summarize what they have learned about congruence in right triangles. Tell students to list each method with a brief description. X Y X Y X Y X 214 Investigating Slope-Intercept Form 214 Chapter 4 Congruent Triangles Resource Manager 214 Chapter 4 Congruent Triangles Teaching Geometry with Manipulatives Glencoe Mathematics Classroom Manipulative Kit • p. 78 (student recording sheet) • p. 16 (protractor) • p. 17 (ruler) • protractor • ruler 7 cm Y A Follow-Up of Lesson 4-5 Answers Analyze 4. Does the model yield a unique triangle? yes 5. Can you use the lengths of the hypotenuse and a leg to show right triangles are 7. Given: DEF and RST are right triangles. E and S are right angles. EF ST ED SR Prove: DEF RST congruent? yes 6. Make a conjecture about the case of SSA that exists for right triangles. SSA is a valid test of congruence for right triangles. The two activities provide evidence for four ways to prove right triangles congruent. F T E R S Right Triangle Congruence Theorem Abbreviation 4.6 Leg-Leg Congruence If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. LL 4.7 Hypotenuse-Angle Congruence If the hypotenuse and acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the two triangles are congruent. HA 4.8 Leg-Angle Congruence If one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent. LA D Example Proof: We are given that EF ST, ED SR , and E and S are right angles. Since all right angles are congruent, E S. Therefore, by SAS, DEF RST. 8. Given: ABC and XYZ are right triangles. A and X are right angles. BC YZ B Y Prove: ABC XYZ Postulate 4.4 PROOF Hypotenuse-Leg Congruence If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. C Z A B X Y Proof: We are given that ABC and XYZ are right triangles with right angles A and X, BC YZ, and B Y. Since all right angles are congruent, A X. Therefore, ABC XYZ by AAS. HL Write a paragraph proof of each theorem. 7–9. See margin. 7. Theorem 4.6 8. Theorem 4.7 9. Theorem 4.8 (Hint: There are two possible cases.) Use the figure to write a two-column proof. 10–11. See p. 233F. 10. Given: ML MK KM 11. Given: JK KM , JM KL , JK J L ML JK Prove: JM KL Prove: ML JK J M L K Geometry Activity Congruence in Right Triangles 215 9. Case 1: C F Given: ABC and DEF are right triangles. A B D E AC DF, C F Prove: ABC DEF Proof: It is given that ABC and DEF are right triangles, AC DF, C F. By the definition of right triangles, A and D are right angles. Thus, A D since all right angles are congruent. ABC DEF by ASA. Case 2: C F Given: ABC and DEF are right triangles. A B D E C A DF, B E Prove: ABC DEF Proof: If is given that ABC and DEF are right triangles, AC DF, and B E. By the definition of right triangle, A and D are right angles. Thus, A D since all right angles are congruent. ABC DEF by AAS. Geometry Activity Congruence in Right Triangles 215 Lesson Notes Isosceles Triangles • Use properties of isosceles triangles. 1 Focus 5-Minute Check Transparency 4-6 Use as a quiz or review of Lesson 4-5. • Use properties of equilateral triangles. are triangles used in art? Vocabulary • vertex angle • base angles Mathematical Background notes are available for this lesson on p. 176D. are triangles used in art? Ask students: • How would the painting’s overall appearance change if you removed or covered the triangles? The painting would appear more blended and based on curves, circles, and ovals without the stark triangles. • Describe where isosceles triangles appear in the art. Accept all reasonable answers. The art of Lois Mailou Jones, a twentieth-century artist, includes paintings and textile design, as well as book illustration. Notice the isosceles triangles in this painting, Damballah. PROPERTIES OF ISOSCELES TRIANGLES In Lesson 4-1, you learned that isosceles triangles have two congruent sides. Like the right triangle, the parts of an isosceles triangle have special names. The angle formed by the congruent sides is called the vertex angle. leg leg base The two angles formed by the base and one of the congruent sides are called base angles. In this activity, you will investigate the relationship of the base angles and legs of an isosceles triangle. Isosceles Triangles C Model • Draw an acute triangle on patty paper with A C B C . • Fold the triangle through C so that A and B coincide. Analyze 2, 3. They are congruent. A 1. What do you observe about A and B? A B 2. Draw an obtuse isosceles triangle. Compare the base angles. 3. Draw a right isosceles triangle. Compare the base angles. B The results of the Geometry Activity suggest Theorem 4.9. Theorem 4.9 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Example: If AB CB , then A C. 216 B A Chapter 4 Congruent Triangles Marvin T. Jones Resource Manager Workbook and Reproducible Masters Chapter 4 Resource Masters • Study Guide and Intervention, pp. 213–214 • Skills Practice, p. 215 • Practice, p. 216 • Reading to Learn Mathematics, p. 217 • Enrichment, p. 218 • Assessment, p. 240 School-to-Career Masters, p. 8 Teaching Geometry With Manipulatives Masters, pp. 8, 80, 81 Transparencies 5-Minute Check Transparency 4-6 Answer Key Transparencies Technology Interactive Chalkboard C Example 1 Proof of Theorem Write a two-column proof of the Isosceles Triangle Theorem. Q R Q Given: PQR, P Prove: PR R Proof: Statements 1. Let S be the midpoint of P R . 2. Draw an auxiliary segment Q S . 3. P S R S 4. Q S Q S 5. P Q R Q 6. PQS RQS 7. PR S 2 Teach P PROPERTIES OF ISOSCELES TRIANGLES Q Reasons 1. Every segment has exactly one midpoint. 2. Two points determine a line. 3. Midpoint Theorem 4. Congruence of segments is reflexive. 5. Given 6. SSS 7. CPCTC Diagrams Label the diagram with the given information. Use your drawing to plan the next step in solving the problem. Given: AB CB BD, ACB BCD Prove: A D D B Multiple-Choice Test Item A If GH HJ JK H K , , and mGJK100, what is the measure of HGK? A 10 B 15 C 20 D 25 K G Test-Taking Tip Power Point® 1 Write a two-column proof. C Example 2 Find the Measure of a Missing Angle Standardized Test Practice In-Class Examples H J Read the Test Item GHK is isosceles with base GK . Likewise, HJK is isosceles with base H K . Solve the Test Item Step 1 The base angles of HJK are congruent. Let x mKHJ mHKJ. mKHJ mHKJmHJK 180 Angle Sum Theorem x x 100 180 Substitution 2 If DE CD , BC AC , and 2x 100 180 Add. 2x80 x 40 Step 2 Subtract 100 from each side. So, mKHJ mHKJ 40. mCDE 120, what is the measure of BAC? D D GHK and KHJ form a linear pair. Solve for mGHK. mKHJmGHK 180 Linear pairs are supplementary. 40 mGHK 180 Substitution mGHK 140 Step 3 Proof: Statements (Reasons) 1. AB CB BD (Given) 2. A B C B B D (Def. of seg.) 3. ABC and BCD are isosceles. (Def. of isos. ) 4. A ACB, BCD D (Isos. Th.) 5. ACB BCD (Given) 6. A D (Transitive Prop.) C B Subtract 40 from each side. The base angles of GHK are congruent. Let y represent mHGK and mGKH. mGHK mHGK mGKH 180 Angle Sum Theorem 140 y y 180 Substitution 140 2y180 Add. 2y 40 Subtract 140 from each side. y 20 Divide each side by 2. E A A 45.5 C 68.5 B 57.5 D 75 The measure of HGK is 20. Choice C is correct. www.geometryonline.com/extra_examples Lesson 4-6 Isosceles Triangles 217 Geometry Activity Materials: paper, scissors, ruler • You may wish to provide students with rectangular dot paper to help them draw accurate isosceles triangles. • Ask students to name the legs, base, vertex angle, and base angles of their triangles. • Have students repeat the activity starting with a line segment and two congruent angles drawn at each end of the segment. Lesson 4-6 Isosceles Triangles 217 In-Class Example 3 The converse of the Isosceles Triangle Theorem is also true. Power Point® Theorem 4.10 M If two angles of a triangle are congruent, then the sides opposite those angles are congruent. P D Abbreviation: Conv. of Isos. Th. L Example: If D F, then DE F E . N a. Name two congruent angles. MLN and MNL F You will prove Theorem 4.10 in Exercise 33. b. Name two congruent L and P M segments. P Example 3 Congruent Segments and Angles PROPERTIES OF EQUILATERAL TRIANGLES In-Class Example E Power Point® 4 Copy the figure in Example 4 and then draw EJ so that EJ bisects 2, and J lies on F G . You can use properties of triangles to prove Thales of Miletus’ important geometric ideas. Visit www.geometryonline. com/webquest to continue work on your WebQuest project. a. Name two congruent angles. AFC is opposite AC and ACF is opposite A F , so AFCACF. C A B H F b. Name two congruent segments. By the converse of the Isosceles Triangle Theorem, the sides opposite congruent angles are congruent. So, BC B F . PROPERTIES OF EQUILATERAL TRIANGLES Recall that an equilateral triangle has three congruent sides. The Isosceles Triangle Theorem also applies to equilateral triangles. This leads to two corollaries about the angles of an equilateral triangle. a. Find mHEJ and mEJH. 15; 75 Corollaries b. Find mEJG. 105 4.3 4.4 A triangle is equilateral if and only if it is equiangular. Each angle of an equilateral triangle measures 60°. 60˚ 60˚ 60˚ You will prove Corollaries 4.3 and 4.4 in Exercises 31 and 32. Example 4 Use Properties of Equilateral Triangles EFG is equilateral, and EH bisects E. a. Find m1 and m2. Each angle of an equilateral triangle measures 60°. So, m1 m2 60. Since the angle was bisected, m1 m2. Thus, m1m230. b. ALGEBRA Find x. mEFHm1mEHF180 60 30 15x 180 90 15x180 15x 90 x 6 E 1 2 15x˚ F Angle Sum Theorem mEFH 60, m1 30, mEFH 15x Add. Subtract 90 from each side. Divide each side by 15. 218 Chapter 4 Congruent Triangles Differentiated Instruction Interpersonal Have groups of students work Exercise 7 on p. 219. Some group members can provide the statements, and other group members can provide corresponding reasons. Encourage groups to discuss the properties of isosceles and equilateral triangles while they are figuring out the proofs. 218 Chapter 4 Congruent Triangles H G Concept Check 3 Practice/Apply 1. Explain how many angles in an isosceles triangle must be given to find the measures of the other angles. 1–3. See margin. 2. Name the congruent sides and angles of isosceles WXZ with base WZ . 3. OPEN ENDED Guided Practice GUIDED PRACTICE KEY Exercises Examples 4–5 6 7 8 3 4 1 2 Study Notebook Describe a method to construct an equilateral triangle. Refer to the figure. D 4. If AD A H , name two congruent angles. ADH AHD 5. If BDH BHD, name two congruent segments. BH BD C B Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 4. • include a concept map with all methods of proving triangle congruence accompanied by helpful theorems, postulates, properties, and formulas. • include any other item(s) that they find helpful in mastering the skills in this lesson. A H 6. ALGEBRA Triangle GHF is equilateral with mF3x 4, mG6y, and 6 , 10, 3 mH19z3. Find x, y, and z. 5 3 Write a two-column proof. 7. Given: CTE is isosceles with vertex C. mT60 Prove: CTE is equilateral. T 60˚ C See p. 233F. E Standardized Test Practice 8. If P QS Q R R Q , S , and mPRS 72, what is the measure of QPS? A A 27 B 54 C 63 D 72 S R Q P ★ indicates increased difficulty About the Exercises… Practice and Apply For Exercises See Examples 9–14 15–22, 27–28, 34–37 23–26, 38–39 29–33 3 4 2 1 Extra Practice See page 762. Refer to the figure. 9. If L T L R , name two congruent angles. LTR LRT 10. If L X L W, name two congruent angles. LXW LWX 11. If S L Q L, name two congruent angles. LSQ LQS T 12. If LXY LYX, name two congruent segments. LXLY 13. If LSR LRS, name two congruent segments. LSLR X 14. If LYW LWY, name two congruent segments. LYLW KLN and LMN are isosceles and mJKN 130. Find each measure. 16. mM 140 15. mLNM 20 17. mLKN 81 18. mJ 106 L S Y R Q Z W K N J 18˚ 25˚ M 20˚ F 28˚ D Odd/Even Assignments Exercises 9–39 are structured so that students practice the same concepts whether they are assigned odd or even problems. Assignment Guide L DFG and FGH are isosceles, mFDH 28 and FG FH DG . Find each measure. 20. mDGF 124 19. mDFG 28 21. mFGH 56 22. mGFH 68 Organization by Objective • Properties of Isosceles Triangles: 9–14, 23–26, 29–33, 38–39 • Properties of Equilateral Triangles: 15–22, 27–28, 34–37 G H Lesson 4-6 Isosceles Triangles 219 Basic: 9–39 odd, 40–54 Average: 9–39 odd, 40–54 Advanced: 10–40 even, 41–51 (optional: 52–54) All: Quiz 2 (1–5) Answers 1. The measure of only one angle must be given in an isosceles triangle to determine the measures of the other two angles. 2. W X ZX , W Z 3. Sample answer: Draw a line segment. Set your compass to the length of the line segment and draw an arc from each endpoint. Draw segments from the intersection of the arcs to each endpoint. Lesson 4-6 Isosceles Triangles 219 NAME ______________________________________________ DATE In the figure, JM ML P M and P L . 23. If mPLJ34, find mJPM. 36.5 24. If mPLJ58, find mPJL. 30.5 ____________ PERIOD _____ Study Guide andIntervention Intervention, 4-6 Study Guide and Properties of Isosceles Triangles An isosceles triangle has two congruent sides. The angle formed by these sides is called the vertex angle. The other two angles are called base angles. You can prove a theorem and its converse about isosceles triangles. A • If two sides of a triangle are congruent, then the angles opposite those sides are congruent. (Isosceles Triangle Theorem) • If two angles of a triangle are congruent, then the sides opposite those angles are congruent. B C Lesson 4-6 p. 213 (shown) Isosceles Triangles and p. 214 Example 1 Example 2 Find x. B R BC BA, so mA mC. 5x 10 4x 5 x 10 5 x 15 Substitution Subtract 4x from each side. Add 10 to each side. G Triangle LMN is equilateral, and M P bisects L N . 27. Find x and y. x3; y18 28. Find the measure of each side of LMN. 10 Converse of Isos. Thm. Substitution Add 13 to each side. Subtract 2x from each side. Exercises K 35 R P 40 2x 2. S 2x 6 T 3x 6 12 3. 5y ˚ P L 15 W 4x 2 3x 1 N V Q 4. D P K T (6x 6) 2x 12 5. G 3x Y 20 B Z 6. PROOF 36 T 30 L D x R S B A 1 3 D 2 C E Statements Reasons 1.1 2 1.Given 2.2 3 2.Vertical angles are congruent. 3.1 3 3.Transitive Property of B CB 4.A 4.If two angles of a triangle are , then the sides opposite the angles are . Write a two-column proof. 29–33. See pp. 233F–233G. 29. Given: XKF is equilateral. XJ bisects X. Prove: J is the midpoint of K F . 3x 3x Q 7. Write a two-column proof. Given: 1 2 Prove: AB CB Gl J M Find x. 1. L T 2x mS mT, so SR TR. 3x 13 2x 3x 2x 13 x 13 Isos. Triangle Theorem M H 3x 13 (5x 10) A Find x. S (4x 5) J In the figure, G K G H and H K K J. 25. If mHGK28, find mHJK. 38 26. If mHGK42, find mHJK. 34.5 If AB CB , then A C. If A C, then A B CB . C P ★ 30. Given: MLP is isosceles. N is the midpoint of M P . Prove: L MP N X NAME ______________________________________________ DATE /M G Hill 213 Skills Practice, 4-6 Practice (Average) 1 ____________ Gl PERIOD G _____ p. 215 and Practice, p. 216 (shown) Isosceles Triangles 2 K Refer to the figure. R 1. If R V RT , name two congruent angles. RTV RVT L J F 31. Corollary 4.3 M 32. Corollary 4.4 N P 33. Theorem 4.10 S V 2. If R S SV , name two congruent angles. SVR SRV T U 34. DESIGN The basic structure covering Spaceship Earth at the Epcot Center in Orlando, Florida, is a triangle. Describe the minimum requirement to show that these triangles are equilateral. The minimum requirement is that two angles 3. If SRT STR, name two congruent segments. S T SR 4. If STV SVT, name two congruent segments. S T SV Triangles GHM and HJM are isosceles, with G H M H and H J M J . Triangle KLM is equilateral, and mHMK 50. Find each measure. J measure 60°. L K M H 5. mKML 60 6. mHMG 70 7. mGHM 40 ALGEBRA 35. G 8. If mHJM 145, find mMHJ. 17.5 9. If mG 67, find mGHM. 46 2x 5 10. Write a two-column proof. E || B C Given: DE 1 2 Prove: AB AC 2 3 C D 4 B Spaceship Earth is a completely spherical geodesic dome that is covered with 11,324 triangular aluminum and plastic alloy panels. Proof: Statements 1. D E Reasons || B C 1. Given 2. 1 4 2 3 2. Corr. are . 3. 1 2 3. Given 4. 3 4 4. Congruence of is transitive. 5. AB AC 5. If 2 of a are , then the sides opposite those are . 11. SPORTS A pennant for the sports teams at Lincoln High School is in the shape of an isosceles triangle. If the measure of the vertex angle is 18, find the measure of each base angle. Lin 81, 81 NAME ______________________________________________ DATE /M G Hill 216 Gl coln Haw Source: disneyworld.disney.go.com ks ELL 37. 30 symmetry is pleasing to the eye. • Why might isosceles right triangles be used in art? Sample answer: Two congruent isosceles right triangles can be placed together to form a square. m217, m326, m417, m518 Reading the Lesson R a. What kind of triangle is QRS? isosceles (2x 25)˚ (x 5)˚ C • Why do you think that isosceles and equilateral triangles are used more often than scalene triangles in art? Sample answer: Their S 1 2 3 4 5 42˚ A 77˚ D F G E Q d. Name the vertex angle of QRS. S e. Name the base angles of QRS. Q, R 220 Chapter 4 Congruent Triangles Dallas & John Heaton/Stock Boston C04-251C-829637 13 2 7 2. Determine whether each statement is always, sometimes, or never true. a. If a triangle has three congruent sides, then it has three congruent angles. always b. If a triangle is isosceles, then it is equilateral. sometimes c. If a right triangle is isosceles, then it is equilateral. never d. The largest angle of an isosceles triangle is obtuse. sometimes e. If a right triangle has a 45° angle, then it is isosceles. always f. If an isosceles triangle has three acute angles, then it is equilateral. sometimes g. The vertex angle of an isosceles triangle is the largest angle of the triangle. sometimes 3. Give the measures of the three angles of each triangle. a. an equilateral triangle 60, 60, 60 b. an isosceles right triangle 45, 45, 90 c. an isosceles triangle in which the measure of the vertex angle is 70 70, 55, 55 d. an isosceles triangle in which the measure of a base angle is 70 70, 70, 40 e. an isosceles triangle in which the measure of the vertex angle is twice the measure of one of the base angles 90, 45, 45 Helping You Remember NAME ______________________________________________ DATE 4-6 Enrichment Enrichment, ____________ PERIOD _____ p. 218 Triangle Challenges Some problems include diagrams. If you are not sure how to solve the problem, begin by using the given information. Find the measures of as many angles as you can, writing each measure on the diagram. This may give you more clues to the solution. 1. Given: BE BF, BFG BEF BED, mBFE 82 and ABFG and BCDE each have opposite sides parallel and congruent. Find m ABC. 148 A 4. If a theorem and its converse are both true, you can often remember them most easily by combining them into an “if-and-only-if” statement. Write such a statement for the Isosceles Triangle Theorem and its converse. Sample answer: Two sides of a triangle are B 2. Given: AC AD, and A B B D , mDAC 44 and C E bisects ACD. Find mDEC. 78 A C congruent if and only if the angles opposite those sides are congruent. E B G 220 (3x 8)˚ 18 40. CRITICAL THINKING In the figure, ABC is isosceles, DCE is equilateral, and FCG is isosceles. Find the measures of the five numbered angles at vertex C. m118, Read the introduction to Lesson 4-6 at the top of page 216 in your textbook. c. Name the base of QRS. Q R 36. 38. Trace the figure. Identify and draw one isosceles triangle from each set in the sign. 39. Describe the similarities between the different triangles. How are triangles used in art? b. Name the legs of QRS. Q S , R S 18 ARTISANS For Exercises 38 and 39, use the following information. This geometric sign from the Grassfields area in Western Cameroon (Western Africa) uses approximations of isosceles triangles within and around two circles. 38–39. See p. 233G. ____________ Gl PERIOD G _____ Mathematics, p. 217 Isosceles Triangles 1. Refer to the figure. 3x 13 (2x 20)˚ Reading 4-6 Readingto to Learn Learn Mathematics Pre-Activity 60˚ Design A 1 Find x. Chapter 4 Congruent Triangles D B 41. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How are triangles used in art? 4 Assess Include the following in your answer: • at least three other geometric shapes frequently used in art, and • a description of how isosceles triangles are used in the painting. Standardized Test Practice 42. Given right triangle XYZ with hypotenuse X Y , YP is equal to YZ. If mPYZ 26, find mXZP. A A 13 B 26 C 32 D 64 Open-Ended Assessment PX Y Z 43. ALGEBRA A segment is drawn from (3, 5) to (9, 13). What are the coordinates of the midpoint of this segment? D A (3, 4) B (12, 18) C (6, 8) D (6, 9) Maintain Your Skills Mixed Review Write a paragraph proof. (Lesson 4-5) 44–45. See p. 233G. 44. Given: ND, GI, 45. Given: VR RS UT SU , A N S D Prove: ANG SDI G R S U S Prove: VRS TUS N V R Getting Ready for Lesson 4-7 S S A D Prerequisite Skill Students will learn about triangles and coordinate proof in Lesson 4-7. They will prove congruency using distances and midpoints in coordinate planes. Use Exercises 52–54 to determine your students’ familiarity with finding midpoints. I U T Determine whether QRS EGH given the coordinates of the vertices. Explain. (Lesson 4-4) 46–47. See margin. 46. Q(3, 1), R(1, 2), S(1, 2), E(6, 2), G(2, 3), H(4, 1) 47. Q(1, 5), R(5, 1), S(4, 0), E(4, 3), G(1, 2), H(2, 1) 48–51. See p. 233G. Getting Ready for the Next Lesson Construct a truth table for each compound statement. (Lesson 2-2) 48. a and b 49. p or q 50. k and m 51. y or z PREREQUISITE SKILL Find the coordinates of the midpoint of the segment with the given endpoints. (To review finding midpoints, see Lesson 1-5.) 52. A(2, 15), B(7, 9) 53. C(4, 6), D(2, 12) 54. E(3, 2.5), F(7.5, 4) (4.5, 12) (1, 3) P ractice Quiz 2 Assessment Options (5.25, 3.25) Lessons 4-4 through 4-6 1. Determine whether JMLBDG given that J(4, 5), M(2, 6), A L(1, 1), B(3, 4), D(4, 2), and G(1, 1). (Lesson 4-4) See p. 233G. E 2. Write a two-column proof to prove that A J E H , given AH, AEJHJE. (Lesson 4-5) See p. 233G. J WXY and XYZ are isosceles and mXYZ128. Find each measure. (Lesson 4-6) 3. mXWY 52 4. mWXY 76 5. mYZX 26 H X 128˚ W www.geometryonline.com/self_check_quiz Speaking Have students come up with examples of how isosceles and equilateral triangles are used in paintings, ceramics, and decorative architecture. Students can talk about where the base and legs of isosceles triangles are typically situated in architecture, and they can discuss how geometry can help builders determine how much material they will need. They can also discuss the visual effect that isosceles and equilateral triangles have in different art forms. Y Z Practice Quiz 2 The quiz provides students with a brief review of the concepts and skills in Lessons 4-4 through 4-6. Lesson numbers are given to the right of the exercises or instruction lines so students can review concepts not yet mastered. Quiz (Lessons 4-5 and 4-6) is available on p. 240 of the Chapter 4 Resource Masters. Lesson 4-6 Isosceles Triangles 221 Answer 41. Sample answer: Artists use angles, lines and shapes to create visual images. Answers should include the following. • Rectangle, squares, rhombi, and other polygons are used in many works of art. • There are two rows of isosceles triangles in the painting. One row has three congruent isosceles triangles. The other row has six congruent isosceles triangles. 46. QR 17 , RS 20 , QS 13 , EG 17 , GH 20 , and EH 13. Each pair of corresponding sides have the same measure so they are congruent. QRS EGH by SSS. 47. QR 52 , RS 2, QS 34 , EG 34 , GH 10 , and EH 52. The corresponding sides are not congruent so QRS is not congruent to EGH. Lesson 4-6 Isosceles Triangles 221 Lesson Notes Triangles and Coordinate Proof • Position and label triangles for use in coordinate proofs. 1 Focus • Write coordinate proofs. 5-Minute Check Transparency 4-7 Use as a quiz or review of Lesson 4-6. Vocabulary • coordinate proof Mathematical Background notes are available for this lesson on p. 176D. can the coordinate plane be useful in proofs? Ask students: • If d 3 and BC 5, what are the coordinates of B and C? Classify ABC. (6, 0); (3, 4); isosceles • If you draw C D where D is the B, what method(s) midpoint of A could you use to prove ACD BCD? the Distance Formula and SSS Study Tip Placement of Figures The guidelines apply to any polygon placed on the coordinate plane. y x A (0, 0) O B (2d , 0 ) Coordinate proof uses figures in the coordinate plane and algebra to prove geometric concepts. The first step in writing a coordinate proof is the placement of the figure on the coordinate plane. Placing Figures on the Coordinate Plane 1. Use the origin as a vertex or center of the figure. 2. Place at least one side of a polygon on an axis. 3. Keep the figure within the first quadrant if possible. Example 1 Position and Label a Triangle Position and label isosceles triangle JKL on a coordinate plane so that base JK is a units long. • Use the origin as vertex J of the triangle. y L ( a2 , b ) • Place the base of the triangle along the positive x-axis. • Position the triangle in the first quadrant. • Since K is on the x-axis, its y-coordinate is 0. Its x-coordinate x O J (0, 0) is a because the base of the triangle is a units long. K (a , 0) • Since JKL is isosceles, the x-coordinate of L is halfway a between 0 and a or . We cannot determine the y-coordinate 2 in terms of a, so call it b. Power Point® 1 Position and label right Z triangle XYZ with leg X d units long on the coordinate plane. Example 2 Find the Missing Coordinates Name the missing coordinates of isosceles right EFG. y y E (0, a ) Vertex F is positioned at the origin; its coordinates are (0, 0). Vertex E is on the y-axis, and vertex G is on the x-axis. So EFG is a right angle. Since EFG is isosceles, EF G F . The distance from E to F is a units. The distance from F to F (?, ?) O G (?, ? ) G must be the same. So, the coordinates of G are (a, 0). Y(0, b) Z(d, 0) x 222 Chapter 4 Congruent Triangles Resource Manager Workbook and Reproducible Masters Chapter 4 Resource Masters • Study Guide and Intervention, pp. 219–220 • Skills Practice, p. 221 • Practice, p. 222 • Reading to Learn Mathematics, p. 223 • Enrichment, p. 224 • Assessment, p. 240 C (d , e ) 4. Use coordinates that make computations as simple as possible. POSITION AND LABEL TRIANGLES X(0, 0) In this chapter, we have used several methods of proof. You have also used the coordinate plane to identify characteristics of a triangle. We can combine what we know about triangles in the coordinate plane with algebra in a new method of proof called coordinate proof. POSITION AND LABEL TRIANGLES 2 Teach In-Class Example can the coordinate plane be useful in proofs? Teaching Geometry With Manipulatives Masters, pp. 1, 2, 8 Transparencies 5-Minute Check Transparency 4-7 Answer Key Transparencies Technology Interactive Chalkboard WRITE COORDINATE PROOFS After the figure has been placed on the coordinate plane and labeled, we can use coordinate proof to verify properties and to prove theorems. The Distance Formula, Slope Formula, and Midpoint Formula are often used in coordinate proof. In-Class Example Power Point® 2 Name the missing coordinates of isosceles right QRS. y Example 3 Coordinate Proof S(?, ?) Write a coordinate proof to prove that the measure of the segment that joins the vertex of the right angle in a right triangle to the midpoint of the hypotenuse is one-half the measure of the hypotenuse. y The first step is to position and label a right triangle on the B (0, 2b ) coordinate plane. Place the right angle at the origin and label it A. Use coordinates that are multiples of 2 because the P Midpoint Formula takes half the sum of the coordinates. Given: right ABC with right BAC A (0, 0) O Q(?, ?) Q(0, 0); S(c, c) C (2c , 0 ) x WRITE COORDINATE PROOFS P is the midpoint of BC . 1 2 Prove: AP BC Proof: In-Class Examples 0 2c 2b 0 2 2 By the Midpoint Formula, the coordinates of P are , or (c, b). Use the Distance Formula to find AP and BC. AP (c 0 )2 (b 0)2 prove that the segment that joins the vertex angle of an isosceles triangle to the midpoint of its base is perpendicular to the base. 2 4b 2 b2 BC 4c 2 or 2 c 1 BC 2 2b2 c 1 2 Therefore, AP BC. y Example 4 Classify Triangles Vertex Angle Remember from the Geometry Activity on page 216 that an isosceles triangle can be folded in half. Thus, the x-coordinate of the vertex angle is the same as the x-coordinate of the midpoint of the base. ARROWHEADS Write a coordinate proof to prove that this arrowhead is shaped like an isosceles triangle. The arrowhead is 3 inches long and 1.5 inches wide. The first step is to label the coordinates of each vertex. Q is at the origin, and T is at (1.5, 0). The y-coordinate of R is 3. The x-coordinate is halfway between 0 and 1.5 or 0.75. So, the coordinates of R are (0.75, 3). y Z (2a, 0) x Midpoint of XZ is (a, 0). Slope of Y W is undefined. Slope of XZ is 0. So, YW ⊥ XZ. Teaching Tip Advise students that they may want to place a figure using numeric coordinates first and then translate to variable coordinates to write their proofs. Q DRAFTING Write a coordinate Tx 4 proof to prove that this drafter’s tool is shaped like a right triangle. The length of one side is 10 inches and the length of another side is 5.75 inches. RT (1.5 0.75)2 (0 3)2 0.5625 9 or 9.5625 Since each leg is the same length, QRT is isosceles. The arrowhead is shaped like an isosceles triangle. www.geometryonline.com/extra_examples W X(0, 0) O 0.5625 9 or 9.5625 Y(a, b) R If the legs of the triangle are the same length, the triangle is isosceles. Use the Distance Formula to determine the lengths of QR and RT. 0)2 (3 0 )2 QR (0.75 Power Point® 3 Write a coordinate proof to BC (2c 0)2 (0 2 b)2 2 b2 c Study Tip R(c, 0) x y E Lesson 4-7 Triangles and Coordinate Proof 223 (0, 10) Francois Gohier/Photo Researchers Differentiated Instruction D Kinesthetic You can mark a coordinate plane on a corkboard. You can demonstrate for students, or volunteers can practice placing different figures on the coordinate plane using pushpins for vertices and string for sides. F (5.75, 0) Slope of ED is undefined. Slope of DF is 0. ED ⊥ DF DEF is a right triangle. The drafter’s tool is shaped like a right triangle. Lesson 4-7 Triangles and Coordinate Proofs 223 Concept Check Groups of students can work together writing coordinate proofs with one student placing the triangle, another student labeling the coordinates, and so on. Tell groups to discuss labeling variables with multiples of two to make midpoint computation easier and using points on the coordinate axes as much as possible to simplify formulas. Groups can also discuss limitations of coordinate proofs such as the difficulty of placing and labeling an equilateral triangle, and so on. Concept Check 2. OPEN ENDED Draw a scalene right triangle on the coordinate plane for use in a coordinate proof. Label the coordinates of each vertex. 1–2. See margin. Guided Practice GUIDED PRACTICE KEY Exercises Examples 3–4 5–7 8 9 1 2 3 4 Application y P (0, c) P (?, ?) Q (a, 0) N (?, ?) x Q (?, ?) O x N (2a, 0) R (–a, 0) O Q (?, ?) 9. TEPEES Write a coordinate proof to prove that the tepee is shaped like an isosceles triangle. Suppose the tepee is 8 feet tall and 4 feet wide. Practice and Apply For Exercises See Examples 10–15 16–24 25–29 30–33 1 2 3 4 About the Exercises… Position and label each triangle on the coordinate plane. 10. isosceles 䉭QRT with base Q 苶R 苶 that is b units long 10–15. See p. 233H. 11. equilateral 䉭MNP with sides 2a units long M and legs c units long 12. isosceles right 䉭JML with hypotenuse J苶苶 1 2 13. equilateral 䉭WXZ with sides ᎏᎏb units long 14. isosceles 䉭PWY with a base P 苶W 苶 that is (a ⫹ b) units long XY 15. right 䉭XYZ with hypotenuse X 苶Z 苶, ZY ⫽2(XY), and 苶 苶 b units long Find the missing coordinates of each triangle. y y 16. 17. Organization by Objective • Position and Label Triangles: 10–24 • Write Coordinate Proofs: 25–33 224 Chapter 4 Congruent Triangles N(0, b), Q(a, 0) See p. 233H. See page 762. Basic: 11–37 odd, 38–47 Average: 11–37 odd, 38–47 Advanced: 10–36 even, 37–47 7. 8. Write a coordinate proof for the following statement. The midpoint of the hypotenuse of a right triangle is equidistant from each of the vertices. See p. 233H. Extra Practice Assignment Guide Find the missing coordinates of each triangle. y y Q(⫺2a, 0) P(0, b) 6. 5. x Study Notebook Odd/Even Assignments Exercises 10–33 are structured so that students practice the same concepts whether they are assigned odd or even problems. Position and label each triangle on the coordinate plane. 3–4. See margin. 3. isosceles 䉭FGH with base 苶 FH 苶 that is 2b units long 4. equilateral 䉭CDE with sides a units long O R ( 0, 0) 3 Practice/Apply Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 4. • include any other item(s) that they find helpful in mastering the skills in this lesson. 1. Explain how to position a triangle on the coordinate plane to simplify a proof. 18. R (?, b) N (?, ?) Q (a, ?) x O P ( 0, 0) Q (2a, 0) x O L ( 0, 0) P (?, ?) R(a, b) 224 Answers 2. Sample answer: y C(0, b) A(0, 0) x Q(a, a), P(a, 0) Chapter 4 Congruent Triangles John Elk III/Stock Boston 1. Place one vertex at the origin, place one side of the triangle on the positive xaxis. Label the coordinates with expressions that will simplify the computations. y B(a, 0) x O J ( 0, 0) K (2a, 0) N(0, 2a) 19. 20. F (b, b √3) y 21. y y NAME ______________________________________________ DATE ____________ PERIOD _____ Study Guide andIntervention Intervention, 4-7 Study Guide and P (?, ?) E (?, ?) p. 219 (shown) and p. 220 Triangles and Coordinate Proof Position and Label Triangles A coordinate proof uses points, distances, and slopes to prove geometric properties. The first step in writing a coordinate proof is to place a figure on the coordinate plane and label the vertices. Use the following guidelines. O C ( 0, 0) B (?, ?) O D (?, ?) D(2b, 0) 22. M (–2b, 0) O B(a, 0), E(0, b) 23. y y 1. 2. 3. 4. x x C (a , 0 ) N (?, ?) Example Position an equilateral triangle on the coordinate plane so that its sides are a units long and one side is on the positive x-axis. Start with R(0, 0). If RT is a, then another vertex is T(a, 0). P(0, c), N(2b, 0) 24. J (?, ?) Use the origin as a vertex or center of the figure. Place at least one side of the polygon on an axis. Keep the figure in the first quadrant if possible. Use coordinates that make the computations as simple as possible. y P (a , a ) G (?, ?) y a 2 For vertex S, the x-coordinate is . Use b for the y-coordinate, so the vertex is S, b. a 2 S a–2, b Lesson 4-7 x T (a, 0) x R (0, 0) Exercises Find the missing coordinates of each triangle. 1. J (–b, 0) O x H (?, ?) x L (2c, 0) O K ( 0, 0) y x Q (?, ?) O N ( 0, 0) 2. C (?, q) J(c, b) y F (?, b) T (?, ?) A(0, 0) B(2p, 0) x R(0, 0) S(2a, 0) x C(p, q) G(0, c), H(b, 0) 3. y G(2g, 0) x E(?, ?) T(2a, 2a) E(2g, 0); F(0, b) Q(a, 0) Sample answers Position and label each triangle on the coordinate plane. are given. Write a coordinate proof for each statement. 25–28. See pp. 233H–233I. 25. The segments joining the vertices to the midpoints of the legs of an isosceles triangle are congruent. y Gl S(4a, 0) x D(0, 0) Q(0, a) E(e, 0) x I (b, 0) x E(–b, 0) NAME ______________________________________________ DATE /M G Hill 219 ____________ Gl PERIOD G _____ p. 221 and Practice, p. Coordinate 222 (shown) Triangles and Proof 1. equilateral SWY with 1 sides a long 2. isosceles BLP with base BL 3b units long 4 y y Y 1–8a, b W 1–4a, 0 x S(0, 0) 29. STEEPLECHASE Write a coordinate proof to prove that triangles ABD and is perpendicular FBD are congruent. BD , and B is the midpoint of the upper to AF bar of the hurdle. See p. 233I. y F (e, e) Sample answers are given. Position and label each triangle on the coordinate plane. 28. If a line segment joins the midpoints of two sides of a triangle, then its length is equal to one-half the length of the third side. Source: www.steeplechasetimes. com 6. equilateral triangle EQI with vertex Q(0, a) and sides 2b units long Skills Practice, 4-7 Practice (Average) 27. If a line segment joins the midpoints of two sides of a triangle, then it is parallel to the third side. The Steeplechase is a horse race two to four miles long that focuses on jumping hurdles. The rails of the fences vary in height. y T(2a, b) R(0, 0) 26. The three segments joining the midpoints of the sides of an isosceles triangle form another isosceles triangle. Steeplechase 5. isosceles right DEF with legs e units long 4. isosceles triangle RST with base R S 4a units long 3. isosceles right DGJ with hypotenuse D J and legs 2a units long y P 3–2b, c D (0, 2a) L(3b, 0) x B(0, 0) G(0, 0) J (2a, 0) x Find the missing coordinates of each triangle. B 4. y 5. S (?, ?) 6. y y M (0, ?) E (0, ?) 4 ft A F D R 1–3b, 0 x J (0, 0) 16 B (–3a, 0) S b, c 1 ft 6 ft C (?, 0) x P (2b, 0) x N (?, 0) C(3a, 0), E(0, c) M(0, c), N(2b, 0) NEIGHBORHOODS For Exercises 7 and 8, use the following information. Karina lives 6 miles east and 4 miles north of her high school. After school she works part time at the mall in a music store. The mall is 2 miles west and 3 miles north of the school. 7. Write a coordinate proof to prove that Karina’s high school, her home, and the mall are at the vertices of a right triangle. NAVIGATION For Exercises 30 and 31, use the following information. A motor boat is located 800 yards east of the port. There is a ship 800 yards to the east, and another ship 800 yards to the north of the motor boat. 30. Write a coordinate proof to prove that the port, motor boat, and the ship to the north form an isosceles right triangle. Given: SKM Prove: SKM is a right triangle. y K (6, 4) M (–2, 3) Proof: 40 60 30 2 0 2 3 Slope of SK or x S (0, 0) 3 2 Slope of SM or Since the slope of S M is the negative reciprocal of the slope of SK , SM ⊥ SK . Therefore, SKM is right triangle. 31. Write a coordinate proof to prove that the distance between the two ships is the same as the distance from the port to the northern ship. 30–31. See p. 233I. 8. Find the distance between the mall and Karina’s home. KM (2 6)2 (3 4)2 64 1 65 or 8.1 miles NAME ______________________________________________ DATE /M G Hill 222 Gl ____________ Gl PERIOD G _____ Reading 4-7 Readingto to Learn Learn Mathematics Mathematics, p. 223 Triangles and Coordinate Proof HIKING For Exercises 32 and 33, use the following information. Tami and Juan are hiking. Tami hikes 300 feet east of the camp and then hikes 500 feet north. Juan hikes 500 feet west of the camp and then 300 feet north. 32. Write a coordinate proof to prove that Juan, Tami, and the camp form a right triangle. See p. 233I. ELL How can the coordinate plane be useful in proofs? Read the introduction to Lesson 4-7 at the top of page 222 in your textbook. From the coordinates of A, B, and C in the drawing in your textbook, what do you know about ABC? Sample answer: ABC is isosceles with C as the vertex angle. Reading the Lesson Lesson 4-7 33. Find the distance between Tami and Juan. Pre-Activity 1. Find the missing coordinates of each triangle. 680,00 0 or about 824.6 ft a. b. y R (?, b) y F (?, ?) E (?, a) T(a, ?) www.geometryonline.com/self_check_quiz S (?, ?) Lesson 4-7 Triangles and Coordinate Proof 225 x D (?, ?) x b 2 R(0, b), S(0, 0), T a, Christopher Morrow/Stock Boston D(0, 0), E(0, a), F(a, a) 2. Refer to the figure. y R and the slope of S T . 1; 1 a. Find the slope of S NAME ______________________________________________ DATE Answers 3. Sample answer: y G(b, c) 4-7 Enrichment Enrichment, 4. Sample answer: y E(a2 , b) ____________ PERIOD _____ c. What does your answer from part b tell you about RST ? p. 224 O (0, 0) T (a, 0) x SR 2a 2 or a2 ; ST 2a 2 or a2 ; SR ST e. What does your answer from part d tell you about RST? Sample answer: RST is isosceles with RST as the vertex angle. Each puzzle below contains many triangles. Count them carefully. Some triangles overlap other triangles. f. Combine your answers from parts c and e to describe RST as completely as possible. Sample answer: RST is an isosceles right triangle. RST is the right angle and is also the vertex angle. How many triangles are there in each figure? 8 R (–a, 0) Sample answer: RST is a right triangle with S as the right angle. d. Find SR and ST. What does this tell you about S R and S T ? How Many Triangles? 1. S (0, a) b. Find the product of the slopes of S R and S T . What does this tell you about SR and ST ? 1; SR ⊥ ST 2. 40 3. 35 g. Find mSRT and mSTR. 45; 45 h. Find mOSR and mOST. 45; 45 Helping You Remember 4. F(0, 0) H(2b, 0) x 5 5. 13 6. C(0, 0) D (a, 0) x 27 3. Many students find it easier to remember mathematical formulas if they can put them into words in a compact way. How can you use this approach to remember the slope and midpoint formulas easily? Sample answer: Slope Formula: change in y over change in x; Midpoint Formula: average of x-coordinates, average of y-coordinates How many triangles can you form by joining points on each circle? Lesson 4-7 Triangles and Coordinate Proofs 225 4 Assess Open-Ended Assessment Speaking Have students speak about how they would place certain figures in a coordinate plane and how they would label the vertices. Students can discuss different ideas about placement and how they can simplify coordinate proofs by using the origin and simple labeling techniques. 34. (a, 0) or (0, b) 35. (2a, 0) Find the coordinates of point Z so XYZ is the indicated type of triangle. Point X has coordinates (0, 0) and Y has coordinates (a, b). 34. right triangle 35. isosceles triangle 36. scalene triangle Sample answer: (c, 0) with right angle Z with base XZ 37. CRITICAL THINKING Classify ABC by its angles y C (0, 2a) and its sides. Explain. See margin. 38. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can the coordinate plane be useful in proofs? x A (–2a, 0) O B (2a, 0) Include the following in your answer: • types of proof, and • a theorem from this chapter that could be proved using a coordinate proof. Standardized Test Practice Assessment Options 39. What is the length of the segment whose endpoints are at (1,2) and (3, 1)? C A 3 B 4 C 5 D 6 40. ALGEBRA What are the coordinates of the midpoint of the line segment whose endpoints are (5, 4) and (2, 1)? B A (3, 3) B (3.5, 1.5) C (1.5, 2.5) D (3.5, 2.5) Quiz (Lesson 4-7) is available on p. 240 of the Chapter 4 Resource Masters. Maintain Your Skills Answers 37. AB 4a AC (0 ( 2a))2 (2a 0)2 Mixed Review 4a 2 4a 2 or 8a 2 Write a two-column proof. (Lessons 4-5 and 4-6) 41–44. See pp. 233I–233J. 41. Given: 3 4 42. Given: isosceles triangle JKN LM with vertex N, JK Prove: QR Q S Prove: NML is isosceles. Q K CB (0 2 a)2 (2a 0)2 2 4a 2 4a 2 or 8a 2 3 2a 0 0 (2a) 2a 0 slope of C B or 1. 0 2a Slope of AC or 1; AB ⊥ CB and A C CB , so ABC is a right isosceles triangle. 38. Sample answer: Placing the figures on the coordinate plane is useful in proofs. We can use coordinate geometry to prove theorems and verify properties. Answers should include the following. • flow proof, two-column proofs, paragraph proofs, informal proofs, and coordinate proofs • Sample answer: The Isosceles Triangle Theorem can be proved using coordinate proof. 226 Chapter 4 Congruent Triangles 1 4 4 2 N 3 S R M 1 J CE 43. Given: AD AD C E , Prove: ABDEBC L 44. Given: W X X Y , VZ Prove: WV Y Z A W C V B X Z D Y E AD 45. BC ; if alt. int. are , lines are 46. h j; Sample answer: cons. int. suppl. 47. ᐉ m; 2 lines to same line are . State which lines, if any, are parallel. State the postulate or theorem that justifies your answer. (Lesson 3-5) g j B C h 45. 46. 47. k f 24˚ 71˚ 111˚ j 24˚ A 226 Chapter 4 Congruent Triangles D 69˚ m Study Guide and Review Vocabulary and Concept Check acute triangle (p. 178) base angles (p. 216) congruence transformations (p. 194) congruent triangles (p. 192) coordinate proof (p. 222) corollary (p. 188) equiangular triangle (p. 178) equilateral triangle (p. 179) exterior angle (p. 186) flow proof (p. 187) included angle (p. 201) included side (p. 207) Vocabulary and Concept Check isosceles triangle (p. 179) obtuse triangle (p. 178) remote interior angles (p. 186) right triangle (p. 178) scalene triangle (p. 179) vertex angle (p. 216) A complete list of theorems and postulates can be found on pages R1–R8. Exercises Choose the letter of the word or phrase that best matches each statement. 1. A triangle with an angle whose measure is greater than 90 is a(n) ? a. acute triangle. h b. AAS Theorem 2. A triangle with exactly two congruent sides is a(n) ? triangle. g c. ASA Theorem 3. The ? states that the sum of the measures of the angles of a d. Angle Sum Theorem triangle is 180. d e. equilateral 4. If BE, AB D E , and B C E F , then ABCDEF by ? . j f. exterior 5. In an equiangular triangle, all angles are ? angles. a g. isosceles 6. If two angles of a triangle and their included side are congruent to h. obtuse two angles and the included side of another triangle, this is called the ? . c i. right 7. If AF, BG, and AC F H , then ABCFGH, by ? . b j. SAS Theorem 8. A(n) ? angle of a triangle has a measure equal to the measures of k. SSS Theorem the two remote interior angles of the triangle. f 4-1 Classifying Triangles See pages 178–183. Example y U V (2)]2 [4 ( 2)]2 TU [5 9 36 or 45 UV [3 ( 5)]2 (1 4)2 O x 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 VT (2 3)2 (2 1)2 25 9 or 34 Since none of the side measures are equal, TUV is scalene. Chapter 4 Study Guide and Review 227 TM For more information about Foldables, see Teaching Mathematics with Foldables. For each lesson, • the main ideas are summarized, • additional examples review concepts, and • practice exercises are provided. T 64 9 or 73 www.geometryonline.com/vocabulary_review Lesson-by-Lesson Review Vocabulary PuzzleMaker Concept Summary • Triangles can be classified by their angles as acute, obtuse, or right. • Triangles can be classified by their sides as scalene, isosceles, or equilateral. Find the measures of the sides of TUV. Classify the triangle by sides. Use the Distance Formula to find the measure of each side. • This alphabetical list of vocabulary terms in Chapter 4 includes a page reference where each term was introduced. • Assessment A vocabulary test/review for Chapter 4 is available on p. 238 of the Chapter 4 Resource Masters. Have students look through the chapter to make sure they have included notes and examples in their Foldables for each lesson of Chapter 4. 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 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 4 Study Guide and Review 227 Study Guide and Review Chapter 4 Study Guide and Review Exercises Classify each triangle by its angles and by its sides if mABC 100. See Examples 1 and 2 on Answers 15. E D, F C, G B, EF DC , FG CB , GE BD 16. FGC DLC, GCF LCD, GFC LDC, GC LC , C F CD , FG DL 17. KNC RKE, NCK KER, CKN ERK, NC KE, C K ER , KN RK B pages 178 and 179. 9. ABC 10. BDP 11. BPQ 60˚ A P D Q C 9. obtuse, isosceles 10. right, scalene 11. equiangular, equilateral 4-2 Angles of Triangles See pages 185–191. Example Concept Summary • The sum of the measures of the angles of a triangle is 180. • The measure of an exterior angle is equal to the sum of the measures of the two remote interior angles. If T UV UV VW U and , find m1. m1 72 mTVW 180 Angle Sum Theorem U T 2 m1 72 (90 27) 180 mTVW 9027 m1 135 180 Simplify. m1 45 Exercises Subtract 135 from each side. 1 27˚ 72˚ V Find each measure. 13. m2 25 W 2 See Example 1 on page 186. 12. m1 85 S 45˚ 14. m3 95 70˚ 1 3 40˚ 4-3 Congruent Triangles See pages 192–198. Example Concept Summary • Two triangles are congruent when all of their corresponding parts are congruent. If EFGJKL, name the corresponding congruent angles and sides. EJ, FK, GL, EF FG KL JK , , and E G JL . Exercises Name the corresponding angles and sides for each pair of congruent triangles. See Example 1 on page 193. 15–17. See margin. 15. EFGDCB 16. LCD GCF 17. NCK KER 4-4 Proving Congruence—SSS, SAS See pages 200–206. Concept Summary • If all of the corresponding sides of two triangles are congruent, then the triangles are congruent (SSS). • If two corresponding sides of two triangles and the included angle are congruent, then the triangles are congruent (SAS). 228 Chapter 4 Congruent Triangles 228 Chapter 4 Congruent Triangles Chapter 4 Study Guide and Review Example Determine whether ABC TUV. Explain. [1 (2)]2 (1 0)2 AB TU (3 4 )2 ( 1 0)2 1 1 or 2 BC [0 ( 1)]2 (1 1)2 B UV (2 3 )2 [1 ( 1)]2 1 4 or 5 V T x O A C 1 4 or 5 CA (2 0)2 [0 ( 1)]2 Answers y 1 1 or 2 U VT (4 2 )2 (0 1)2 4 1 or 5 4 1 or 5 Exercises Determine whether MNP QRS given the coordinates of the vertices. Explain. See Example 2 on page 201. 18 – 19. See margin. 18. M(0, 3), N(4, 3), P(4, 6), Q(5, 6), R(2, 6), S(2, 2) 19. M(3, 2), N(7, 4), P(6, 6), Q(2, 3), R(4, 7), S(6, 6) 4-5 Proving Congruence—ASA, AAS Example D Concept Summary • If two pairs of corresponding angles and the included sides of two triangles are congruent, then the triangles are congruent (ASA). • If two pairs of corresponding angles and a pair of corresponding nonincluded sides of two triangles are congruent, then the triangles are congruent (AAS). Write a proof. KM MN Given: JK ; L is the midpoint of . Prove: J C K JK || MN JKL LMN Given Alt. int. s are . L is the midpoint of KM. KL ML JLK NLM Given Midpoint Theorem ASA N JLK NLM Vertical s are . Exercises For Exercises 20 and 21, use the figure. Write a two-column proof for each of the following. See Example 2 on page 209. D 20–21. See margin. 20. Given: D F bisects CDE. DF CE Prove: DGC DGE 21. Given: DGC DGE GCF GEF Prove: DFC DFE G C E Proof: Statements (Reasons) L Flow proof: G F M JLKNLM 18. MN 4, NP 3, MP 5, QR 3, RS 4, and QS 5. Each pair of corresponding sides does not have the same measure. Therefore, MNP is not congruent to QRS. MNP is congruent to SRQ. 19. MN 20, NP 5, MP 5, QR 20, RS 5, and QS 5. Each pair of corresponding sides has the same measure. Therefore, MNP QRS by SSS. 20. Given: DF bisects CDE, CE ⊥ DF. Prove: DGC DGE By the definition of congruent segments, all corresponding sides are congruent. Therefore, ABCTUV by SSS. See pages 207–213. Study Guide and Review E 1. D F bisects CDE, CE ⊥ DF. (Given) 2. DG DG (Reflexive Prop.) 3. CDF EDF (Def. of bisector) 4. DGC is a rt. ; DGE is a rt. (Def. of ⊥ segments) 5. DGC DGE (All rt. are .) 6. DGC DGE (ASA) 21. Given: DGC DGE, GCF GEF Prove: DFC DFE D F Chapter 4 Study Guide and Review 229 C G E F Proof: Statements (Reasons) 1. DGC DGE, GCF GEF (Given) 2. CDG EDG, CD ED , and CFD EFD (CPCTC) 3. DFC DFE (AAS) Chapter 4 Study Guide and Review 229 • Extra Practice, see pages 760–762. • Mixed Problem Solving, see page 785. Study Guide and Review 4-6 Isosceles Triangles Answers See pages 216–221. 26. Sample answer: y R(2a, b) Example I(4a, 0) x T(0, 0) GJ, GJ JH FH If FG , F J , and mGJH 40, find mH. GHJ is isosceles with base GH , so JGHH by the Isosceles Triangle Theorem. Thus, mJGH mH. 2(mH) 140 Subtract 40 from each side. mH 70 Exercises D(6m, 0) x 22. 23. 24. 25. y J(0, b) L(a, 0) See pages 222–226. Example Answers (p. 231) 10. D P, E Q, F R, DE PQ , EF QR , DF PR 11. F H, M N, G J, FM H, N MG NJ, FG HJ 12. X Z, Y Y, Z X, XY ZY , YZ YX , XZ ZX K J L N 230 For Exercises 22–25, refer to the figure at the right. If P Q U Q and mP 32, find mPUQ. 32 If P Q U Q , P R R T , and mPQU 40, find mR. 40 If R Q R S and mRQS 75, find mR. 30 If R RP Q R S , R T , and mRQS 80, find mP. 80 Chapter 4 Congruent Triangles H R Q P S U T Concept Summary • Coordinate proofs use algebra to prove geometric concepts. • The Distance Formula, Slope Formula, and Midpoint Formula are often used in coordinate proof. Position and label isosceles right triangle ABC with legs of length a units on the coordinate plane. • Use the origin as the vertex of ABC that has the right angle. A(0, 0) O • Place each base along an axis. • Since B is on the x-axis, its y-coordinate is 0. Its x-coordinate C (0, – a ) is a because the leg A B of the triangle is a units long. • Since ABC is isosceles, C should also be a distance of a units from the origin. Its coordinates are (0, a). Exercises Position and label each triangle on the coordinate plane. See Example 1 on page 222. 26– 28. See margin. 26. isosceles TRI with base T I 4a units long 27. equilateral BCD with side length 6m units long 28. right JKL with leg lengths of a units and b units 230 Chapter 4 Congruent Triangles Proof: JKM JNM Given JK JN KJL NJL M J 4-7 Triangles and Coordinate Proof x 13. JK 10 , KL 17 , JL 5, MN 80, NP 53, 221. Corresponding sides MP are not congruent, so JKL is not congruent to MNP. 14. Given: JKM JNM Prove: JKL JNL Divide each side by 2. See Example 2 on page 217. 28. Sample answer: K(0, 0) G 40 2(mH) 180 Substitution C(3m, n) B(0, 0) F mGJH mJGH mH 180 Angle Sum Theorem 27. Sample answer: y Concept Summary • Two sides of a triangle are congruent if and only if the angles opposite those sides are congruent. • A triangle is equilateral if and only if it is equiangular. CPCTC JKL JNL JL JL SAS Reflexive Prop. B (a , 0) Practice Test Vocabulary and Concepts Assessment Options Choose the letter of the type of triangle that best matches each phrase. 1. triangle with no sides congruent b a. isosceles 2. triangle with at least two sides congruent a b. scalene 3. triangle with all sides congruent c c. equilateral Vocabulary Test A vocabulary test/review for Chapter 4 can be found on p. 238 of the Chapter 4 Resource Masters. Chapter Tests There are six Chapter 4 Tests and an OpenEnded Assessment task available in the Chapter 4 Resource Masters. Skills and Applications Identify the indicated triangles in the figure AD PA PC if PB and . 6. PBA, PBC, PBD 4. obtuse PCD 5. isosceles PAC 6. right P 1 3 Find the measure of each angle in the figure. 7. m1 80 8. m2 105 9. m3 25 A B C 100˚ 2 75˚ D Questions 4–6 Form Questions 7–9 1 2A 2B 2C 2D 3 Name the corresponding angles and sides for each pair of congruent triangles. 10–12. See margin. 10. DEF PQR 11. FMG HNJ 12. XYZ ZYX 13. Determine whether JKL MNP given J(1, 2), K(2, 3), L(3, 1), M(6, 7), N(2, 1), and P(5, 3). Explain. See margin. 14. Write a flow proof. See margin. Given: JKMJNM Prove: JKL JNL K F J In the figure, FJ FH GF GH and . 15. If mJFH 34, find mJ. 73 16. If mGHJ 152 and mG 32, find mJFH. 24 G J N H Question 14 Questions 15–16 17. LANDSCAPING A landscaper designed a garden shaped as shown in the figure. The landscaper has decided to place point B 22 feet east of point A, point C 44 feet east of point A, point E 36 feet south of point A, and point D 36 feet south of point C. The angles at points A and C are right angles. Prove that ABECBD. See p. 233J. B A B 62 C 56 D D H J 28 F www.geometryonline.com/chapter_test Pages 225–226 227–228 229–230 231–232 233–234 235–236 MC = multiple-choice questions FR = free-response questions ExamView® Pro 18. STANDARDIZED TEST PRACTICE In the figure, FGH is a right triangle with hypotenuse F H and GJ GH. What is the measure of JGH? C 104 basic average average average average advanced C E A MC MC MC FR FR FR Open-Ended Assessment Performance tasks for Chapter 4 can be found on p. 237 of the Chapter 4 Resource Masters. A sample scoring rubric for these tasks appears on p. A28. L M Chapter 4 Tests Type Level 28˚ G Chapter 4 Practice Test 231 Portfolio Suggestion Introduction As students progress with writing geometric proofs, it can help for them to have an example of their own work to refer to for self-encouragement or to review for method or style. Ask Students Go back through your notes and problems that you worked and find a well-organized, clear and concise proof that you composed. Place an example of this proof in your portfolio, and write a paragraph explaining why you selected this particular example. Highlight important concepts and methods that you used to form your proof. 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 4 Practice Test 231 Standardized Test Practice These two pages contain practice questions in the various formats that can be found on the most frequently given standardized tests. Part 1 Multiple Choice Record your answers on the answer sheet provided by your teacher or on a sheet of paper. (Lesson 3-3) 1. In 2002, Capitol City had a population of 2010, and Shelbyville had a population of 1040. If Capitol City grows at a rate of 150 people a year and Shelbyville grows at a rate of 340 people a year, when will the population of Shelbyville be greater than that of Capitol City? (Prerequisite Skill) B A practice answer sheet for these two pages can be found on p. A1 of the Chapter 4 Resource Masters. O Practice 4Standardized Standardized Test Test Practice Student Record Sheet (Use with Sheet, pages 232–233 of Student Recording p.the Student A1 Edition.) Part 1 Multiple Choice A 2004 B 2008 C 2009 D 2012 Select the best answer from the choices given and fill in the corresponding oval. 1 A 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 For Questions 12 and 14, also enter your answer by writing each number or symbol in a box. Then fill in the corresponding oval for that number or symbol. 10 11 12 (grid in) 13 14 (grid in) B x 2y1 8 C x2y 1 4 D 2x y 1 2 . / . / . . 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 A 35 B 70 A grams B feet C 90 C liters D meters D 110 Part 3 Extended Response 3. A 9-foot tree casts a shadow on the ground. The distance from the top of the tree to the end of the shadow is 12 feet. To the nearest foot, how long is the shadow? (Lesson 1-3) B Record your answers for Questions 15–16 on the back of this paper. 9 ft 12 ft 10 6 0 1 14 Answers 12 12 2x y 1 2 3 4 5 6 6. What is mEFG? 2. Which unit is most appropriate for measuring liquid in a bottle? (Lesson 1-2) C Solve the problem and write your answer in the blank. D A (Lesson 4-2) Part 2 Short Response/Grid In 9 5. Students in a math classroom simulated stock trading. Kris drew the graph below to model the value of his shares at closing. The graph that modeled the value of Mitzi’s shares was parallel to the one Kris drew. Which equation might represent the line for Mitzi’s graph? A 7 ft B 8 ft C 10 ft D 13 ft G B F (9x 7)˚ 5x˚ 5x˚ D E 7. In the figure, ABDCBD. If A has the coordinates (2, 4), what are the coordinates of C? (Lesson 4-3) C A (4, 2) B (4, 2) C (2, 4) D (2, 4) y A B D x O C ? ft Additional Practice See pp. 243–244 in the Chapter 4 Resource Masters for additional standardized test practice. 4. Which of the following is the inverse of the statement If it is raining, then Kamika carries an umbrella? (Lesson 2-2) D A If Kamika carries an umbrella, then it is raining. B If Kamika does not carry an umbrella, then it is not raining. C If it is not raining, then Kamika carries an umbrella. D If it is not raining, then Kamika does not carry an umbrella. 232 Chapter 4 Standardized Test Practice 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. 232 Chapter 4 Congruent Triangles 8. The wings of some butterflies can be modeled by triangles as shown. If AC D C and ACB ECD, which additional statements are needed to prove that ACB ECD? (Lesson 4-4) A A B C C E B A B E D C BAC CED D ABCCDE D A C E B 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. 9. Find the product (Prerequisite Skill) 3s2(2s3 6s5 21s2 Test-Taking Tip • If you are not permitted to write in your test booklet, make a sketch of the figure on scrap paper. 7). • Mark the figure with all of the information you know so that you can determine the congruent triangles more easily. 10. After a long workout, Brian noted, “If I do not drink enough water, then I will become dehydrated.” He then made another statement, “If I become dehydrated, then I did not drink enough water.” How is the second statement related to the original statement? (Lesson 2-2) converse • Make a list of postulates or theorems that you might use for this case. Part 3 Extended Response Record your answers on a sheet of paper. Show your work. 11. On a coordinate map, the towns of Creston and Milford are located at (1, 1) and (1, 3), respectively. A third town, Dixville, is located at (x, 1) so that Creston and Dixville are endpoints of the base of the isosceles triangle formed by the three locations. What is the value of x? (Lesson 4-1) 3 15. Train tracks a and b are parallel lines although they appear to come together to give the illusion of distance in a drawing. All of the railroad ties are parallel to each other. x˚ 12. A watchtower, built to help prevent forest fires, was designed as an isosceles triangle. If the side of the tower meets the ground at a 105° angle, what is the measure of the angle at the top of the tower? (Lesson 4-2) 1 105˚ 2 b. What is the relationship between the tracks and the ties that run across the tracks? (Lesson 1-5) perpendicular lines 1 c. What is the relationship between 1 and 2? Explain. (Lesson 3-2) See margin. 0 16. The measures of the angles of ABC are 5x, 4x 1, and 3x13. a. Draw a figure to illustrate ABC. D E b a. What is the value of x? (Lesson 3-1) 90 ASA C 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 90˚ (Lesson 4-1) B Criteria 4 3 a 30 A Score 2 13. During a synchronized flying show, airplane A and airplane D are equidistant from the ground. They descend at the same angle to land at points B and E, respectively. Which postulate would prove that ABC DEF? (Lesson 4-4) Extended Response questions are graded by using a multilevel rubric that guides you in assessing a student’s knowledge of a particular concept. Goal: Students find angle measures, describe angle relationships, and prove that a triangle is isosceles. 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. Question 8 See margin. b. Find the measure of each angle of ABC. Explain. (Lesson 4-2) 70, 55, and 55 F 14. ABC is an isosceles triangle with AB B C , and the measure of vertex angle B is three times mA. What is mC? (Lesson 4-6) 36 See margin for explanation. c. Prove that ABC is an isosceles triangle. (Lesson 4-6) www.geometryonline.com/standardized_test See margin. Chapter 4 Standardized Test Practice 233 Answers 15c. They are congruent. Sample answers: Both are right angles; they are supplementary angles; they are corresponding angles. 16a. Sample answer: A 5x B (4x 1) (3x 13) C 16b. From the Angle Sum Theorem, we know that mA mB mC 180. Substituting the given measures, 5x 4x 1 3x 13 180. Solve for x to find that x 14. Substitute 14 for x to find the measures: 5x 5(14) or 70, 4x 1 4(14) 1 or 55, and 3x 13 3(14) 13 or 55. 16c. If two angles of a are , then the sides opposite those angles are (Converse of the Isosceles Theorem). Since two sides of this are , it is an isosceles (Definition of Isosceles ). Chapter 4 Standardized Test Practice 233 Page 180–184 Lesson 4-1 38. Given: EUI is equiangular. QL || U I Prove: EQL is equiangular. Pages 188–191, Lesson 4-2 39. Given: FGI IGH GI ⊥ FH Prove: F H Proof: E Q L G F GI ⊥ FH U Proof: Statements (Reasons) 1. EUI is equiangular. QL || U I (Given) 2. E EUI EIU (Def. of equiangular ) 3. EUI EQL; EIU ELQ (Corr. Post.) 4. E E (Reflexive Prop.) 5. E EQL ELQ (Trans. Prop.) 6. EQL is equiangular. (Def. of equiangular ) 39. Given: mNPM 33 Prove: RPM is obtuse. M I N 33 Additional Answers for Chapter 4 P R Proof: NPM and RPM form a linear pair. NPM and RPM are supplementary because if two angles form a linear pair, then they are supplementary. So, mNPM mRPM 180. It is given that mNPM 33. By substitution, 33 mRPM 180. Subtract to find that mRPM 147. RPM is obtuse by definition. RPM is obtuse by definition. 40. TS (7 (4))2 (8 14 )2 9 36 or 45 SR (10 (7 ))2 ( 2 8 )2 9 36 or 45 S is the midpoint of R T . UT (0 ( 4))2 (8 14)2 16 36 or 52 VU (4 0 )2 (2 8)2 16 36 or 52 U is the midpoint of TV . 41. AD CD 0 a2 (0 b)2 a2 (b)2 2 2 a b 4 2 2 a a2 (0 b)2 a2 (b)2 2 2 a b 4 2 2 AD CD, so A D C D . ADC is isosceles by definition. 233A Chapter 4 Additional Answers I H Given GIF and GIH are right angles. ⊥ lines form rt. . GIF GIH All rt. are . FGI IGH Given F H Third Angle Theorem 40. Given: ABCD is a quadrilateral. A B 1 2 Prove: mDAB mB mBCD mD 360 3 4 C Proof: D Statements (Reasons) 1. ABCD is a quadrilateral. (Given) 2. m2 m3 mB 180; m1 m4 mD 180 ( Sum Theorem) 3. m2 m3 mB m1 m4 mD 360 (Addition Prop.) 4. mDAB m1 m2; mBCD m3 m4 ( Addition) 5. mDAB mB mBCD mD 360 (Substitution) 41. Given: ABC C Prove: mCBD mA mC D A Proof: B Statements (Reasons) 1. ABC (Given) 2. CBD and ABC form a linear pair. (Def. of linear pair) 3. CBD and ABC are supplementary. (If 2 form a linear pair, they are suppl.) 4. mCBD mABC 180 (Def. of suppl.) 5. mA mABC mC 180 ( Sum Theorem) 6. mA mABC mC mCBD mABC (Substitution) 7. mA mC mCBD (Subtraction Prop.) 42. Given: RST R is a right angle. Prove: S and T are complementary. Proof: S R T Pages 195–198, Lesson 4-3 33. Given: RST XYZ Prove: XYZ RST R S X R is a rt. . T Given Proof: mR 90 Def. of rt. mR mS mT 180 Y Z RST XYZ Given Angle Sum Theorem R X, S Y, T Z, RS XY, ST YZ, RT XZ 90 mS mT 180 Substitution CPCTC X R, Y S, Z T, XY RS, YZ ST, XZ RT Congruence of and mS mT 90 Subtraction Prop. segments is symmetric. S and T are complementary Def. of complementary XYZ RST Def. of Pages 203–206, Lesson 4-4 and BC bisect each other. 5. Given: DE Prove: DGB EGC Proof: B D G E DG and GE bisect each other. Given C DE BC, BG GC Def. of bisector of segments DGB EGC SAS DGB EGC Vertical are . 6. Given: KM || JL , KM JL Prove: JKM MLJ K J Proof: Statements (Reasons) 1. 2. 3. 4. M L KM || JL , KM JL (Given) KMJ LJM (Alt. Int. Theorem) JM JM (Reflexive Prop.) JKM MLJ (SAS) Chapter 4 Additional Answers 233B Additional Answers for Chapter 4 43. Given: MNO N M is a right angle. Prove: There can be at most one M O right angle in a triangle. Proof: In MNO, M is a right angle. mM mN mO 180. mM 90, so mN mO 90. If N were a right angle, then mO 0. But that is impossible, so there cannot be two right angles in a triangle. Given: PQR Q P is obtuse. Prove: There can be at most one R obtuse angle in a triangle. P Proof: In PQR, P is obtuse. So mP 90. mP mQ mR 180. It must be that mQ m R 90. So, Q and R must be acute. 44. Given: A D F B E C D E Prove: C F A B Proof: Statements (Reasons) 1. A D, B E (Given) 2. mA mD, mB mE (Def. of ) 3. mA mB mC 180, mD mE mF 180 (Angle Sum Theorem) 4. mA mB mC mD mE mF (Transitive Prop.) 5. mD mE mC mD mE mF (Substitution Prop.) 6. mC mF (Subtraction Prop.) 7. C F (Def. of ) 14. Given: A E F C , AB B C , BE B F Prove: AFB CEB GC 18. Given: AC C E bisects AG . Prove: GEC AEC B F E Given AE FC Def. of seg. AE EF EF FC EF EF Addition Prop. AE EF AF EF FC EC Substitution Seg. Addition Post. Additional Answers for Chapter 4 AF EC Def. of seg. AFB CEB AB BC BE BF SAS Given 15. Given: RQ TQ YQ WQ RQY WQT Prove: QWT QYR R Y Q W Proof: Given Proof: Statements (Reasons) T RQY WQT QWT QYR SAS 233C Chapter 4 Additional Answers J R 1. R S PN , RT MP (Given) 2. S N, and T M (Given) 3. R P (Third Angle Theorem) 4. RST MNP (SAS) 21. Given: EF H F G is the midpoint of EH . F Prove: EFG HFG Given 16. Given: CDE is an isosceles D triangle. G is the midpoint of CE . Prove: CDG EDG Proof: C G Statements (Reasons) 1. CDE is an isosceles triangle, G is the midpoint of CE . (Given) 2. C D DE (Def. of isos. ) 3. CG GE (Midpoint Th.) 4. DG DG (Reflexive Prop.) 5. CDG EDG (SSS) 17. Given: MRN QRP M MNP QPN R Prove: MNP QPN Proof: N P Statements (Reasons) 1. MRN QRP, MNP QPN (Given) 2. MN QP (CPCTC) 3. N P NP (Reflexive Prop.) 4. MNP QPN (SAS) K Proof: G L Statements (Reasons) 1. GHJ LKJ (Given) 2. HJ KJ, GJ L J, GH LK (CPCTC) 3. HJ KJ, GJ LJ (Def. of segments) 4. HJ LJ KJ JG (Addition Prop.) 5. KJ GJ KG; HJ LJ HL (Segment Addition) 6. KG HL (Substitution) 7. KG HL (Def. of segments) 8. GL GL (Reflexive Prop.) 9. GHL LKG (SSS) N S 20. Given: RS , RT PN MP S N, T M M T Prove: RST PNM P Reflexive Prop. AE EC C G 1. A C GC , EC bisects A G . (Given) 2. AE EG (Def. of segment bisector) 3. E C EC (Reflexive Prop.) 4. GEC AEC (SSS) 19. Given: GHJ LKJ H Prove: GHL LKG Proof: AE FC RQ TQ YQ WQ E Proof: Statements (Reasons) C A A E Q Proof: Statements (Reasons) 1. E F H F ; G is the midpoint of EH . (Given) 2. E G GH (Def. of midpoint) 3. FG FG (Reflexive Prop.) 4. EFG HFG (SSS) 26. Given: T S S F F H H T S TSF, SFH, FHT, and HTS are right angles. P T Prove: HS TF H G E F Proof: H Statements (Reasons) 1. T S S F F H H T (Given) 2. TSF, SFH, FHT, and HTS are right angles. (Given) 3. STH THF (All right are .) 4. STH THF (SAS) 5. HS TF (CPCTC) 27. Given: TS S F F H H T TSF, SFH, FHT, and HTS are right angles. Prove: SHT SHF S T Proof: Statements (Reasons) P F GK || HJ Given G H K KGJ HJG Alt. int. are . J XWY ZYW Alt. int. are . GJ GJ Reflexive Property W Z WY WY Reflexive Property WXY YZW AAS 6. Given: QS bisects RST; R R T. Prove: QRS QTS S Q Proof: We are given that R T and QS bisects T RST, so by definition of angle bisector, RSQ TSQ. By the Reflexive Property, QS QS . QRS QTS by AAS. 7. Given: E K, D DGH DHG, EG KH Prove: EGD KHD E G H K Proof: Since EGD and DGH are linear pairs, the angles are supplementary. Likewise, KHD and DHG are supplementary. We are given that DGH DHG. Angles supplementary to congruent angles are congruent so EGD KHD. Since we are given that E K and EG KH , EGD KHD by ASA. 9. Given: EF || G H , EF GH F E Prove: EK KH Proof: K EF || GH Given E H Alt. int. are . H EF GH G Given EKF HKG AAS EK KH CPCTC 10. Given: D E || JK K D bisects JE . Prove: EGD JGK Proof: WXY YZW ASA Y Given GKH EKF Vert. are . Given HGJ KJG Alt. int. are . X Z X J D G DK bisects JE. DE || JK Given E J Alt. int. are . DGE KGJ Vert. are . K E Given EG GJ Def. of seg. bisector EGD JGK ASA Chapter 4 Additional Answers 233D Additional Answers for Chapter 4 1. TS S F F H H T (Given) 2. TSF, SFH, FHT, and HTS are right angles. (Given) 3. STH SFH (All right are .) 4. STH SFH (SAS) 5. SHT SHF (CPCTC) 28. Given: DE FB , AE FC , A B AE ⊥ DB , CF ⊥ DB F Prove: ABD CDB Plan: First use SAS to E show that ADE CBF. D C Next use CPCTC and Reflexive Property for segments to show ABD CDB. Proof: Statements (Reasons) 1. D E , AE (Given) FB FC 2. A E ⊥ DB , CF ⊥ DB (Given) 3. AED is a right angle. CFB is a right angle. (⊥ lines form right .) 4. AED CFB (All right angles are .) 5. ADE CBF (SAS) 6. AD BC (CPCTC) 7. D B DB (Reflexive Prop. for Segments) 8. CBD ADB (CPCTC) 9. ABD CDB (SAS) GH || KS XW || YZ Given H Pages 210–213, Lesson 4-5 4. Given: GH K J, GK H J Prove: GJK JGH Proof: 5. Given: XW || Y Z , X Z Prove: WXY YZW Proof: 11. Given: V S, TV Q S Prove: VR SR Proof: V S TV QS 1 2 R Q V 1 2 Vert. are . Given S T 14. Given: Z is the midpoint of CT . CY || T E Prove: YZ E Z Proof: Given TZ CZ Midpt. Th. CPCTC 12. Given: E J || F K , JG || K H , E F GH Prove: EJG FKH Proof: J Additional Answers for Chapter 4 Substitution AAS E F G H FG FG Reflexive Prop. EF FG EG FG GH FH Seg. Addition Post. EG FH Def. of seg. EJ || FK, JG || KH EJG FKH JEG KFH JGE HFK Corr. are . SAS Given 13. Given: MN PQ , M Q 2 3 Prove: MLP QLN Proof: MN PQ L 1 M MN PQ Def. of seg. Addition Prop. MP NQ Substitution NP NP Reflexive Prop. MN NP MP NP PQ NQ Seg. Addition Post. MP NQ Def. of seg. MLP QLN ASA M Q 2 3 Given 233E N 2 3 4 P Q 19. Given: MYT NYT MTY NTY Prove: RYM RYN M R T Y Proof: N Statements (Reasons) 1. MYT NYT, MTY NTY (Given) 2. YT Y T , RY (Reflexive Property) RY 3. MYT NYT (ASA) 4. MY (CPCTC) NY 5. RYM and MYT are a linear pair; RYN and NYT are a linear pair (Def. of linear pair) 6. RYM and MYT are supplementary and RYN and NYT are supplementary. (Suppl.Th.) 7. RYM RYN ( suppl. to are .) 8. RYM RYN (SAS) 20. Given: BMI KMT I B P IP PT Prove: IPK TPB M T Given MN NP NP PQ Given EZT YZC K EF GH Def. of seg. EG FH Y CPCTC Given Addition Prop. T YZ EZ EF GH EF FG FG GH Z ETC YCT TEY CYE Alt. int. are . AAS VR SR C CY ⊥ TE Z is the midpoint of CT. TRV QRS E Chapter 4 Additional Answers Proof: K Statements (Reasons) 1. BMI KMT (Given) 2. B K (CPCTC) 3. IP PT (Given) 4. P P (Reflexive Prop.) 5. IPK TPB (AAS) 21. CD , because the segments have the same GH measure. CFD HFG because vertical angles are congruent. Since F is the midpoint of DG , DF FG . It cannot be determined whether CFD HFG. The information given does not lead to a unique triangle. 22. Since F is the midpoint of DG , DF FG . F is also the midpoint of CH , so C F FH . Since D G C H , DF CF and F G FH . CFD HFG because vertical angles are congruent. CFD HFG by SAS. 23. Since N is the midpoint of J L, J N N L. JNK LNK because perpendicular lines form right angles and right angles are congruent. By the Reflexive Property, KN . JKN LKN by SAS. KN 24. It is given that JM L M and NJM NLM. By the Reflexive Property, NM N M . It cannot be determined whether JNM LNM. The information given does not lead to a unique triangle. 33. Given: BA DE , DA BE B D C Prove: BEA DAE Proof: A E DA BE 11. Given: JK ⊥ KM , JM K L , ML || JK Prove: ML JK Proof: Statements (Reasons) 1. 2. 3. 4. 5. 6. 7. M J L K JK ⊥ KM , JM K L , ML || JK (Given) JKM is a rt. (⊥ lines form rt. .) KM ⊥ ML (Perpendicular Transversal Th.) LMK is a rt. (⊥ lines form rt. .) MK MK (Reflexive Property) JMK LMK (HL) ML JK (CPCTC) Given BA DE BEA DAE Given ASA AE AE Reflexive Prop. W Z Y X XZ ⊥ WY Given WZX and YZX are rt . Def. of ⊥ lines. WZX YZX All rt. are . XZ XZ XZ bisects WY. Given WZ ZY Def. of seg. bisector WZX YZX SAS Reflexive Prop. Pages 214–215, Geometry Activity 10. Given: ML ⊥ MK , JK ⊥ KM , J L Prove: JM KL Proof: Statements (Reasons) 1. 2. 3. 4. 5. 6. M J L 1 K ML ⊥ MK , JK ⊥ KM , J L (Given) LMK and JKM are rt. (⊥ lines form 4 rt. .) LMK and JKM are rt. s (Def. of rt. ) MK (Reflexive Property) MK LMK JKM (LA) JM KL (CPCTC) Proof: K J Statements (Reasons) 1. XKF is equilateral. (Given) 2. 1 2 (Equilateral s are equiangular.) 3. KX FX (Definition of equilateral ) 4. XJ bisects X (Given) 5. KXJ FXJ (Def. of bisector) 6. KXJ FXJ (ASA) 7. KJ JF (CPCTC) 8. J is the midpoint of K F . (Def. of midpoint) 2 F Chapter 4 Additional Answers 233F Additional Answers for Chapter 4 34. Given: XZ ⊥ WY , X Z bisects W Y . Prove: WZX YZX Proof: Pages 219–221, Lesson 4-6 7. Given: CTE is isosceles T with vertex C. 60 mT 60 C Prove: CTE is equilateral. E Proof: Statements (Reasons) 1. CTE is isosceles with vertex C. (Given) 2. C T (Def. of isosceles ) CE 3. E T (Isosceles Th.) 4. mE mT (Def. of ) 5. mT 60 (Given) 6. mE 60 (Substitution) 7. mC mE mT 180 (Angle Sum Theorem) 8. mC 60 60 180 (Substitution) 9. mC 60 (Subtraction) 10. CTE is equiangular. (Def. of equiangular ) 11. CTE is equilateral. (Equiangular s are equilateral.) 29. Given: XKF is equilateral. X XJ bisects X. Prove: J is the midpoint of KF . Additional Answers for Chapter 4 30. Given: MLP is isosceles. N is the midpoint of MP . Prove: LN ⊥ MP L Proof: M N P Statements (Reasons) 1. MLP is isosceles. (Given) 2. M L L P (Definition of isosceles ) 3. M P (Isosceles Theorem) 4. N is the midpoint of MP . (Given) 5. MN NP (Midpoint Theorem) 6. MNL PNL (SAS) 7. LNM LNP (CPCTC) 8. mLNM mLNP ( have equal measures.) 9. LNM and LNP are a linear pair (Def. of linear pair) 10. mLNM mLNP 180 (Sum of measures of linear pair of 180) 11. 2mLNM 180 (Substitution) 12. mLNM 90 (Division) 13. LNM is a right angle. (Definition of right ) 14. LN ⊥ MP (Definition of ⊥) 31. Case I: Given: ABC is an equilateral triangle. B Prove: ABC is an equiangular triangle. A C Proof: Statements (Reasons) 1. ABC is an equilateral triangle. (Given) 2. AB AC B C (Def. of equilateral ) 3. A B C (Isosceles Th.) 4. ABC is an equiangular triangle. (Def. of equiangular ) Case II: Given: ABC is an equiangular B triangle. Prove: ABC is an equilateral triangle. A C Proof: Statements (Reasons) 1. ABC is an equiangular triangle. (Given) 2. A B C (Def. of equiangular ) 3. A B AC B C (If 2 of a are , then the sides opp. those are .) 4. ABC is an equilateral triangle. (Def. of equiangular ) 233G Chapter 4 Additional Answers 32. Given: MNO is an equilateral triangle. Prove: mM mN mO 60 O Proof: M N Statements (Reasons) 1. MNO is an equilateral triangle. (Given) 2. MN MO NO (Def. of equilateral ) 3. M N O (Isosceles Th.) 4. mM mN mO (Def. of ) 5. mM mN mO 180 ( Sum Theorem) 6. 3mM 180 (Substitution) 7. mM 60 (Division prop.) 8. mM mN mO 60 (Substitution) 33. Given: ABC B A C Prove: AB CB A C D Proof: Statements (Reasons) bisect ABC. (Protractor Post.) 1. Let BD 2. ABD CBD (Def. bisector) 3. A C (Given) (Reflexive Prop.) 4. BD BD 5. ABD CBD (AAS) 6. AB (CPCTC) CB 38. There are two sets of 12 isosceles triangles. One black set forms a circle with their bases on the outside of the circle. Another black set encircles a circle in the middle. 39. The triangles in each set appear to be acute. 44. Given: N D, G I, AN S D Prove: ANG SDI G N S A D I Proof: We are given N D and G I. By the Third Angle Theorem, A S. We are also given AN SD . ANG SDI by ASA. 45. Given: VR ⊥ RS , UT ⊥S U , R V RS US S Prove: VRS TUS Proof: We are given that T U VR ⊥ RS , UT ⊥ SU , and R S U S . Perpendicular lines form four right angles, so R and U are right angles. R U because all right angles are congruent. RSV UST since vertical angles are congruent. Therefore, VRS TUS by ASA. 48. 50. 49. p q p q p or q T T T a b a and b T T F F F T F F T F F T T F T F F T T F T F F F F F T T T k m m k and m T T F T F F F 51. y y or z y z F T T F T T T T F F F T F F F T T T F T F F F T T 9. Given: ABC Prove: ABC is isosceles. Proof: Use the Distance Formula to find AB and BC. B(2, 8) AB (2 0 )2 (8 0)2 4 64 or 68 BC (4 2 )2 (0 8)2 A(0, 0) C (4, 0) x 4 64 or 68 Since AB BC, AB B C . Since the legs are congruent, ABC is isosceles. 10. Sample answer: 11. Sample answer: y y T (b–2, c) 12. Sample answer: N (2a, 0) x M (0, 0) R(b, 0) x Q (0, 0) P (a, b) 13. Sample answer: y y X(1–4b, c) J(0, c) E A J Proof: Statements (Reasons) 1. A H, AEJ HJE (Given) 2. EJ E J (Reflexive Prop.) 3. AEJ HJE (AAS) 4. AJ E H (CPCTC) Pages 224–226, Lesson 4-7 y 8. Given: ABC is a right B(0, 2b) triangle with hypotenuse B C . M M is the midpoint of BC . A(0, 0) C(2c, 0) x Prove: M is equidistant from the vertices. Proof: The coordinates of M, the midpoint of BC , will 22c 2b 2 be , (c, b). The distance from M to each of the vertices can be found using the Distance Formula. 2 b2 MB (c 0 )2 (b 2b )2 c MC (c 2 c)2 (b 0 )2 (c 0 )2 (b 0)2 2 b2 c 2 b2 MA c Thus, MB MC MA, and M is equidistant from the vertices. 14. Sample answer: y 15. Sample answer: ab 2 ,c Y( Z (1–2b, 0) x W(0, 0) M (c, 0) x L(0, 0) H y ) X(0, b) P(0, 0) W(a b, 0) x 25. Given: isosceles ABC with A C BC R and S are midpoints of legs A C and BC . Prove: AS BR Proof: The coordinates of R 2a 2 0 2b 0 2 Z(2b, 0) x Y (0, 0) y C(2a, 2b) S R B(4a, 0) x A(0, 0) are , or (a, b). 2a 2 4a 2b 0 2 The coordinates of S are , or (3a, b). BR (4a a)2 (0 b)2 (3a)2 (b)2 or 9a 2 b2 AS (3a 0)2 (b 0)2 (3a)2 (b)2 or 9a 2 b2 Since BR AS, AS B R . Chapter 4 Additional Answers 233H Additional Answers for Chapter 4 Page 221, Practice Quiz 2 1. JM 5 , ML 26 , JL 5, BD 5 , DG 26 , and BG 5. Each pair of corresponding sides have the same measure so they are congruent. JML BDG by SSS. 2. Given: A H, AEJ HJE Prove: AJ E H y y 26. Given: isosceles triangle ABC AB BC A C R, S, and T are R midpoints of their respective sides. A(0, 0) Prove: RST is isosceles. Proof: a0 b0 a b Midpoint R is , or , . C(a, b) S Additional Answers for Chapter 4 2 B(a, 0) x ST and AB have the same slope so S T || A B. y 28. Given: ABC C(b, c) S is the midpoint of AC . T is the midpoint of B C . T S 1 Prove: ST AB 2 c c 2 ab b 2 2 2 2 2 a 2 02 2 a 2 a or 2 2 ST AB (a 0 )2 (0 0)2 a2 02 or a 1 2 ST AB 233I Chapter 4 Additional Answers x B(800, 800) A(1600, 0) P(0, 0) R(800, 800) x Proof: Since PR and BR have the same measure, PR . BR 00 800 0 800 0 The slope of BR , which is undefined. 800 800 0 a Proof: y The slope of PR or 0. Slope of A B or 0. A(0, 0) O BF (6 3 )2 (1 4)2 9 9 or 32 Since AB BF, AB BF . ABD FBD by SSS. 30. Given: BPR PR 800, BR 800 Prove: BPR is an isosceles right triangle. Slope of S T or 0. ab b 00 a0 F(6, 1) AB (3 0 )2 (4 1)2 9 9 or 32 2 2 2 2 ab 0c ab c Midpoint T is , or , . 2 2 2 2 2 2 D(3, 1) Proof: BD B D by the Reflexive Property. DF (6 3 )2 (4 1)2 9 0 or 3 Since AD DF, AD DF . Proof: b0 c0 b c Midpoint S is , or , . 0 a B(3, 4) AD (3 0 )2 (1 1)2 9 0 or 3 RT ST and RT S T and RST is isosceles. y 27. Given: ABC C(b, c) S is the midpoint of AC . T is the midpoint of B C . T S Prove: ST || A B c c 2 2 y A(0, 1) T B(2a, 0) x 2 2 2 2 a 2a b 0 3a b Midpoint S is , or , . 2 2 2 2 2a 0 0 0 Midpoint T is , or (a, 0). 2 2 2 2 2 2 a b RT 2 a 2 0 a2 2b 2 2 a2 2b 2 2 ST 32a a b2 0 2 2 a2 b2 A(0, 0) 29. Given: ABD, FBD AF 6, BD 3 Prove: ABD FBD B(a, 0) x PR ⊥ BR , so PRB is a right angle. BPR is an isosceles right triangle. 31. Given: BPR, BAR PR 800, BR 800, RA 800 Prove: P B BA y B(800, 800) A(1600, 0) P(0, 0) R(800, 800) x Proof: PB (800 0)2 (800 0)2 or 1,280 ,000 BA (800 1600 )2 (8 00 0)2 or 1,280 ,000 PB BA, so P B BA . 32. Given: JCT Prove: JCT is a right triangle. y 43. Given: A D CE , A D || C E Prove: ABD EBC T(300, 500) J A Proof: Statements (Reasons) (–500, 300) C(0, 0) x Proof: 300 0 3 The slope of JC or . 500 0 5 500 0 5 The slope of T C or . 300 0 3 The slope of T C is the opposite reciprocal of the slope of JC . JC ⊥ TC , so TCJ is a right angle. So JCT is a right triangle. 41. Given: 3 4 Q Prove: QR QS 3 1 2 4 1 3 L Proof: J Statements (Reasons) 1. isosceles triangle JKN with vertex N (Given) 2. NJ NK (Def. of isosceles triangle) 3. 2 1 (Isosceles Triangle Theorem) 4. JK || L M (Given) 5. 1 3, 4 2 (Corr. Post.) 6. 2 3, 4 1 (Congruence of is transitive. Statements 3 and 5) 7. 4 3 (Congruence of is transitive. Statements 3 and 6) 8. LN M N (If 2 of a are , then the sides opp. those are .) 9. NML is an isosceles triangle. (Def. of isosceles ) D E 1. A D || C E (Given) 2. A E, D C (Alt. Int. Theorem) 3. AD CE (Given) 4. ABD EBC (ASA) 44. Given: W X XY , V Z V Prove: WV Y Z W X Z Proof: Statements (Reasons) 1. 2. 3. 4. Y WX X Y , V Z (Given) WXV YXZ (Vertical ) WXV YXZ (AAS) WV Y Z (CPCTC) Page 231, Chapter 4 Practice Test 17. Given: ABE, BCE AB 22, AC 44, AE 36, CD 36 A and C are right angles. Prove: ABE CBD A B E C D Proof: We are given that AB 22 and AC 44. By the Segment Addition Postulate, AB BC AC 22 BC 44 BC 22 Substitution Subtract 22 from each side. Since AB BC, then by the definition of congruent segments, AB BC . We are given that AE 36 and CD 36. Then also by the definition of congruent segments, AE CD . We are additionally given that both A and C are right angles. Since all right angles are congruent, A C. Since AB B C , A C, and AE C D , then by SAS, ABE CBD. Chapter 4 Additional Answers 233J Additional Answers for Chapter 4 Proof: S R Statements (Reasons) 1. 3 4 (Given) 2. 2 and 4 form a linear pair. 1 and 3 form a linear pair. (Def. of linear pair) 3. 2 and 4 are supplementary. 1 and 3 are supplementary. (If 2 form a linear pair, then they are suppl.) 4. 2 1 (Angles that are suppl. to are .) 5. Q R QS (If 2 of a are , then the sides opposite those are .) 42. Given: isosceles JKN K M 2 with vertex N, JK || L M 4 Prove: NML is isosceles N C B