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................................. 880 READY TO GO ON? QUIZ................................. 881 EXT Using Patterns to Generate Fractals..................... 882 G.5.C Study Guide: Preview.................................... 822 Reading and Writing Math............................... 823 Study Guide: Review.................................... 884 Chapter Test........................................... 888 Problem Solving on Location............................. 894 Tools for Success Writing Math 829, 836, 844, 852, 861, Study Strategy 823 868, 878, 883 Vocabulary 821, 822, 827, 851, 859, 866, 875, 884 Know-It Notes 825, 826, 832, 833, 840, 841, 848, 849, 850, 856, 857, 858, 866, 873, 874 Graphic Organizers 826, 833, 841, 850, 858, 866, 874 Test Prep Exercises 829–830, 836–837, 845, 853, 862, 869, 878 Multi-Step TAKS Prep 829, 835, 843, 853, 854, 861, 868, 876, 880 College Entrance Exam |
Practice 889 TAKS Tackler 890 Homework Help Online 827, 834, 842, TAKS Prep 892 851, 859, 866, 875 WHO USES MATHEMATICS? The Career Path features are a set of interviews with young adults who are either preparing for or just beginning in different career fields. These people share what math courses they studied in high school, how math is used in their field, and what options the future holds. Also, many exercises throughout the book highlight skills used in various career fields. KEYWORD: MG7 Career ELECTRICIAN p. 320 Electricians install and maintain the systems that provide many of the modernday comforts we rely on, such as climate control, lighting, and technology. Look on page 320 to find out how Alex Peralta got started on this career path. TECHNICAL WRITER p. 612 Have you ever wondered who writes manuals for operating televisions or stereos? A technical writer not only writes manuals for operating electronics, but also documents maintenance procedures for airplanes. Look at the Career Path on page 612 to find out how to become a technical writer. FURNITURE MAKER p. 805 A furniture maker must take precise measurements and be aware of spatial relationships in order to build a quality finished product. The Career Path on page 805 describes the kind of experience needed to be successful as a furniture maker. Career Applications Advertising 499 Agriculture 765 Animation 53, 835, 842 Anthropology 802 Archaeology 262, 787, 793 Architecture 47, 467, 667 Art 483, 657, 873 Aviation 277, 546, 564 Business 108, 194, 625 Carpentry 168, 418, 836 City Planning 305, 827 Communication 634, 802 Computer Graphics 495 Design 311, 317, 318 Electronics 692 Engineering 260, 554 Finance 108, 522 Forestry 548 Graphic Design 498, 752 Health 343 Industry 344 Interior Decorating 609, 867 Landscaping 607, 686, 702 Manufacturing 38, 754 Marine Biology 698, 720 Mechanics 434 Meteorology 801 Music 24 Navigation 228, 567, 729 Nutrition 107 Oceanography 174 Optometry 877 Photography 385, 459, 475 Political Science 79, 93 Real Estate 486 Surveying 25, 263, 556 xviii xviii Who Uses Mathematics? WHY LEARN MATHEMATICS? Links to |
interesting topics, including some in Texas, may accompany realworld applications in the text. These links help you see how math is used in the real world. For a complete list of all applications in Holt Geometry, see page S162 in the Index. Real-World Animation 835 Archaeology 787 Architecture 159, 220, 695 Astronomy 752 Bicycles 337 Bird-Watching 401 Biology 100, 604 Chemistry 828 Conservation 271 Design 313 Ecology 248 Electronics 692 Engineering 115, 233 Entertainment 149, 683, 803, 833 Fitness 539 Navigation 278 Oceanography 174 Pets 361 Racing 392 Recreation 92, 674 Food 195 Geography 626 Geology 86, 804 History 48, 413, 531, 566, 595 Kites 428 Landscaping 607 Marine Biology 698, 720 Math History 41, 78, 257, 318, 493, 611, 703, 768 Measurement 404 Mechanics 434 Meteorology 476, 675, 797 Monument 466 Mosaics 876 Shuffleboard 305 Space Shuttle 548 Sports 19, 530, 635 Surveying 353, 556 Transportation 183 Travel 458 Why Learn Mathematics xix xix Each frame of a computer-animated feature represents 1__24of a second of film.Source: www.pixar.comAnimationThe Top Thrill Dragster is 420 feet tall and includes a 400-foot vertical drop. It twists 270° as it drops. It is one of 16 roller coasters at Cedar Point amusement park.RecreationThis mosaic of the seal of the Republic of Texas is one of six tile mosaics that were installed on the front façade of the Sam Houston Regional Library and Research Center in Liberty, Texas, in fall 2001.MosaicsEach frame of a computer-animated feature represents 1__24of a second of film.Source: www.pixar.comAnimationThe Top Thrill Dragster is 420 feet tall and includes a 400-foot vertical drop. It twists 270° as it drops. It is one of 16 roller coasters at Cedar Point amusement park.RecreationThis mosaic of the seal of the Republic of Texas is one of six tile mosaics that were installed on the front façade of the Sam Houston Regional Library and Research Center in Liberty, Texas, in fall 2001.MosaicsEach frame of a computer-animated feature represents 1__24of a second of film.Source: www.pixar |
.comAnimationThe Top Thrill Dragster is 420 feet tall and includes a 400-foot vertical drop. It twists 270° as it drops. It is one of 16 roller coasters at Cedar Point amusement park.RecreationThis mosaic of the seal of the Republic of Texas is one of six tile mosaics that were installed on the front façade of the Sam Houston Regional Library and Research Center in Liberty, Texas, in fall 2001.Mosaics476Chapter 7 Similarity 26.Critical Thinking�ABC is not similar to �DEF, and �DEF is not similar to�XYZ. Could �ABC be similar to �XYZ? Why or why not? Make a sketch to support your answer. 27.Recreation To play shuffleboard, two teams take ���������������������turns sliding disks on a court. The dimensions of the scoring area for a standard shuffleboard court are shown. What are JK and MN? 28. Prove the Transitive Property of Similarity. Given:�ABC∼�DEF,�DEF∼�XYZProve:�ABC∼�XYZ 29. Draw and label �PQR and �STU such that PQ___ST= QR___TUbut�PQR is NOT similar to �STU. 30.Given:�KNJ is isosceles with �������∠N as the vertex angle.∠H�∠LProve:�GHJ∼�MLK31.MeteorologySatellite photography makes it possible ������������to measure the diameter of a hurricane. The figure shows that a camera’s aperture YX is 35 mm and its focal lengthWZ is 50 mm. The satellite W holding the camera is 150 mi above the hurricane, centered at C. a. Why is �XYZ∼�ABZ? What assumption must you make about the position of the camera in order to make this conclusion? b. What other triangles in the figure must be similar? Why? c. Find the diameter AB of the hurricane. 32./////ERROR ANALYSIS///// Which solution for the �����������value of y is incorrect? Explain the error.�������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� |
����������������������������������������� 33.Write About It Two isosceles triangles have congruent vertex angles. Explain why the two triangles must be similar.476 25. This problem will prepare you for the ���������������������������������������Multi-Step TAKS Prep on page 478. The set for an animated film includes three small triangles that represent pyramids. a. Which pyramids are similar? Why? b. What is the similarity ratio of the similar pyramids?This satellite image shows Hurricane Lili as it moves across the Gulf of Mexico. In October 2002, an estimated 500,000 people evacuated in advance of Lili’s hitting Texas.Meteorology HOW TO STUDY GEOMETRY This book has many features designed to help you learn and study effectively. Becoming familiar with these features will prepare you for greater success on your exams. Learn The vocabulary is listed at the beginning of every lesson. Look for the Know-It-Note icons to identify important information. Practice Use a graphic organizer to summarize each lesson. Refer to the examples from the lesson to solve the Guided Practice exercises. Review Study and review vocabulary from the entire chapter. xx xx How To Study Geometry 4-4 Triangle Congruence: SSS and SAS TEKS G.10.B Congruence and the geometry of size: justify and apply triangle congruence relationships. Also G.2.A, G.3.B, G.3.E Objectives Apply SSS and SAS to construct triangles and to solve problems. Prove triangles congruent by using SSS and SAS. Vocabulary triangle rigidity included angle Who uses this? Engineers used the property of triangle rigidity to design the internal support for the Statue of Liberty and to build bridges, towers, and other structures. (See Example 2.) In Lesson 4-3, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent. The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape. For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate. Postulate 4-4-1 Side-Side-Side (SSS) Congruence Side-Side-Side (SSS) Congruence POST |
ULATE POSTULATE POSTULATE POSTULATE HYPOTHESIS HYPOTHESIS HYPOTHESIS HYPOTHESIS CONCLUSION CONCLUSION CONCLUSION CONCLUSION If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. � ���� ���� � ���� � � ���� ���� ���� ���� � � � FDE �ABC � �FDE �ABC �FDE � � � E X A M P L E 1 Using SSS to Prove Triangle Congruence Adjacent triangles Adjacent triangles share a side, so share a side, so you can apply the Reflexive Property to get a pair of congruent parts. Use SSS to explain why �PQR � �PSR. −− −− SR. By QR � It is given that −− PR � the Reflexive Property of Congruence, Therefore �PQR � �PSR by SSS. −− PS and that −− PQ � −− PR. 1. Use SSS to explain why �ABC � �CDA. � � � � � An included angle is an angle formed by two adjacent sides of a polygon. ∠B is the included angle between sides −− AB and −− BC. � � � � 242 242 Chapter 4 Triangle Congruence THINK AND DISCUSS 1. Describe three ways you could prove that �ABC � �DEF.DEF.DEF prove that �ABC � �DEF. 2. Explain why the SSS and SAS 2. Explain why the SSS and SAS � � Postulates are shortcuts for Postulates are shortcuts for proving triangles congruent. proving triangles congruent. 3. GET ORGANIZED Copy and 3. GET ORGANIZED Copy and complete the graphic organizer. complete the graphic organizer. Use it to compare the SSS and Use it to compare the SSS and SAS postulates. SAS postulates. � � � � � ��� ��� ������������������� ������������������� ����������������������� 4-4 4-4 Exercises Exercises Exercises Exercises Exercises Exercises Exercises KEYWORD: MG7 4-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In �RST which angle is the included angle of sides −− ST and −− TR Use SSS to explain why |
the triangles in each pair are congruent. p. 242 2. �ABD � �CDB 3. �MNP � �MQP � � � � � � � �. Design This Texas flag consists of a blue, p. 243 perpendicular stripe with a white star in the center. The star consists of five triangles. GJ = LG = 20 in., and GK = GH = 13 in. Use SAS to explain why �JGK � �LGH Show that the triangles are congruent for the given value of the variable. p. 244 5. �GHJ � �IHJ, x = 4 6. �RST � �TUR, x = 18 � � ������ � � � ������ � �� � ��� � � ��� ������� � Study the examples to apply new concepts and skills. Examples include stepped out solutions. Test your understanding of examples by trying the Check It Out problems. Check your work in the Selected Answers. If you get stuck, use the internet for Homework Help Online. 4-4 Triangle Congruence: SSS and SAS 245 245 For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary acute triangle acute triangle.............. 216 CPCTC..................... 260 isosceles triangle........... 217 auxiliary line auxiliary line............... 223 equiangular triangle........ 216 legs of an isosceles triangle.. 273 base base....................... 273 equilateral triangle......... 217 obtuse triangle............. 216 base angle base angle.................. 273 exterior.................... 225 remote interior angle....... 225 congruent polygons congruent polygons......... 231 exterior angle.............. 225 |
right triangle............... 216 coordinate proof coordinate proof............ 267 included angle.............. 242 scalene triangle............. 217 corollary corollary................... 224 included side............... 252 triangle rigidity............. 242 corresponding angles corresponding angles....... 231 interior.................... 225 vertex angle................ 273 corresponding sides corresponding sides......... 231 interior angle............... 225 Complete the sentences below with vocabulary words from the list above. Complete the sentences below with vocabulary words from the list above. 1. A(n) 1. A(n)? is a triangle with at least two congruent sides. −−−− 2. A name given to matching angles of congruent triangles is?. −−−− 3. A(n)? is the common side of two consecutive angles in a polygon. −−−− 4-1 Classifying Triangles (pp. 216–221) TEKS G.1.A TEKS G.1.A E X A M P L E EXERCISES � Classify the triangle by its angle measures and side lengths. isosceles right triangle Classify each triangle by its angle measures and side lengths. 4. 5. ��� ��� ��� ���� 4-2 Angle Relationships in Triangles (pp. 223–230) TEKS G.1.A, G.2.B TEKS G.1.A, G.2.B E X A M P L E � Find m∠S. 12x = 3x + 42 + 6x ���������� � 12x = 9x + 42 ���� � ��� � 3x = 42 x = 14 m∠S = 6(14) = 84° EXERCISES Find m∠N. 6. � �� �� |
� ���� � 284 284 Chapter 4 Triangle Congruence 7. In�LMN, m∠L = 8x °, m∠M = (2x + 1)°, and m∠N = (6x - 1)°. Use the list on p. S82 to review the postulates and theorems found in the chapter. Test yourself with practice problems from every lesson in the chapter. TOOLS OF GEOMETRY In geometry, it is important to use tools correctly in order to measure accurately and produce accurate figures. One important tool is your pencil. Always use a sharp pencil with a good eraser. Ruler The ruler shown has a mark every 1 __ 8 inch, so the accuracy is to the nearest 1 __ 8 inch. Protractor To use a protractor to measure an angle, you may need to extend the sides of the angle. For acute angles, use the smaller measurement. For obtuse angles, use the larger measurement. � � � � � � � � � � � �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� � � � � � Line up one ray with 0. � � � � �� �� �� Line up one end with 0, not the edge. Choose the measurement that is the closest. Place the center of your protractor on the vertex. Compass A compass is used to draw arcs and circles. If you have trouble keeping the point in place, try keeping the compass still and turning the paper. Straightedge A straightedge is used to draw a line through two points. If you use a ruler as a straightedge, do not use the marks on the ruler. Keep your wrist flexible. Turn the compass with your index finger and thumb. Tilt the compass slightly. First place your pencil on one points. Place the straightedge against your pencil and the other point. Draw the line. � � � � � � � � � � � � � � � � � � � �� �� �� � � Geometry Software Geometry software can be used to create figures and explore their properties. Use the toolbar to select, draw; and label figures. Drag points to explore properties of a figure. Use the menus to construct, transform, and measure figures. The parts of each figure are linked. To avoid deleting a whole figure, hide parts |
instead of deleting them. Tools of Geometry xxixxi �� �� �� � �� �� �� � � �� �� �� �� �� �� � � �� �� �� �� �� �� �� �� �� �� Scavenger H Use this scavenger hunt to discover a few of the many tools in the Texas Edition of Holt Geometry that you can use to become an independent learner. On a separate sheet of paper, write the answers to each question below. Within each answer, one letter will be in a yellow box. After you have answered every question, identify the letters that would be in yellow boxes and rearrange them to reveal the answer to the question at the bottom of the page. What theorem are you asked about in the Know-It Note on page 352? What keyword should you enter for Homework Help for Lesson 3-3? What is the first Vocabulary term in the Study Guide: Preview for Chapter 1? In Lesson 8-2, what is Example 4 teaching you to find? 1. ■ ■ ■ ■ ■ 2. ■ ■ ■ ■ ■ ■ 3. ■ ■ ■ ■ ■ ■ ■ 4. ■ ■ ■ ■ ■ ■ 8. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ In the Study Guide: Review for Lesson 11-1, what do the lines intersect? Whose job is described in the Career Path on page 612? What advice does Chapter 1’s TAKS Tackler give about how to answer a multiple choice test item you don’t know how to solve? What mathematician is featured in the Math History link on page 318? What did the little acorn say when it grew up? ■ ■ ■ ■ ■ ■ ■ ■ xxii xxii Scavenger Hunt You can practice using the four-step Problem Solving Plan to solve problems in the Problem Solving on Location feature located at the end of selected chapters. Each page focuses on interesting people, and facts from the Lone Star State. You can follow the flight path of the hot air balloons in the Great Texas Balloon Race that starts in Longview. ������� ������� ��������� ������ ������� ������������ ������������ ���������� ���������� ������ ������ Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an |
Organized List The Great Texas Balloon Race The Great Texas Balloon Race The Great Texas Balloon Race The annual Great Texas Balloon Race is one of the most The annual Great Texas Balloon Race is one of the most The annual Great Texas Balloon Race is one of the most exciting hot air balloon events in Texas. “Balloon Glow,” exciting hot air balloon events in Texas. “Balloon Glow,” exciting hot air balloon events in Texas. “Balloon Glow,” exciting hot air balloon events in Texas. “Balloon Glow,” in which balloons are tethered and illuminated in an in which balloons are tethered and illuminated in an in which balloons are tethered and illuminated in an evening display, was begun in Longview, the race’s evening display, was begun in Longview, the race’s evening display, was begun in Longview, the race’s starting point, in 1980. Traditionally held in July, the starting point, in 1980. Traditionally held in July, the starting point, in 1980. Traditionally held in July, the race attracts balloonists who compete to fly the race attracts balloonists who compete to fly the race attracts balloonists who compete to fly the obstacle course the most accurately. obstacle course the most accurately. Choose one or more strategies to solve each problem. Choose one or more strategies to solve each problem. 1. The event starts in Longview, and ends near Estes, 1. The event starts in Longview, and ends near Estes, Texas. The balloons do not fly from the start to the Texas. The balloons do not fly from the start to the finish in a straight line. They follow a zigzag course finish in a straight line. They follow a zigzag course finish in a straight line. They follow a zigzag course finish in a straight line. They follow a zigzag course to take advantage of the wind. Suppose one of the to take advantage of the wind. Suppose one of the to take advantage of the wind. Suppose one of the balloons leaves Longview at a bearing of N 50° E balloons leaves Longview at a bearing of N 50° E and follows the course shown. At what bearing does the balloon approach Estes? � � ��� �������� � � � ���� � � ��� � ����� 2. The speed of the balloon depends on the current wind speed. One event in The Great Texas Balloon Race requires the balloonist to fly to a pole |
that is 2 mi from the starting point. The balloonist must drop a small ring around the pole, which is 20 ft tall. A second target is 1 mi from the first, a third target is another 3 mi from the second, and a final target is 5 mi farther. If the wind speed is 3.5 mi/h, how long will it take the balloonist to finish the course? Round to the nearest hundredth 3. During the race, one of the balloons leaves Longview L, flies to X, and then flies to Y. The team discovers a problem with the balloon, so it must return directly to Longview. Does the table contain enough information to determine the return course to L? Explain. of an hour. � � � � L to X X to Y Y to L Bearing N 42° E S 59° E Distance (mi) 3.1 2.4 Problem Solving on Location 295 295 T E X A S TAKS Grades 9–11 Obj. 10 Port Isabel Point Isabel Lighthouse Point Isabel Lighthouse The Point Isabel Lighthouse was built in 1853 on a prominent The Point Isabel Lighthouse was built in 1853 on a prominent bluff on the mainland. Today, the fully restored lighthouse is bluff on the mainland. Today, the fully restored lighthouse is the only one in Texas that is open for climbing and viewing. the only one in Texas that is open for climbing and viewing. Choose one or more strategies to solve each problem. Choose one or more strategies to solve each problem. 1. Suppose the beam from the lighthouse is visible for up to 1. Suppose the beam from the lighthouse is visible for up to 15 miles at sea. To the nearest square mile, what is the area of water covered by the beam as it rotates by an angle of 60°? of water covered by the beam as it rotates by an angle of 60°? 2. Given that Earth’s radius is approximately 4000 miles 2. Given that Earth’s radius is approximately 4000 miles and that the top of the tower of a lighthouse is 65 ft above and that the top of the tower of a lighthouse is 65 ft above sea level, find the distance from the top of the tower to sea level, find the distance from the top of the tower to the horizon. Round your answer to the nearest mile. the horizon. Round your answer to the nearest mile. (Hint: (Hint: 65 feet = 0.01 miles) For 3, use the table |
. For 3, use the table. 3. Most lighthouses use Fresnel lenses, named after their 3. Most lighthouses use inventor, Augustine Fresnel. The table shows the sizes, or inventor, Augustine Fresnel. The table shows the sizes, or orders orders, of the circular lenses. The diagram shows some measurements of the Fresnal lens used in the Point Isabel measurements of the Fresnal lens used in the Point Isabel Lighthouse. What is the order of the lens? Lighthouse. What is the order of the lens? Fresnel Lenses Order ����� ������� Lens Diameter 6 ft 1 in. 4 ft 7 in. 3 ft 3 in. 1 ft 8 in. 1 ft 3 in. 1 ft 0 in. First Second Third Fourth Fifth Sixth 894 894 Chapter 12 Extending Transformational Geometry 11 4 1 Learn about the sizes and order of lighthouse lenses, such as the one at the Point Isabel Lighthouse. Take a tour of the show caves that are located in the Texas Hill Country between San Antonio and Austin Te Te Ta Ve To ow n a s t h e Te ow ow ow c a v e i n Te Ja 5_ Ja ow Te x a s s h ow ’’t I f y o u d o n’ Te? ����������������� �������� �������� ������� Problem Solving on Location xxixxxix Foundations for Geometry 1A Euclidean and Construction Tools 1-1 Understanding Points, Lines, and Planes Lab Explore Properties Associated with Points 1-2 Measuring and Constructing Segments 1-3 Measuring and Constructing Angles 1-4 Pairs of Angles 1B Coordinate and Transformation Tools 1-5 Using Formulas in Geometry 1-6 Midpoint and Distance in the Coordinate Plane 1-7 Transformations in the Coordinate Plane Lab Explore Transformations KEYWORD: MG7 ChProj The lights encasing the geodesic dome at the top of Reunion Tower are a familiar feature of the Dallas skyline. 2 2 Chapter 1 Vocabulary Match each term on the left with a definition on the right. 1. coordinate A. a mathematical phrase that contains operations, numbers, 2. metric system of measurement 3. expression 4. order of operations and/or variables B. the measurement system often used in the United States C. one of the numbers of an ordered pair that locates a point on a coordinate graph D. a list of rules for evaluating expressions |
E. a decimal system of weights and measures that is used universally in science and commonly throughout the world Measure with Customary and Metric Units For each object tell which is the better measurement. 5. length of an unsharpened pencil 2 in. or 9 3__ 7 1__ 4 in. 7. length of a soccer field 100 yd or 40 yd 9. height of a student’s desk 30 in. or 4 ft Combine Like Terms Simplify each expression. 11. -y + 3y - 6y + 12y 13. -5 - 9 - 7x + 6x 1 m or 2 1__ 6. the diameter of a quarter 2 cm 8. height of a classroom 5 ft or 10 ft 10. length of a dollar bill 15.6 cm or 35.5 cm 12. 63 + 2x - 7 - 4x 14. 24 - 3 + y + 7 Evaluate Expressions Evaluate each expression for the given value of the variable. 15. x + 3x + 7x for x = -5 16. 5p + 10 for p = 78 17. 2a - 8a for a = 12 18. 3n - 3 for n = 16 Ordered Pairs Write the ordered pair for each point. 19. A 20. B 21. C 23. E 22. D 24. F Foundations for Geometry 3 3 ������������������� Key Vocabulary/Vocabulario angle area ángulo área coordinate plane plano cartesiano line perimeter plane point línea perímetro plano punto transformation transformación undefined term término indefinido Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. A definition is a statement that gives the meaning of a word or phrase. What do you think the phrase undefined term means? 2. Coordinates are numbers used to describe a location. A plane is a flat surface. How can you use these meanings to understand the term coordinate plane? 3. A point is often represented by a dot. What real-world items could represent points? 4. Trans- is a prefix that means “across,” as in movement. A form is a shape. How can you use these meanings to understand the term transformation? Geometry TEKS G.1.A Geometric structure* develop an |
awareness of the structure of a mathematical system, connecting definitions, postulates... 1-2 Tech. Lab Les. 1-1 ★ Les. 1-2 Les. 1-3 Les. 1-4 Les. 1-5 Les. 1-6 Les. 1-7 ★ ★ ★ ★ ★ 1-7 Tech. Lab G.1.B Geometric structure* recognize the historical development of ★ ★ geometric systems and know mathematics is developed for a variety of purposes G.2.A Geometric structure* use constructions to explore attributes ★ ★ ★ of geometric figures... G.2.B Geometric structure* make conjectures about angles, lines... ★ ★ ★ ★ and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational... G.3.B Geometric structure* construct and justify statements about ★ ★ ★ geometric figures and their properties G.5.C Geometric patterns* use properties of transformations... to make connections between mathematics and the real world... G.7.A Dimensionality and the geometry of location* use one- and ★ two-dimensional coordinate systems to represent points, lines, rays, line segments... G.7.C Dimensionality and the geometry of location* develop and use formulas involving length, slope, and midpoint ★ G.8.A Congruence and the geometry of size* find areas of regular ★ polygons, circles... G.8.C Congruence and the geometry of size* derive, extend, and use the Pythagorean Theorem ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 4 4 Chapter 1 Reading Strategy: Use Your Book for Success Understanding how your textbook is organized will help you locate and use helpful information. As you read through an example problem, pay attention to the notes in the margin. These notes highlight key information about the concept and will help you to avoid common mistakes. The Glossary is found in the back of your textbook. Use it when you need a definition of an unfamiliar word or phrase. The Index is located at the end of your textbook. If you need to locate the page where a particular concept is explained, use the Index to find the corresponding page number. The Skills Bank is located in the back of your textbook. Look in the Skills Bank for help with math topics that were taught in previous courses, such as the order of operations. Try This Use your textbook for the following problems. 1. Use the |
index to find the page where right angle is defined. 2. What formula does the Know-It Note on the first page of Lesson 1-6 refer to? 3. Use the glossary to find the definition of congruent segments. 4. In what part of the textbook can you find help for solving equations? Foundations for Geometry 5 5 1-1 Understanding Points, Lines, and Planes TEKS G.7.A Dimensionality and the geometry of location: use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments.... Also G.1.A Objectives Identify, name, and draw points, lines, segments, rays, and planes. Apply basic facts about points, lines, and planes. Who uses this? Architects use representations of points, lines, and planes to create designs of buildings. Light, color, and geometric shapes are reflected in the design of the Central Library in San Antonio, Texas. Vocabulary undefined term point line plane collinear coplanar segment endpoint ray opposite rays postulate The most basic figures in geometry are undefined terms, which cannot be defined by using other figures. The undefined terms point, line, and plane are the building blocks of geometry. Undefined Terms TERM NAME DIAGRAM A point names a location and has no size. It is represented by a dot. A capital letter point P A line is a straight path that has no thickness and extends forever. A lowercase letter or two points on the line line ℓ, XY or YX A plane is a flat surface that has no thickness and extends forever. A script capital letter or three points not on a line plane R or plane ABC Points that lie on the same line are collinear. K, L, and M are collinear. K, L, and N are noncollinear. Points that lie in the same plane are coplanar. Otherwise they are noncoplanar. E X A M P L E 1 Naming Points, Lines, and Planes A plane may be named by any three noncollinear points on that plane. Plane ABC may also be named BCA, CAB, CBA, ACB, or BAC. Refer to the architectural design of the Central Library building. A Name four coplanar points. K, L, M, and N all lie in plane R. B Name three |
lines. AB, BC, and CA. 1. Use the diagram to name two planes Chapter 1 Foundations for Geometry ������������ Segments and Rays DEFINITION NAME DIAGRAM A segment, or line segment, is the part of a line consisting of two points and all points between them. The two endpoints ̶̶ BA ̶̶ AB or An endpoint is a point at one end of a segment or the starting point of a ray. A capital letter C and D A ray is a part of a line that starts at an endpoint and extends forever in one direction. Its endpoint and any other point on the ray RS Opposite rays are two rays that have a common endpoint and form a line. The common endpoint and any other point on each ray EF and EG E X A M P L E 2 Drawing Segments and Rays Draw and label each of the following. A a segment with endpoints U and V B opposite rays with a common endpoint Q 2. Draw and label a ray with endpoint M that contains N. A postulate, or axiom, is a statement that is accepted as true without proof. Postulates about points, lines, and planes help describe geometric properties. Postulates Points, Lines, and Planes 1-1-1 Through any two points there is exactly one line. 1-1-2 Through any three noncollinear points there is exactly one plane containing them. 1-1-3 If two points lie in a plane, then the line containing those points lies in the plane. E X A M P L E 3 Identifying Points and Lines in a Plane Name a line that passes through two points. There is exactly one line n passing through G and H. 3. Name a plane that contains three noncollinear points. 1- 1 Understanding Points, Lines, and Planes 7 7 ��������������������� Recall that a system of equations is a set of two or more equations containing two or more of the same variables. The coordinates of the solution of the system satisfy all equations in the system. These coordinates also locate the point where all the graphs of the equations in the system intersect. An intersection is the set of all points that two or more figures have in common. The next two post |
ulates describe intersections involving lines and planes. Postulates Intersection of Lines and Planes 1-1-4 If two lines intersect, then they intersect in exactly one point. 1-1-5 If two planes intersect, then they intersect in exactly one line. Use a dashed line to show the hidden parts of any figure that you are drawing. A dashed line will indicate the part of the figure that is not seen. E X A M P L E 4 Representing Intersections Sketch a figure that shows each of the following. A A line intersects a plane, but does not lie in the plane. B Two planes intersect in one line. 4. Sketch a figure that shows two lines intersect in one point in a plane, but only one of the lines lies in the plane. THINK AND DISCUSS 1. Explain why any two points are collinear. 2. Which postulate explains the fact that two straight roads cannot cross each other more than once? 3. Explain why points and lines may be coplanar even when the plane containing them is not drawn. 4. Name all the possible lines, segments, and rays for the points A and B. Then give the maximum number of planes that can be determined by these points. 5. GET ORGANIZED Copy and complete the graphic organizer below. In each box, name, describe, and illustrate one of the undefined terms. 8 8 Chapter 1 Foundations for Geometry ������������������ 1-1 Exercises Exercises KEYWORD: MG7 1-1 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Give an example from your classroom of three collinear points. 2. Make use of the fact that endpoint is a compound of end and point and name the endpoint of ST. Use the figure to name each of the following. p. 6 3. five points 4. two lines 5. two planes 6. point on BD Draw and label each of the following. p. 7 7. a segment with endpoints M and N 8. a ray with endpoint F that passes through Use the figure to name each of the following. p. 7 9. a line that contains A and C 10. a plane that contains A, D, and Sketch a figure that shows each of the following. p. 8 11. three coplanar lines |
that intersect in a common point 12. two lines that do not intersect PRACTICE AND PROBLEM SOLVING Independent Practice Use the figure to name each of the following. For See Exercises Example 13. three collinear points 13–15 16–17 18–19 20–21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S4 Application Practice p. S28 14. four coplanar points 15. a plane containing E Draw and label each of the following. 16. a line containing X and Y 17. a pair of opposite rays that both contain R Use the figure to name each of the following. 18. two points and a line that lie in plane T 19. two planes that contain ℓ Sketch a figure that shows each of the following. 20. a line that intersects two nonintersecting planes 21. three coplanar lines that intersect in three different points 1- 1 Understanding Points, Lines, and Planes 9 9 �������������������������� 22. This problem will prepare you for the Multi-Step TAKS Prep on page 34. Name an object at the archaeological site shown that is represented by each of the following. a. a point b. a segment c. a plane Draw each of the following. 23. plane H containing two lines that intersect at M 24. ST intersecting plane M at R Use the figure to name each of the following. 25. the intersection of TV and US 26. the intersection of 27. the intersection of US and plane R ̶̶ TU and ̶̶ UV Write the postulate that justifies each statement. 28. The line connecting two dots on a sheet of paper lies on the same sheet of paper as the dots. 29. If two ants are walking in straight lines but in different directions, their paths cannot cross more than once. 30. Critical Thinking Is it possible to draw three points that are noncoplanar? Explain. Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 31. If two planes intersect, they intersect in a straight line. 32. If two lines intersect, they intersect at two different points. 33. AB is another name for |
BA. 34. If two rays share a common endpoint, then they form a line. 35. Art Pointillism is a technique in which tiny dots of complementary colors are combined to form a picture. Which postulate ensures that a line connecting two of these points also lies in the plane containing the points? 36. Probability Three of the labeled points are chosen at random. What is the probability that they are collinear? 37. Campers often use a cooking stove with three legs. Which postulate explains why they might prefer this design to a stove that has four legs? 38. Write About It Explain why three coplanar lines may have zero, one, two, or three points of intersection. Support your answer with a sketch. 10 10 Chapter 1 Foundations for Geometry ��������� 39. Which of the following is a set of noncollinear points? P, R, T Q, R, S P, Q, R S, T, U 40. What is the greatest number of intersection points four coplanar lines can have? 6 4 2 0 41. Two flat walls meet in the corner of a classroom. Which postulate best describes this situation? Through any three noncollinear points there is exactly one plane. If two points lie in a plane, then the line containing them lies in the plane. If two lines intersect, then they intersect in exactly one point. If two planes intersect, then they intersect in exactly one line. 42. Gridded Response What is the greatest number of planes determined by four noncollinear points? CHALLENGE AND EXTEND Use the table for Exercises 43–45. Figure Number of Points Maximum Number of Segments 2 1 3 3 4 43. What is the maximum number of segments determined by 4 points? 44. Multi-Step Extend the table. What is the maximum number of segments determined by 10 points? 45. Write a formula for the maximum number of segments determined by n points. 46. Critical Thinking Explain how rescue teams could use two of the postulates from this lesson to locate a distress signal. SPIRAL REVIEW 47. The combined age of a mother and her twin daughters is 58 years. The mother was 25 years old when the twins were born. Write and solve an equation to find the age of each of the three people. (Previous course) Determine whether each set of ordered pairs is a function. (Previous course) ⎬ � |
� (0, 1), (1, -1), (5, -1), (-1, 2) 48. Find the mean, median, and mode for each set of data. (Previous course) ⎬ ⎨ (3, 8), (10, 6), (9, 8), (10, -6) 49. 50. 0, 6, 1, 3, 5, 2, 7, 10 51. 0.47, 0.44, 0.4, 0.46, 0.44 1- 1 Understanding Points, Lines, and Planes 11 11 ������ 1-2 Use with Lesson 1-2 Activity Explore Properties Associated with Points The two endpoints of a segment determine its length. Other points on the segment are between the endpoints. Only one of these points is the midpoint of the segment. In this lab, you will use geometry software to measure lengths of segments and explore properties of points on segments. TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.2.B, G.3.B KEYWORD: MG7 Lab1 1 Construct a segment and label its endpoints A and C. 2 Create point B on ̶̶ AC. 3 Measure the distances from A to B and from B to C. Use the Calculate tool to calculate the sum of AB and BC. 4 Measure the length of ̶̶ AC. What do you notice about this length compared with the measurements found in Step 3? 5 Drag point B along ̶̶ AC. Drag one of the endpoints ̶̶ AC. What relationships do you think are true of about the three measurements? 6 Construct the midpoint of ̶̶ AC and label it M. 7 Measure ̶̶̶ AM and ̶̶̶ MC. What relationships do you ̶̶̶ AM, and think are true about the lengths of Use the Calculate tool to confirm your findings. ̶̶ AC, ̶̶̶ MC? 8 How many midpoints of ̶̶ AC exist? Try This 1. Repeat the activity with a new segment. Drag each of the points in your figure (the endpoints, the point on the segment, and the midpoint). Write down any relationships you observe about the measurements. ̶̶ AC |
. Measure ̶̶ AD, What do you think has to be true about D for the relationship to always be true? ̶̶ AC. Does AD + DC = AC? 2. Create a point D not on ̶̶ DC, and 12 12 Chapter 1 Foundations for Geometry 1-2 Measuring and Constructing Segments TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.2.A, G.2.B, G.7.C Objectives Use length and midpoint of a segment. Construct midpoints and congruent segments. Why learn this? You can measure a segment to calculate the distance between two locations. Maps of a race are used to show the distance between stations on the course. (See Example 4.) Vocabulary coordinate distance length congruent segments construction between midpoint bisect segment bisector A ruler can be used to measure the distance between two points. A point corresponds to one and only one number on the ruler. This number is called a coordinate. The following postulate summarizes this concept. Postulate 1-2-1 Ruler Postulate The points on a line can be put into a one-to-one correspondence with the real numbers. The distance between any two points is the absolute value of the difference of the coordinates. If the coordinates of points A and B are a and b, then the distance between A and B is ⎜a - b⎟ or ⎜b - a⎟. The distance between A and B is also called the length of ̶̶ AB, or AB. E X A M P L E 1 Finding the Length of a Segment � � �� �� �� �� � � � � � � � � ��� B EF EF = ⎜-4 - (-1) ⎟ = ⎜-4 + 1⎟ = ⎜-3⎟ = 3 Find each length. A DC DC = ⎜4.5 - 2⎟ = ⎜2.5⎟ = 2.5 Find each length. 1a. XY 1b. XZ _ PQ PQ represents a number, while represents a geometric figure. Be sure to use equality for numbers (PQ = RS) and congruence for _ PQ ≅ figures ( _ RS ). Congruent segments are segments that have the same length. In |
the diagram, PQ = RS, so you can write is congruent to segment RS.” Tick marks are used in a figure to show congruent segments. _ RS. This is read as “segment PQ _ PQ ≅ 1- 2 Measuring and Constructing Segments 13 13 ��������������������������������������������������������������������������� You can make a sketch or measure and draw a segment. These may not be exact. A construction is a way of creating a figure that is more precise. One way to make a geometric construction is to use a compass and straightedge. Construction Congruent Segment Construct a segment congruent to _ AB. Draw ℓ. Choose a point on ℓ and label it C. Open the compass to distance AB. Place the point of the compass at C and make an arc through ℓ. Find the point where the arc and ℓ intersect and label it D. _ CD ≅ _ AB E X A M P L E 2 Copying a Segment Sketch, draw, and construct a segment congruent to _ MN. Step 1 Estimate and sketch. _ MN and sketch Estimate the length of _ PQ approximately the same length. Step 2 Measure and draw. Use a ruler to measure _ MN. MN appears _ XY to to be 3.1 cm. Use a ruler and draw have length 3.1 cm. Step 3 Construct and compare. Use a compass and straightedge to construct _ ST congruent to _ MN. _ PQ and _ XY are A ruler shows that approximately the same length as _ ST is precisely the same length. but _ MN, 2. Sketch, draw, and construct a segment congruent to _ JK. In order for you to say that a point B is between two points A and C, all three of the points must lie on the same line, and AB + BC = AC. Postulate 1-2-2 Segment Addition Postulate If B is between A and C, then AB + BC = AC. 14 14 Chapter 1 Foundations for Geometry ���������������������� E X A M P L E 3 Using the Segment Addition Postulate A B is between A and C, AC = 14, and BC = 11.4. Find AB. AC = AB + BC 14 = AB + 11.4 - 11 |
.4 ̶̶̶̶̶ - 11.4 ̶̶̶̶̶̶̶ 2.6 = AB Seg. Add. Post. Substitute 14 for AC and 11.4 for BC. Subtract 11.4 from both sides. Simplify. B S is between R and T. Find RT. - 2x ̶̶̶̶̶̶̶ RT = RS + ST 4x = (2x + 7) + 28 4x = 2x + 35 - 2x ̶̶̶̶ 2x = 35 = 35 _ 2x _ 2 2 x = 35 _ 2 RT = 4x, or 17.5 Seg. Add. Post. Substitute the given values. Simplify. Subtract 2x from both sides. Simplify. Divide both sides by 2. Simplify. = 4 (17.5) = 70 Substitute 17.5 for x. 3a. Y is between X and Z, XZ = 3, and XY = 1 1 __ 3 3b. E is between D and F. Find DF.. Find YZ. _ AB is the point that bisects, or divides, the segment into The midpoint M of two congruent segments. If M is the midpoint of AM = MB. So if AB = 6, then AM = 3 and MB = 3. _ AB, then E X A M P L E 4 Recreation Application The map shows the route for a race. You are 365 m from drink station R and 2 km from drink station S. The first-aid station is located at the midpoint of the two drink stations. How far are you from the first-aid station? Let your current location be X and the location of the first-aid station be Y. XR + RS = XS 365 + RS = 2000 - 365 ̶̶̶̶̶ RS = 1635 RY = 817.5 - 365 ̶̶̶̶̶̶̶̶ Seg. Add. Post. Substitute 365 for XR and 2000 for XS. Subtract 365 from both sides. Simplify. Y is the mdpt. of _ RS, so RY = 1 __ RS. 2 XY = XR + RY = 365 + 817.5 = 1182.5 m Substitute 365 for XR and 817.5 for RY. You are 1182.5 m from the first-aid station. 4. What is the distance |
to a drink station located at the midpoint between your current location and the first-aid station? 1- 2 Measuring and Constructing Segments 15 15 XR365 mYS2 kmKaren Minot(415)883-6560Final art file 11/18/04Marathon RouteHolt Rinehart WinstonGeometry SE 2007 Texasge07sec01l02002a�������������������������� A segment bisector is any ray, segment, or line that intersects a segment at its midpoint. It divides the segment into two equal parts at its midpoint. Construction Segment Bisector Draw _ XY on a sheet of paper. Fold the paper so that Y is on top of X. Unfold the paper. The line represented by the crease bisects _ XY. Label the midpoint M. XM = MY E X A M P L E 5 Using Midpoints to Find Lengths B is the midpoint of ̶̶ AC, AB = 5x, and BC = 3x + 4. Find AB, BC, and AC. Step 1 Solve for x. AB = BC 5x = 3x + 4 - 3x ̶̶̶̶ 2x = 4 = 4 _ 2x _ 2 2 x = 2 - 3x ̶̶̶̶̶̶ B is the mdpt. of _ AC. Substitute 5x for AB and 3x + 4 for BC. Subtract 3x from both sides. Simplify. Divide both sides by 2. Simplify. Step 2 Find AB, BC, and AC. AB = 5x = 5 (2) = 10 BC = 3x + 4 AC = AB + BC = 3 (2) + 4 = 10 = 10 + 10 = 20 5. S is the midpoint of RT, RS = -2x, and ST = -3x - 2. Find RS, ST, and RT. THINK AND DISCUSS 1. Suppose R is the midpoint of _ ST. Explain how SR and ST are related. 2. GET ORGANIZED Copy and complete the graphic organizer. Make a sketch and write an equation to describe each relationship. 16 16 Chapter 1 Foundations for Geometry �������������������������������������������������������������������� 1-2 Exercises Exercises KEYWORD: MG7 1-2 KEYWORD: MG7 Parent GUID |
ED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. _ XY into two equal parts. Name a pair of congruent _ XY at M and divides 1. Line ℓ bisects segments. 2. __?__ is the amount of space between two points on a line. It is always expressed as a nonnegative number. (distance or midpoint Find each length. p. 13 3. AB 4. BC. 14 5. Sketch, draw, and construct a segment congruent to _ RS.. B is between A and C, AC = 15.8, and AB = 9.9. Find BC. p. 15 7. Find MP. 15 8. Travel If a picnic area is located at the midpoint between Lubbock and Amarillo, find the distance to the picnic area from the sign. 16 9. Multi-Step K is the midpoint of and JK = 7. Find x, KL, and JL. _ JL, JL = 4x - 2, 10. E bisects _ DF, DE = 2y, and EF = 8y - 3. Find DE, EF, and DF. Independent Practice For See Exercises Example 11–12 13 14–15 16 17–18 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S4 Application Practice p. S28 PRACTICE AND PROBLEM SOLVING Find each length. 11. DB 12. CD 13. Sketch, draw, and construct a segment twice the length of _ AB. 14. D is between C and E, CE = 17.1, and DE = 8. Find CD. 15. Find MN. 16. Sports During a football game, a quarterback standing at the 9-yard line passes the ball to a receiver at the 24-yard line. The receiver then runs with the ball halfway to the 50-yard line. How many total yards (passing plus running) did the team gain on the play? 17. Multi-Step E is the midpoint of _ DF, DE = 2x + 4, and EF = 3x - 1. Find DE, EF, and DF. _ PR, PQ = 3y, and PR = 42. Find y and QR. 18. Q bisects 1- 2 Measuring and Constructing Segments 17 17 �������������������������������������������������������������������� |
������������������������������������������������������������������������������������������ 19. This problem will prepare you for the Multi-Step TAKS Prep on page 34. Archaeologists at Valley Forge were eager to find what remained of the winter camp that soldiers led by George Washington called home for several months. The diagram represents one of the restored log cabins. a. How is C related to b. If AC = 7 ft, EF = 2 (AC) + 2, and AB = 2 (EF) - 16, _ AE? what are AB and EF? Use the diagram for Exercises 20–23. 20. GD = 4 2 _ 3 _ _ CD ≅ DF, E bisects. Find GH. 21. _ DF, and CD = 14.2. Find EF. 22. GH = 4x - 1, and DH = 8. Find x. _ GH bisects _ CF, CF = 2y - 2, and CD = 3y - 11. Find CD. 23. Tell whether each statement is sometimes, always, or never true. Support each of your answers with a sketch. 24. Two segments that have the same length must be congruent. 25. If M is between A and B, then M bisects _ AB. 26. If Y is between X and Z, then X, Y, and Z are collinear. 27. /////ERROR ANALYSIS///// Below are two statements about the midpoint of _ AB. Which is incorrect? Explain the error. 28. Carpentry A carpenter has a wooden dowel that is 72 cm long. She wants to cut it into two pieces so that one piece is 5 times as long as the other. What are the lengths of the two pieces? 29. The coordinate of M is 2.5, and MN = 4. What are the possible coordinates for N? 30. Draw three collinear points where E is between D and F. Then write an equation using these points and the Segment Addition Postulate. Suppose S is between R and T. Use the Segment Addition Postulate to solve for each variable. 31. RS = 7y - 4 ST = y + 5 RT = 28 32. RS = 3x + 1 ST = 1 _ 2 RT = 18 x + 3 33. RS = 2z + 6 ST = 4z - 3 RT = 5z + 12 34. Write |
About It In the diagram, B is not between A and C. Explain. 35. Construction Use a compass and straightedge to construct a segment whose length is AB + CD. 18 18 Chapter 1 Foundations for Geometry ����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� 36. Q is between P and R. S is between Q and R, and R is between Q and T. PT = 34, QR = 8, and PQ = SQ = SR. What is the length of 18 10 9 _ RT? 37. C is the midpoint of What is the length of _ AD? _ AD. B is the midpoint of _ AC. BC = 12. 12 24 36 38. Which expression correctly states that XY ≅ VW _ XY ≅ _ VW _ XY is congruent to _ VW _ XY = _ VW? 22 48 XY = VW 39. A, B, C, D, and E are collinear points. AE = 34, BD = 16, and AB = BC = CD. What is the length of _ CE? 10 16 18 24 CHALLENGE AND EXTEND 40. 40. HJ is twice JK. J is between H and K. If HJ = 4x and HK = 78, find JK. 41. 41. A, D, N, and X are collinear points. D is between N and A. NA + AX = NX. Draw a diagram that represents this information. Sports Use the following information for Exercises 42 and 43. The table shows regulation distances between hurdles in women’s and men’s races. In both the women’s and men’s events, the race consists of a straight track with 10 equally spaced hurdles. Distance of Race 100 m 110 m Distance from Start to First Hurdle 13.00 m 13.72 m Distance Between Hurdles 8.50 m 9.14 m Event Women’s Men’s Distance from Last Hurdle to Finish 42. Find the distance from the last hurdle to the finish line for the women’s race. 43. Find the distance from the last hurdle to the finish line for the men’s race. 44. Critical Thinking Given that J, K, and L are collinear and that K is between |
J and L, is it possible that JK = JL? If so, draw an example. If not, explain. Sports Joanna Hayes, of the United States, clears a hurdle on her way to winning the gold medal in the women’s 100 m hurdles during the 2004 Olympic Games. SPIRAL REVIEW Evaluate each expression. (Previous course) 45. ⎜20 - 8⎟ 46. ⎜-9 + 23⎟ 47. - ⎜4 - 27⎟ Simplify each expression. (Previous course) 48. 8a - 3 (4 + a) - 10 49. x + 2 (5 - 2x) - (4 + 5x) Use the figure to name each of the following. (Lesson 1-1) 50. two lines that contain B 51. two segments containing D 52. three collinear points 53. a ray with endpoint C 1- 2 Measuring and Constructing Segments 19 19 ������������� 1-3 Measuring and Constructing Angles TEKS G.3.B Geometric structure: construct and justify statements about geometric figures and their properties. Also G.1.A, G.1.B, G.2.A, G.2.B Objectives Name and classify angles. Measure and construct angles and angle bisectors. Who uses this? Surveyors use angles to help them measure and map the earth’s surface. (See Exercise 27.) Vocabulary angle vertex interior of an angle exterior of an angle measure degree acute angle right angle obtuse angle straight angle congruent angles angle bisector A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a surveyor can measure the angle formed by his or her location and two distant points. An angle is a figure formed by two rays, or sides, with a common endpoint called the vertex (plural: vertices). You can name an angle several ways: by its vertex, by a point on each ray and the vertex, or by a number. The set of all points between the sides of the angle is the interior of an angle. The exterior of an angle is the set of all points outside the angle. Angle Name ∠R, ∠SRT, ∠TRS, or ∠1 You cannot name an angle just by its vertex if the point is the vertex of more than one |
angle. In this case, you must use all three points to name the angle, and the middle point is always the vertex. E X A M P L E 1 Naming Angles A surveyor recorded the angles formed by a transit (point T) and three distant points, Q, R, and S. Name three of the angles. ∠QTR, ∠QTS, and ∠RTS 1. Write the different ways you can name the angles in the diagram. The measure of an angle is usually given in degrees. Since there are 360° in a circle, one degree is 1 ___ 360 of a circle. When you use a protractor to measure angles, you are applying the following postulate. Postulate 1-3-1 ___ ___ ‹ › ‹ › Given AB, all rays that can be drawn from O can be put into AB and a point O on a one-to-one correspondence with the real numbers from 0 to 180. Protractor Postulate 20 20 Chapter 1 Foundations for Geometry �������������������������� Using a Protractor Most protractors have two sets of numbers around the edge. When I measure an angle and need to know which number to use, I first ask myself whether the angle is acute, right, or obtuse. For example, ∠RST looks like it is obtuse, so I know its measure must be 110°, not 70°. � � � José Muñoz Lincoln High School You can use the Protractor Postulate to help you classify angles by their measure. The measure of an angle is the absolute value of the difference of the real numbers that the rays correspond OC corresponds with on a protractor. If with c and m∠DOC = ⎜d - c⎟ or ⎜c - d⎟. OD corresponds with d, Types of Angles Acute Angle Right Angle Obtuse Angle Straight Angle Measures greater than 0° and less than 90° Measures 90° Measures greater than 90° and less than 180° Formed by two opposite rays and meaures 180° E X A M P L E 2 Measuring and Classifying Angles Find the measure of each angle. Then classify each as acute, right, or obtuse. A ∠AOD m∠AOD = 165° ∠AOD is obtuse. B |
∠COD m∠COD = ⎜165 - 75⎟ = 90° ∠COD is a right angle. Use the diagram to find the measure of each angle. Then classify each as acute, right, or obtuse. 2a. ∠BOA 2b. ∠DOB 2c. ∠EOC 1- 3 Measuring and Constructing Angles 21 21 ������������� Congruent angles are angles that have the same measure. In the diagram, m∠ABC = m∠DEF, so you can write ∠ABC ≅ ∠DEF. This is read as “angle ABC is congruent to angle DEF.” Arc marks are used to show that the two angles are congruent. Construction Congruent Angle Construct an angle congruent to ∠A. Use a straightedge to draw a ray with endpoint D. Place the compass point at A and draw an arc that intersects both sides of ∠A. Label the intersection points B and C. Using the same compass setting, place the compass point at D and draw an arc that intersects the ray. Label the intersection E. Place the compass point at B and open it to the distance BC. Place the point of the compass at E and draw an arc. Label its intersection with the first arc F. Use a straightedge to DF. draw ∠D ≅ ∠A The Angle Addition Postulate is very similar to the Segment Addition Postulate that you learned in the previous lesson. Postulate 1-3-2 Angle Addition Postulate If S is in the interior of ∠PQR, then m∠PQS + m∠SQR = m∠PQR. (∠ Add. Post.) E X A M P L E 3 Using the Angle Addition Postulate m∠ABD = 37° and m∠ABC = 84°. Find m∠DBC. m∠ABC = m∠ABD + m∠DBC ∠ Add. Post. 84° = 37° + m∠DBC - 37 ̶̶̶̶ - 37 ̶̶̶̶̶̶̶̶̶̶̶ 47° = m∠DBC Substitute the given values. Subtract |
37 from both sides. Simplify. 3. m∠XWZ= 121° and m∠XWY = 59°. Find m∠YWZ. 22 22 Chapter 1 Foundations for Geometry ��������������������������������������� An angle bisector is a ray that divides an angle into two JK bisects ∠LJM; thus ∠LJK ≅ ∠KJM. congruent angles. Construction Angle Bisector Construct the bisector of ∠A. Use a straightedge to draw ___ › AD bisects ∠A. ___ › AD. Place the point of the compass at A and draw an arc. Label its points of intersection with ∠A as B and C. Without changing the compass setting, draw intersecting arcs from B and C. Label the intersection of the arcs as D. E X A M P L E 4 Finding the Measure of an Angle BD bisects ∠ABC, m∠ABD = (6x + 3) °, and m∠DBC = (8x - 7) °. Find m∠ABD. Step 1 Find x. m∠ABD = m∠DBC (6x + 3) ° = (8x - 7) ° + 7 + 7 ̶̶̶̶̶̶ ̶̶̶̶̶̶̶ 6x + 10 = 8x - 6x ̶̶̶̶̶̶̶ 10 = 2x = 2x _ 10 _ 2 2 5 = x - 6x ̶̶̶̶̶̶ Step 2 Find m∠ABD. m∠ABD = 6x + 3 Def. of ∠ bisector Substitute the given values. Add 7 to both sides. Simplify. Subtract 6x from both sides. Simplify. Divide both sides by 2. Simplify. = 6 (5) + 3 = 33° Substitute 5 for x. Simplify. Find the measure of each angle. 4a. 4b. ___ › QS bisects ∠PQR, m∠PQS = (5y - 1) °, and m∠PQR = (8y + 12) °. Find m∠PQS. __ › JK bisect |
s ∠LJM, m∠LJK = (-10x + 3) °, and m∠KJM = (-x + 21) °. Find m∠LJM. 1- 3 Measuring and Constructing Angles 23 23 �������������������� THINK AND DISCUSS 1. Explain why any two right angles are congruent. ___ › BD bisects ∠ABC. How are m∠ABC, m∠ABD, and m∠DBC related? 2. 3. GET ORGANIZED Copy and complete the graphic organizer. In the cells sketch, measure, and name an example of each angle type. 1-3 Exercises Exercises KEYWORD: MG7 1-3 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. ∠A is an acute angle. ∠O is an obtuse angle. ∠R is a right angle. Put ∠A, ∠O, and ∠R in order from least to greatest by measure. 2. Which point is the vertex of ∠BCD? Which rays form � the sides of ∠BCD. 20 3. Music Musicians use a metronome to keep time as they play. The metronome’s needle swings back and forth in a fixed amount of time. Name all of the angles in the diagram. 21 Use the protractor to find the measure of each angle. Then classify each as acute, right, or obtuse. � � � � � 4. ∠VXW 5. ∠TXW 6. ∠RXU. 22 L is in the interior of ∠JKM. Find each of the following. 7. m∠JKM if m∠JKL = 42° and m∠LKM = 28° 8. m∠LKM if m∠JKL = 56.4° and m∠JKM = 82.5. 23 Multi-Step BD bisects ∠ABC. Find each of the following. 9. m∠ABD if m∠ABD = (6x + 4) ° and m∠DBC = (8x - 4) ° 10. m∠ABC if m∠ABD = |
(5y - 3) ° and m∠DBC = (3y + 15) ° 24 24 Chapter 1 Foundations for Geometry ������������������������������������������������������������������������� Independent Practice For See Exercises Example 11 12–14 15–16 17–18 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S4 Application Practice p. S28 PRACTICE AND PROBLEM SOLVING 11. Physics Pendulum clocks have been used since 1656 to keep time. The pendulum swings back and forth once or twice per second. Name all of the angles in the diagram. � � � � � � Use the protractor to find the measure of each angle. Then classify each as acute, right, or obtuse. 12. ∠CGE 13. ∠BGD 14. ∠AGB T is in the interior of ∠RSU. Find each of the following. 15. m∠RSU if m∠RST = 38° and m∠TSU = 28.6° 16. m∠RST if m∠TSU = 46.7° and m∠RSU = 83.5° SP bisects ∠RST. Find each of the following. Multi-Step 17. m∠RST if m∠RSP= (3x - 2) ° and m∠PST = (9x - 26) ° 18. m∠RSP if m∠RST = 5 __ 2 y ° and m∠PST = (y + 5) ° Estimation Use the following information for Exercises 19–22. Assume the corner of a sheet of paper is a right angle. Use the corner to estimate the measure and classify each angle in the diagram. 19. ∠BOA 21. ∠EOD 20. ∠COA 22. ∠EOB Use a protractor to draw an angle with each of the following measures. 23. 33° 24. 142° 25. 90° 26. 168° 27. Surveying A surveyor at point S discovers that the angle between peaks A and B is 3 times as large as the angle between peaks B and C. The surveyor knows that ∠ASC is a right angle. Find m∠ASB and m∠ |
BSC. 28. Math History As far back as the 5th century B.C., mathematicians have been fascinated by the problem of trisecting an angle. It is possible to construct an angle with 1 __ 4 the measure of a given angle. Explain how to do this. Find the value of x. 29. m∠AOC = 7x - 2, m∠DOC = 2x + 8, m∠EOD = 27 30. m∠AOB = 4x - 2, m∠BOC = 5x + 10, m∠COD = 3x - 8 31. m∠AOB = 6x + 5, m∠BOC = 4x - 2, m∠AOC = 8x + 21 32. Multi-Step Q is in the interior of right ∠PRS. If m∠PRQ is 4 times as large as m∠QRS, what is m∠PRQ? 1- 3 Measuring and Constructing Angles 25 25 ��������������������������������������� 33. This problem will prepare you for the Multi-Step TAKS Prep on page 34. An archaeologist standing at O looks for clues on where to dig for artifacts. a. What value of x will make the angle between the pottery and the arrowhead measure 57°? b. What value of x makes ∠LOJ ≅ ∠JOK? c. What values of x make ∠LOK an acute angle? L O 3xº J (2x + 12)º K Data Analysis Use the circle graph for Exercises 34–36. 34. Find m∠AOB, m∠BOC, m∠COD, and m∠DOA. Classify each angle as acute, right, or obtuse. 35. What if...? Next year, the music store will use some of the shelves currently holding jazz music to double the space for rap. What will m∠COD and m∠BOC be next year? 36. Suppose a fifth type of music, salsa, is added. If the space is divided equally among the five types, what will be the angle measure for each type of music in the circle graph? 37. Critical Thinking Can an obtuse angle be congruent to an acute angle? Why or why not? 38. The measure of an obtuse angle is (5 |
x + 45) °. What is the largest value for x? ___ › FH bisects ∠EFG. Use the Angle Addition Postulate to explain 39. Write About It why m∠EFH = 1 __ 2 m∠EFG. 40. Multi-Step Use a protractor to draw a 70° angle. Then use a compass and straightedge to bisect the angle. What do you think will be the measure of each angle formed? Use a protractor to support your answer. 41. m∠UOW = 50°, and What is m∠VOY? ___ › OV bisects ∠UOW. 25° 65° 130° 155° 42. What is m∠UOX? 50° 115° 140° 165° 43. ___ › BD bisects ∠ABC, m∠ ABC = (4x + 5) °, and m∠ ABD = (3x - 1) °. What is the value of x? 2.2 3 3.5 7 44. If an angle is bisected and then 30° is added to the measure of the bisected angle, the result is the measure of a right angle. What is the measure of the original angle? 30° 60° 75° 120° 45. Short Response If an obtuse angle is bisected, are the resulting angles acute or obtuse? Explain. 26 26 Chapter 1 Foundations for Geometry ��������������������������������������������������������������� CHALLENGE AND EXTEND 46. Find the measure of the angle formed by the hands of a clock when it is 7:00. 47. ___ › QS bisects ∠PQR, m∠PQR = (x2) °, and m∠PQS = (2x + 6) °. Find all the possible measures for ∠PQR. 48. For more precise measurements, a degree can be divided into 60 minutes, and each minute can be divided into 60 seconds. An angle measure of 42 degrees, 30 minutes, and 10 seconds is written as 42°30′10″. Subtract this angle measure from the measure 81°24′15″. 49. If 1 degree equals 60 minutes and 1 minute equals 60 seconds, how many seconds are in 2.25 degrees? 50. ∠ABC ≅ ∠DBC. m∠ABC |
= ( 3x __ 2 + 4) ° and m∠DBC = (2x - 27 1 __ 4 ) °. Is ∠ABD a straight angle? Explain. SPIRAL REVIEW 51. What number is 64% of 35? 52. What percent of 280 is 33.6? (Previous course) Sketch a figure that shows each of the following. (Lesson 1-1) 53. a line that contains _ AB and ___ › CB 54. two different lines that intersect _ MN 55. a plane and a ray that intersect only at Q Find the length of each segment. (Lesson 1-2) _ KL 57. _ JL 58. _ JK 56. Using Technology Segment and Angle Bisectors 1. Construct the bisector _ MN. of 2. Construct the bisector of ∠BAC. a. Draw _ MN and construct the midpoint B. a. Draw ∠BAC. b. Construct a point A not on the segment. ___ _ › ‹ c. Construct bisector MB AB and measure _ NB. and d. Drag M and N and observe MB and NB. b. Construct the angle bisector ___ › AD and measure ∠DAC and ∠DAB. c. Drag the angle and observe m∠DAB and m∠DAC. 1- 3 Measuring and Constructing Angles 27 27 ����������������� 1-4 Pairs of Angles TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates.... Also G.2.B Objectives Identify adjacent, vertical, complementary, and supplementary angles. Find measures of pairs of angles. Vocabulary adjacent angles linear pair complementary angles supplementary angles vertical angles Who uses this? Scientists use properties of angle pairs to design fiber-optic cables. (See Example 4.) A fiber-optic cable is a strand of glass as thin as a human hair. Data can be transmitted over long distances by bouncing light off the inner walls of the cable. Many pairs of angles have special relationships. Some relationships are because of the measurements of the angles in the pair. Other relationships are because of the positions of the angles in the pair. Pairs of Angles Adjacent angles are two angles in the same plane with a common vertex and a common side, but no common interior points. ∠1 and ∠2 are adjacent angles. A linear pair |
of angles is a pair of adjacent angles whose noncommon sides are opposite rays. ∠3 and ∠4 form a linear pair. E X A M P L E 1 Identifying Angle Pairs Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. A ∠1 and ∠2 ∠1 and ∠2 have a common vertex, B, a common BC, and no common interior points. side, Therefore ∠1 and ∠2 are only adjacent angles. B ∠2 and ∠4 ∠2 and ∠4 share vertex, so ∠2 and ∠4 are not adjacent angles. ̶̶ BC but do not have a common C ∠1 and ∠3 ∠1 and ∠3 are adjacent angles. Their noncommon sides, are opposite rays, so ∠1 and ∠3 also form a linear pair. BC and BA, Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 1a. ∠5 and ∠6 1b. ∠7 and ∠SPU 1c. ∠7 and ∠8 28 28 Chapter 1 Foundations for Geometry ����������������������� Complementary and Supplementary Angles Complementary angles are two angles whose measures have a sum of 90°. ∠A and ∠B are complementary. Supplementary angles are two angles whose measures have a sum of 180°. ∠A and ∠C are supplementary. You can find the complement of an angle that measures x° by subtracting its measure from 90°, or (90 - x)°. You can find the supplement of an angle that measures x° by subtracting its measure from 180°, or (180 - x)°. E X A M P L E 2 Finding the Measures of Complements and Supplements Find the measure of each of the following. A complement of ∠M (90 - x)° 90° - 26.8° = 63.2° B supplement of ∠N (180 - x)° 180° - (2y + 20)° = 180° - 2y - 20 = (160 - 2y)° Find the measure of each of the following. 2a. complement of ∠E 2b. supplement of |
∠F E X A M P L E 3 Using Complements and Supplements to Solve Problems An angle measures 3 degrees less than twice the measure of its complement. Find the measure of its complement. Step 1 Let m∠A = x°. Then ∠B, its complement, measures (90 - x)°. Step 2 Write and solve an equation. m∠A = 2m∠B - 3 x = 2 (90 - x) - 3 x = 180 - 2x -3 x = 177 - 2x + 2x ̶ + 2x ̶ 3x = 177 = 177_ 3x_ 3 3 x = 59 Substitute x for m∠A and 90 - x for m∠B. Distrib. Prop. Combine like terms. Add 2x to both sides. Simplify. Divide both sides by 3. Simplify. The measure of the complement, ∠B, is (90 - 59)° = 31°. 3. An angle’s measure is 12° more than 1__ 2 the measure of its supplement. Find the measure of the angle. 1- 4 Pairs of Angles 29 29 ������������������������������������������������ E X A M P L E 4 Problem-Solving Application Light passing through a fiber optic cable reflects off the walls in such a way that ∠1 ≅ ∠2. ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. If m∠1 = 38°, find m∠2, m∠3, and m∠4. Understand the Problem The answers are the measures of ∠2, ∠3, and ∠4. List the important information: • ∠1 ≅ ∠2 • ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. • m∠1 = 38° Make a Plan If ∠1 ≅ ∠2, then m∠1 = m∠2. If ∠3 and ∠1 are complementary, then m∠3 = (90 - 38) °. If ∠4 and ∠2 are complementary, then m∠4 = (90 - 38) °. Solve By the Transitive Property of Equality, if m∠1 = 38° and m∠1 = m∠2, then m� |
�2 = 38°. Since ∠3 and ∠1 are complementary, m∠3 = 52°. Similarly, since ∠2 and ∠4 are complementary, m∠4 = 52°. Look Back The answer makes sense because 38° + 52° = 90°, so ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. Thus m∠2 = 38°, m∠3 = 52°, and m∠4 = 52°. 4. What if...? Suppose m∠3 = 27.6°. Find m∠1, m∠2, and m∠4. Another angle pair relationship exists between two angles whose sides form two pairs of opposite rays. Vertical angles are two nonadjacent angles formed by two intersecting lines. ∠1 and ∠3 are vertical angles, as are ∠2 and ∠4. E X A M P L E 5 Identifying Vertical Angles Name one pair of vertical angles. Do they appear to have the same measure? Check by measuring with a protractor. ∠EDF and ∠GDH are vertical angles and appear to have the same measure. Check m∠EDF ≈ m∠GDH ≈ 135°. 5. Name another pair of vertical angles. Do they appear to have the same measure? Check by measuring with a protractor. 30 30 Chapter 1 Foundations for Geometry 4321Light1234��������� THINK AND DISCUSS 1. Explain why any two right angles are supplementary. 2. Is it possible for a pair of vertical angles to also be adjacent? Explain. 3. GET ORGANIZED Copy and complete the graphic organizer below. In each box, draw a diagram and write a definition of the given angle pair. 1-4 Exercises Exercises KEYWORD: MG7 1-4 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An angle measures x°. What is the measure of its complement? What is the measure of its supplement? 2. ∠ABC and ∠CBD are adjacent angles. Which side do the angles have in common. 28 Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 3. ∠1 and ∠2 4. ∠1 and ∠3 5 |
. ∠2 and ∠4 6. ∠2 and ∠ Find the measure of each of the following. p. 29 7. supplement of ∠A 8. complement of ∠A 9. supplement of ∠B 10. complement of ∠. 29 11. Multi-Step An angle’s measure is 6 degrees more than 3 times the measure of its complement. Find the measure of the angle. 30 12. Landscaping A sprinkler swings back and forth between A and B in such a way that ∠1 ≅ ∠2. ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. If m∠1 = 18.5°, find m∠2, m∠3, and m∠4 13. Name each pair of vertical angles. p. 30 1- 4 Pairs of Angles 31 31 ��������������������������������������������������������������������������������������������������������������������������������� Independent Practice For See Exercises Example 14–17 18–21 22 23 24 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S4 Application Practice p. S28 PRACTICE AND PROBLEM SOLVING Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 14. ∠1 and ∠4 15. ∠2 and ∠3 16. ∠3 and ∠4 17. ∠3 and ∠1 Given m∠A = 56.4° and m∠B = (2x - 4)°, find the measure of each of the following. 18. supplement of ∠A 20. supplement of ∠B 19. complement of ∠A 21. complement of ∠B 22. Multi-Step An angle’s measure is 3 times the measure of its complement. Find the measure of the angle and the measure of its complement. 23. Art In the stained glass pattern, ∠1 ≅ ∠2. ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. If m∠1 = 22.3°, find m∠2, m∠3, and m∠4. 24. Name the pairs of vertical angles. 25. Probability The angle measures 30°, |
60°, 120°, and 150° are written on slips of paper. You choose two slips of paper at random. What is the probability that the angle measures are supplementary? Multi-Step ∠ABD and ∠BDE are supplementary. Find the measures of both angles. 26. m∠ABD = 5x°, m∠BDE = (17x - 18) ° 27. m∠ABD = (3x + 12) °, m∠BDE = (7x - 32) ° 28. m∠ABD = (12x - 12) °, m∠BDE = (3x + 48) ° Multi-Step ∠ABD and ∠BDC are complementary. Find the measures of both angles. 29. m∠ABD = (5y + 1) °, m∠BDC = (3y - 7) ° 30. m∠ABD = (4y + 5) °, m∠BDC = (4y + 8) ° 31. m∠ABD = (y - 30) °, m∠BDC = 2y° 32. Critical Thinking Explain why an angle that is supplementary to an acute angle must be an obtuse angle. 33. This problem will prepare you for the Multi-Step TAKS Prep on page 34. H is in the interior of ∠JAK. m∠JAH = (3x - 8) °, and m∠KAH = (x + 2) °. Draw a picture of each relationship. Then find the measure of each angle. a. ∠JAH and ∠KAH are complementary angles. b. ∠JAH and ∠KAH form a linear pair. c. ∠JAH and ∠KAH are congruent angles. 32 32 Chapter 1 Foundations for Geometry ����ge07sec01l0400a1234������� Determine whether each statement is true or false. If false, explain why. 34. If an angle is acute, then its complement must be greater than its supplement. 35. A pair of vertical angles may also form a linear pair. 36. If two angles are supplementary and congruent, the measure of each angle is 90°. 37. If a ray divides an angle into two complementary angles, then the original angle is a right angle. 38. Write About |
It Describe a situation in which two angles are both congruent and complementary. Explain. 39. What is the value of x in the diagram? 15 30 45 90 40. The ratio of the measures of two complementary angles is 1 : 2. What is the measure of the larger angle? (Hint: Let x and 2x represent the angle measures.) 30° 45° 60° 120° 41. m∠A = 3y, and m∠B = 2m∠A. Which value of y makes ∠A supplementary to ∠B? 10 18 20 36 42. The measures of two supplementary angles are in the ratio 7 : 5. Which value is the measure of the smaller angle? (Hint: Let 7x and 5x represent the angle measures.) 37.5 52.5 75 105 CHALLENGE AND EXTEND 43. How many pairs of vertical angles are in the diagram? 44. The supplement of an angle is 4 more than twice its complement. Find the measure of the angle. 45. An angle’s measure is twice the measure of its complement. The larger angle is how many degrees greater than the smaller angle? 46. The supplement of an angle is 36° less than twice the supplement of the complement of the angle. Find the measure of the supplement. SPIRAL REVIEW Solve each equation. Check your answer. (Previous course) 47. 4x + 10 = 42 49. 2 (y + 3) = 12 48. 5m - 9 = m + 4 50. - (d + 4) = 18 Y is between X and Z, XY = 3x + 1, YZ = 2x - 2, and XZ = 84. Find each of the following. (Lesson 1-2) 51. x 52. XY 53. YZ XY bisects ∠WYZ. Given m∠WYX = 26°, find each of the following. (Lesson 1-3) 54. m∠XYZ 55. m∠WYZ 1- 4 Pairs of Angles 33 33 ��������� SECTION 1A Euclidean and Construction Tools Can You Dig It? A group of college and high school students participated in an archaeological dig. The team discovered four fossils. To organize their search, Sierra used a protractor and ruler to make a diagram of where different members of the group found fossils. She drew the |
locations based on the location of the campsite. The campsite is located at X on XB. The four fossils were found at R, T, W, and M. 1. Are the locations of the campsite at X and the fossils at R and T collinear or noncollinear? 2. How is X related to ̶̶ RT? If RX = 10x - 6 and XT = 3x + 8, what is the distance between the locations of the fossils at R and T? 3. ∠RXB and ∠BXT are right angles. � Find the measure of each angle formed by the locations of the fossils and the campsite. Then classify each angle by its measure. 4. Identify the special angle pairs shown in the diagram of the archaeological dig. � � �� � � � � � � � ����������� ������������� ����������������������� ��������������������� ��������������������� �������������������� ���������������� 34 34 Chapter 1 Foundations for Geometry SECTION 1A Quiz for Lessons 1-1 Through 1-4 1-1 Understanding Points, Lines, and Planes Draw and label each of the following. 1. a segment with endpoints X and Y 2. a ray with endpoint M that passes through P 3. three coplanar lines intersecting at a point 4. two points and a line that lie in a plane Use the figure to name each of the following. 5. three coplanar points 6. two lines 7. a plane containing T, V, and X 8. a line containing V and Z 1-2 Measuring and Constructing Segments Find the length of each segment. ̶̶ SV 9. 10. ̶̶ TR 11. ̶̶ ST 12. The diagram represents a straight highway with three towns, Henri, Joaquin, and Kenard. Find the distance from Henri H to Joaquin J. 13. Sketch, draw, and construct a segment congruent to ̶̶ CD. 14. Q is the midpoint of ̶̶ PR, PQ = 2z, and PR = 8z - 12. Find z, PQ, and PR. 1-3 Measuring and Constructing Angles 15. Name all the angles in the diagram. Classify each angle by its measure. 16. m∠PVQ = 21° 19. RS |
bisects ∠QRT, m∠QRS = (3x + 8) °, and m∠SRT = (9x - 4) °. Find m∠SRT. 20. Use a protractor and straightedge to draw a 130° angle. Then bisect the angle. 18. m∠PVS = 143° 17. m∠RVT = 96° 1-4 Pairs of Angles Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 21. ∠1 and ∠2 22. ∠4 and ∠5 23. ∠3 and ∠4 If m∠T = (5x - 10) °, find the measure of each of the following. 24. supplement of ∠T 25. complement of ∠T Ready to Go On? 35 35 ��������������������������������������������������������� 1-5 Using Formulas in Geometry TEKS G.8.A Congruence and the geometry of size: find areas of regular polygons, circles.... Also G.1.A, G.1.B Objective Apply formulas for perimeter, area, and circumference. Why learn this? Puzzles use geometric-shaped pieces. Formulas help determine the amount of materials needed. (See Exercise 6.) Vocabulary perimeter area base height diameter radius circumference pi The perimeter P of a plane figure is the sum of the side lengths of the figure. The area A of a plane figure is the number of nonoverlapping square units of a given size that exactly cover the figure. Perimeter and Area RECTANGLE SQUARE TRIANGLE P = 2ℓ + 2w or 2 (ℓ + w) A = ℓw P = 4s bh or bh ___ A = 1 __ 2 2 The base b can be any side of a triangle. The height h is a segment from a vertex that forms a right angle with a line containing the base. The height may be a side of the triangle or in the interior or the exterior of the triangle. E X A M P L E 1 Finding Perimeter and Area Find the perimeter and area of each figure. A rectangle in which ℓ = 17 cm B triangle in which a = 8, Perimeter is expressed in linear units, such as inches (in.) or meters (m). Area is expressed in square units, such as square centimeters |
( cm 2 ). and w = 5 cm P = 2ℓ + 2w = 2 (17) + 2 (5) = 34 + 10 = 44 cm A = ℓw = (17) (5) = 85 cm 2 b = (x + 1), c = 4x, and x + 1) + 4x = 5x + x + 1) (6) = 3x + 3 2 bh 1. Find the perimeter and area of a square with s = 3.5 in. 36 36 Chapter 1 Foundations for Geometry Project TitleGeometry 2007 Student EditionSpec Numberge07sec01l05002aCreated ByKrosscore Corporation��������������� E X A M P L E 2 Crafts Application The Texas Treasures quilt block includes 24 purple triangles. The base and height of each triangle are about 3 in. Find the approximate amount of fabric used to make the 24 triangles. The area of one triangle is (3)(3) = 4 1_ bh = 1_ in 2. 2 2 A = 1_ 2 The total area of the 24 triangles is 24(4 1_ 2) = 108 in 2. 2. Find the amount of fabric used to make the four rectangles. Each rectangle has a length of 6 1__ 2 in. and a width of 2 1__ 2 in. In a circle a diameter is a segment that passes through the center of the circle and whose endpoints are on the circle. A radius of a circle is a segment whose endpoints are the center of the circle and a point on the circle. The circumference of a circle is the distance around the circle. Circumference and Area of a Circle The circumference C of a circle is given by the formula C = πd or C = 2πr. The area A of a circle is given by the formula A = π r 2. The ratio of a circle’s circumference to its diameter is the same for all circles. This ratio is represented by the Greek letter π (pi). The value of π is irrational. Pi is often approximated as 3.14 or 22 __ Finding the Circumference and Area of a Circle Find the circumference and area of the circle. C = 2πr = 2π (3) = 6π ≈ 18.8 cm A = π r 2 = π (3) 2 = 9π ≈ 28.3 cm 2 3. Find the circumference |
and area of a circle with radius 14 m. THINK AND DISCUSS 1. Describe three different figures whose areas are each 16 in 2. 2. GET ORGANIZED Copy and complete the graphic organizer. In each shape, write the formula for its area and perimeter. 1- 5 Using Formulas in Geometry 37 37 �������������������������������������������������������������������������� 1-5 Exercises Exercises KEYWORD: MG7 1-5 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Explain how the concepts of perimeter and circumference are related. 2. For a rectangle, length and width are sometimes used in place of __?__. (base and height or radius and diameter Find the perimeter and area of each figure. p. 36 3. 4. 5. 37 6. Manufacturing A puzzle contains a triangular piece with a base of 3 in. and a height of 4 in. A manufacturer wants to make 80 puzzles. Find the amount of wood used if each puzzle contains 20 triangular pieces. 37 Find the circumference and area of each circle. Use the π key on your calculator. Round to the nearest tenth. 7. 8. 9. PRACTICE AND PROBLEM SOLVING Find the perimeter and area of each figure. 10. 11. 12. Independent Practice For See Exercises Example 10–12 13 14–16 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S5 Application Practice p. S28 13. Crafts The quilt pattern includes 32 small triangles. Each has a base of 3 in. and a height of 1.5 in. Find the amount of fabric used to make the 32 triangles. Find the circumference and area of each circle with the given radius or diameter. Use the π key on your calculator. Round to the nearest tenth. 14. r = 12 m 15. d = 12.5 ft 16. d = 1 _ 2 mi Find the area of each of the following. 17. square whose sides are 9.1 yd in length 18. square whose sides are (x + 1) in length 19. triangle whose base is 5 1 __ 2 in. and whose height is 2 1 __ 4 in. 38 38 Chapter 1 Foundations for Geometry ��������������������������������������������������������������� Given the area |
of each of the following figures, find each unknown measure. 20. The area of a triangle is 6.75 m 2. If the base of the triangle is 3 m, what is the height of the triangle? 21. A rectangle has an area of 347.13 cm 2. If the length is 20.3 cm, what is the width of the rectangle? 22. The area of a circle is 64π. Find the radius of the circle. 23. /////ERROR ANALYSIS///// Below are two statements about the area of the circle. Which is incorrect? Explain the error. r Equator Find the area of each circle. Leave answers in terms of π. 24. circle with a diameter of 28 m 25. circle with a radius of 3y 26. Geography The radius r of the earth at the equator is approximately 3964 mi. Find the distance around the earth at the equator. Use the π key on your calculator and round to the nearest mile. 27. Critical Thinking Explain how the formulas for the perimeter and area of a square may be derived from the corresponding formulas for a rectangle. 28. Find the perimeter and area of a rectangle whose length is (x + 1) and whose width is (x - 3). Express your answer in terms of x. 29. Multi-Step If the height h of a triangle is 3 inches less than the length of the base b, and the area A of the triangle is 19 times the length of the base, find b and h. 30. This problem will prepare you for the Multi-Step TAKS Prep on page 58. A landscaper is to install edging around a garden. The edging costs $1.39 for each 24-inch-long strip. The landscaper estimates it will take 4 hours to install the edging. a. If the total cost is $120.30, what is the cost of the material purchased? b. What is the charge for labor? c. What is the area of the semicircle to the nearest tenth? d. What is the area of each triangle? e. What is the total area of the garden to the nearest foot? 1- 5 Using Formulas in Geometry 39 39 ������������������������������������������������������������������������������������������������������������������������������������������ 31. Algebra The large rectangle has length a + b and width c + d. |
Therefore, its area is (a + b) (c + d). a. Find the area of each of the four small rectangles in the figure. Then find the sum of these areas. Explain why this sum must be equal to the product (a + b)(c + d). b. Suppose b = d = 1. Write the area of the large rectangle as a product of its length and width. Then find the sum of the areas of the four small rectangles. Explain why this sum must be equal to the product (a + 1)(c + 1). c. Suppose b = d = 1 and a = c. Write the area of the large rectangle as a product of its length and width. Then find the sum of the areas of the four small rectangles. Explain why this sum must be equal to the product (a + 1) 2. 32. Sports The table shows the minimum and maximum dimensions for rectangular soccer fields used in international matches. Find the difference in area of the largest possible field and the smallest possible field. Length Width Minimum Maximum 100 m 64 m 110 m 75 m Find the value of each missing measure of a triangle. ft; A = 28 ft 2 33. b = 2 ft; h = 34. b = ft; h = 22.6 yd; A = 282.5 yd 2 Find the area of each rectangle with the given base and height. 35. 9.8 ft; 2.7 ft 36. 4 mi 960 ft; 440 ft 37. 3 yd 12 ft; 11 ft Find the perimeter of each rectangle with the given base and height. 38. 21.4 in.; 7.8 in. 39. 4 ft 6 in.; 6 in. 40. 2 yd 8 ft; 6 ft Find the diameter of the circle with the given measurement. Leave answers in terms of π. 41. C = 14 42. A = 100π 43. C = 50π 44. A skate park consists of a two adjacent rectangular regions as shown. Find the perimeter and area of the park. 45. Critical Thinking Explain how you would measure a triangular piece of paper if you wanted to find its area. 46. Write About It A student wrote in her journal, “To find the perimeter of a rectangle, add the length and width together and then double this value.” Does her method work? Explain. 47. Manda made a circular tabletop that has an area of 452 in2. Which is closest to |
the radius of the tabletop? 9 in. 12 in. 24 in. 72 in. 48. A piece of wire 48 m long is bent into the shape of a rectangle whose length is twice its width. Find the length of the rectangle. 8 m 16 m 24 m 32 m 40 40 Chapter 1 Foundations for Geometry ��������������������� 49. Which equation best represents the area A of the triangle? A = 2 x 2 + 4x A = 4x (x + 2 50. Ryan has a 30 ft piece of string. He wants to use the string to lay out the boundary of a new flower bed in his garden. Which of these shapes would use all the string? A circle with a radius of about 37.2 in. A rectangle with a length of 6 ft and a width of 5 ft A triangle with each side 9 ft long A square with each side 90 in. long Math History CHALLENGE AND EXTEND 51. A circle with a 6 in. diameter is stamped out of a rectangular piece of metal as shown. Find the area of the remaining piece of metal. Use the π key on your calculator and round to the nearest tenth. 52. a. Solve P = 2ℓ + 2w for w. ������ ����� b. Use your result from part a to find the width of a rectangle that has a perimeter of 9 ft and a length of 3 ft. 53. Find all possible areas of a rectangle whose sides are natural numbers and whose perimeter is 12. The Ahmes Papyrus is an ancient Egyptian source of information about mathematics. A page of the Ahmes Papyrus is about 1 foot wide and 18 feet long. Source: scholars.nus.edu.sg 54. Estimation The Ahmes Papyrus dates from approximately 1650 B.C.E. Lacking a precise value for π, the author assumed that the area of a circle with a diameter of 9 units had the same area as a square with a side length of 8 units. By what percent did the author overestimate or underestimate the actual area of the circle? 55. Multi-Step The width of a painting is 4 __ 5 the measure of the length of the painting. If the area is 320 in 2, what are the length and width of the painting? SPIRAL REVIEW Determine the domain and range of each function. (Previous course) ⎬ ⎨ (2, 4), (-5, |
8), (-3, 4) 56. ⎬ ⎨ (4, -2), (-2, 8), (16, 0) 57. Name the geometric figure that each item suggests. (Lesson 1-1) 58. the wall of a classroom 59. the place where two walls meet 60. Marion has a piece of fabric that is 10 yd long. She wants to cut it into 2 pieces so that one piece is 4 times as long as the other. Find the lengths of the two pieces. (Lesson 1-2) 61. Suppose that A, B, and C are collinear points. B is the midpoint of of A is -8, and the coordinate of B is -2.5. What is the coordinate of C? (Lesson 1-2) 62. An angle’s measure is 9 degrees more than 2 times the measure of its supplement. Find the measure of the angle. (Lesson 1-4) 1- 5 Using Formulas in Geometry 41 41 _ AC. The coordinate ������� Graphing in the Coordinate Plane Algebra The coordinate plane is used to name and locate points. Points in the coordinate plane are named by ordered pairs of the form (x, y). The first number is the x-coordinate. The second number is the y-coordinate. The x-axis and y-axis intersect at the origin, forming right angles. The axes separate the coordinate plane into four regions, called quadrants, numbered with Roman numerals placed counterclockwise. See Skills Bank page S56 Examples 1 Name the coordinates of P. Starting at the origin (0, 0), you count 1 unit to the right. Then count 3 units up. So the coordinates of P are (1, 3). 2 Plot and label H (-2, -4) on a coordinate plane. Name the quadrant in which it is located. Start at the origin (0, 0) and move 2 units left. Then move 4 units down. Draw a dot and label it H. H is in Quadrant III. You can also use a coordinate plane to locate places on a map. Try This TAKS Grades 9–11 Obj. 6, 7 Name the coordinates of the point where the following streets intersect. 1. Chestnut and Plum 2. Magnolia and Chestnut 3. |
Oak and Hawthorn 4. Plum and Cedar Name the streets that intersect at the given points. 5. (-3, -1) 7. (1, 3) 6. (4, -1) 8. (-2, 1) 42 42 Chapter 1 Foundations for Geometry �������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� 1-6 Midpoint and Distance in the Coordinate Plane TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates.... Also G.7.A, G.7.C, G.8.C Objectives Develop and apply the formula for midpoint. Use the Distance Formula and the Pythagorean Theorem to find the distance between two points. Vocabulary coordinate plane leg hypotenuse Why learn this? You can use a coordinate plane to help you calculate distances. (See Example 5.) Major League baseball fields are laid out according to strict guidelines. Once you know the dimensions of a field, you can use a coordinate plane to find the distance between two of the bases. A coordinate plane is a plane that is divided into four regions by a horizontal line (x-axis) and a vertical line (y-axis). The location, or coordinates, of a point are given by an ordered pair (x, y). Minute Maid Park, Houston You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints. Midpoint Formula _ AB with The midpoint M of endpoints A ( x 1, y 1 ) and B ( x 2, y 2 ) is found by To make it easier to picture the problem, plot the segment’s endpoints on a coordinate plane. Finding the Coordinates of a Midpoint ̶̶ CD M ( Find the coordinates of the midpoint of with endpoints C (-2, -1) and D (4, 21 + 2 -1, 1 _ ) 2 1. Find the coordinates of the midpoint of _ EF with endpoints E (-2, 3) and F (5, - |
3). 1- 6 Midpoint and Distance in the Coordinate Plane 43 43 ����������������������������������������������������������������������������������������������������������������������������������� E X A M P L E 2 Finding the Coordinates of an Endpoint ̶̶ AB. A has coordinates (2, 2), and M has coordinates M is the midpoint of (4, -3). Find the coordinates of B. Step 1 Let the coordinates of B equal (x, y). Step 2 Use the Midpoint Formula: (4, -3 = Step 3 Find the x-coordinate4) = 2 ( Set the coordinates equal. Multiply both sides by 2. Find the y-coordinate. -3 = 2 (-3 Simplify. - 2 - ̶̶̶̶ 2 ̶̶̶̶ 6 = x Subtract 2 from both sides. Simplify. -6 = 2 + y - 2 - ̶̶̶̶ -8 = y 2 ̶̶̶̶ The coordinates of B are (6, -8). 2. S is the midpoint of _ RT. R has coordinates (-6, -1), and S has coordinates (-1, 1). Find the coordinates of T. The Ruler Postulate can be used to find the distance between two points on a number line. The Distance Formula is used to calculate the distance between two points in a coordinate plane. Distance Formula In a coordinate plane, the distance d between two points ( x 1, y 1 ) and ( x 2, y 2 ) is d = √ ( Using the Distance Formula Find AB and CD. Then determine if ̶̶ AB ≅ ̶̶ CD. Step 1 Find the coordinates of each point. A (0, 3), B (5, 1), C (-1, 1), and D (-3, -4) Step 2 Use the Distance Formula. d = √ AB = √ ( 5 - 0) 2 + (1 - 3) 2 CD |
= √ ⎤ ⎦ ⎡ ⎣ 2 + (-4 - 1) 2 -3 - (-1) = √ 5 2 + (-2) 2 = √ (-2) 2 + (-5) 2 = √ 25 + 4 = √ 29 = √ 4 + 25 = √ 29 Since AB = CD, _ AB ≅ _ CD. 3. Find EF and GH. Then determine if _ EF ≅ _ GH. 44 44 Chapter 1 Foundations for Geometry ���������������� You can also use the Pythagorean Theorem to find the distance between two points in a coordinate plane. You will learn more about the Pythagorean Theorem in Chapter 5. In a right triangle, the two sides that form the right angle are the legs. The side across from the right angle that stretches from one leg to the other is the hypotenuse. In the diagram, a and b are the lengths of the shorter sides, or legs, of the right triangle. The longest side is called the hypotenuse and has length c. Theorem 1-6-1 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse Finding Distances in the Coordinate Plane Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from A to B. Method 1 Method 2 Use the Distance Formula. Substitute the values for the coordinates of A and B into the Distance Formula. AB = √ ( = √ ⎤ ⎦ ⎡ ⎣ 2 + (-2 - 3) 2 2 - (-2) = √ |
4 2 + (-5) 2 16 + 25 = √ = √ 41 ≈ 6.4 Use the Pythagorean Theorem. Count the units for sides a and b. a = 4 and b = 5 = 16 + 25 = 41 c = √ 41 c ≈ 6.4 Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from R to S. 4a. R (3, 2) and S (-3, -1) 4b. R (-4, 5) and S (2, -1) 1- 6 Midpoint and Distance in the Coordinate Plane 45 45 ����������������������������������������� E X A M P L E 5 Sports Application The four bases on a baseball field form a square with 90 ft sides. When a player throws the ball from home plate to second base, what is the distance of the throw, to the nearest tenth? Set up the field on a coordinate plane so that home plate H is at the origin, first base F has coordinates (90, 0), second base S has coordinates (90, 90), and third base T has coordinates (0, 90). The distance HS from home plate to second base is the length of the hypotenuse of a right triangle. HS = √ = √ ( 90 - 0) 2 + (90 - 0) 2 = √ 90 2 + 90 2 = √ 8100 + 8100 = √ 16,200 ≈ 127.3 ft 5. The center of the pitching mound has coordinates (42.8, 42.8). When a pitcher throws the ball from the center of the mound to home plate, what is the distance of the throw, to the nearest tenth? THINK AND DISCUSS 1. Can you exchange the coordinates ( x 1, y 1 ) and ( |
x 2, y 2 ) in the Midpoint Formula and still find the correct midpoint? Explain. 2. A right triangle has sides lengths of r, s, and t. Given that s 2 + t 2 = r 2, which variables represent the lengths of the legs and which variable represents the length of the hypotenuse? 3. Do you always get the same result using the Distance Formula to find distance as you do when using the Pythagorean Theorem? Explain your answer. 4. Why do you think that most cities are laid out in a rectangular grid instead of a triangular or circular grid? 5. GET ORGANIZED Copy and complete the graphic organizer below. In each box, write a formula. Then make a sketch that will illustrate the formula. 46 46 Chapter 1 Foundations for Geometry H (0,0)F(90,0)S(90,90)T(0,90)ge07se_c01L06006a�������������������������������������������������������� 1-6 Exercises Exercises KEYWORD: MG7 1-6 KEYWORD: MG7 Parent. 43. 44. 44. 45 GUIDED PRACTICE 1. Vocabulary The right angle. (hypotenuse or leg)? is the side of a right triangle that is directly across from the ̶̶̶̶ Find the coordinates of the midpoint of each segment. _ AB with endpoints A (4, -6) and B (-4, 2) _ CD with endpoints C (0, -8) and D (3, 0) 2. 3. 4. M is the midpoint of _ LN. L has coordinates (-3, -1), and M has coordinates (0, 1). Find the coordinates of N. 5. B is the midpoint of (-1 1 __ 2, 1). Find the coordinates of C. _ AC. A has coordinates (-3, 4), and B has coordinates Multi-Step Find the length of the given segments and determine if they are congruent. _ JK and _ FG 6. _ JK and _ RS 7. Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. 8. A (1, -2) and B (-4, -4) 9. X (-2, 7) and Y (-2, - |
8) 10. V (2, -1) and W (-4, 8. 46 11. Architecture The plan for a rectangular living room shows electrical wiring will be run in a straight line from the entrance E to a light L at the opposite corner of the room. What is the length of the wire to the nearest tenth? Independent Practice For See Exercises Example 12–13 14–15 16–17 18–20 21 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S5 Application Practice p. S28 PRACTICE AND PROBLEM SOLVING Find the coordinates of the midpoint of each segment. 12. _ XY with endpoints X (-3, -7) and Y (-1, 1) _ MN with endpoints M (12, -7) and N (-5, -2) _ QR. Q has coordinates (-3, 5), and M has coordinates (7, -9). 14. M is the midpoint of 13. Find the coordinates of R. 15. D is the midpoint of Find the coordinates of C. _ CE. E has coordinates (-3, -2), and D has coordinates (2 1 __ 2, 1). Multi-Step Find the length of the given segments and determine if they are congruent. _ DE and _ DE and _ FG _ RS 16. 17. 1- 6 Midpoint and Distance in the Coordinate Plane 47 47 ������������������������������ Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. 18. U (0, 1) and V (-3, -9) 19. M (10, -1) and N (2, -5) 20. P (-10, 1) and Q (5, 5) 21. Consumer Application Televisions and computer screens are usually advertised based on the length of their diagonals. If the height of a computer screen is 11 in. and the width is 14 in., what is the length of the diagonal? Round to the nearest inch. 22. Multi-Step Use the Distance Formula to _ EF from shortest to longest. _ CD, and _ AB, order 23. Use the Pythagorean Theorem to find the distance from A to E. Round to the nearest hundredth. 24. X has coordinates (a, 3a), and Y has coordinates (-5a, |
0). Find the coordinates of the midpoint of (XY). 25. Describe a shortcut for finding the midpoint of a segment when one of its endpoints has coordinates (a, b) and the other endpoint is the origin. On the map, each square of the grid represents 1 square mile. Find each distance to the nearest tenth of a mile. tenth of a mile. 26. 26. Find the distance along Highway 201 from Cedar City to Milltown. 27. 27. A car breaks down on Route 1, at the midpoint between Jefferson and Milltown. A tow truck is sent out from Jefferson. How far does the truck travel to reach the car? 28. History The Forbidden City in Beijing, China, is the world’s largest palace complex. Surrounded by a wall and a moat, the rectangular complex is 960 m long and 750 m wide. Find the distance, to the nearest meter, from one corner of the complex to the opposite corner. 29. Critical Thinking Give an example of a line segment with midpoint (0, 0). The coordinates of the vertices of △ABC are A(1, 4), B (-2, -1), and C (-3, -2). 30. Find the perimeter of △ABC to the nearest tenth. _ BC is √ 2, and b is the length of 31. The height h to side _ BC. What is the area of △ABC? History The Forbidden City of Imperial China is replicated in Katy, Texas. The museum has 6000 miniature terra-cotta soldiers. Source: www.forbiddengardens.com 32. Write About It Explain why the Distance Formula is not needed to find the distance between two points that lie on a horizontal or a vertical line. 33. This problem will prepare you for the Multi-Step TAKS Prep on page 58. Tania uses a coordinate plane to map out plans for landscaping a rectangular patio area. On the plan, one square represents 2 feet. She plans to plant a tree at the midpoint of of the patio does she plant the tree? Round to the nearest tenth. _ AC. How far from each corner 48 48 Chapter 1 Foundations for Geometry ������������������������������������������������������������������������������� 34. Which segment has a length closest to 4 units? _ EF _ GH _ JK _ LM 35. Find the distance, to |
the nearest tenth, between the midpoints of _ LM and _ JK. 1.8 3.6 4.0 5.3 36. What are the coordinates of the midpoint of a line segment that connects the points (7, -3) and (-5, 6)? (2, 1 __ ) (1, 1 1 __ ) (6, -4 1 __ ) (2, 3) 2 2 2 37. A coordinate plane is placed over the map of a town. A library is located at (-5, 1), and a museum is located at (3, 5). What is the distance, to the nearest tenth, from the library to the museum? 4.5 5.7 6.3 8.9 CHALLENGE AND EXTEND 38. Use the diagram to find the following. _ AB, and R is the midpoint of a. P is the midpoint of _ BC. Find the coordinates of Q. b. Find the area of rectangle PBRQ. c. Find DB. Round to the nearest tenth. 39. The coordinates of X are (a - 5, -2a). The coordinates of Y are (a + 1, 2a). If the distance between X and Y is 10, find the value of a. 40. Find two points on the y-axis that are a distance of 5 units from (4, 2). 41. Given ∠ACB is a right angle of △ABC, AC = x, and BC = y, find AB in terms of x and y. SPIRAL REVIEW Determine if the ordered pair (-1, 4) satisfies each function. (Previous course) 42. y = 3x - 1 44. g (x) = x 2 - x + 2 43. f (x) = 5 - x 2 BD bisects straight angle ABC, and Find the measure of each angle and classify it as acute, right, or obtuse. (Lesson 1-3) BE bisects ∠CBD. 45. ∠ABD 46. ∠CBE 47. ∠ABE Find the area of each of the following. (Lesson 1-5) 48. square whose perimeter is 20 in. 49. triangle whose height is 2 ft and whose base is twice its height 50. rectangle whose length is x and |
whose width is (4x + 5) 1- 6 Midpoint and Distance in the Coordinate Plane 49 49 ����������������������������������� 1-7 Transformations in the Coordinate Plane TEKS G.5.C Geometric patterns: use properties of transformations... to make connections between mathematics and the real world.... Also G.1.A Objectives Identify reflections, rotations, and translations. Graph transformations in the coordinate plane. Vocabulary transformation preimage image reflection rotation translation Who uses this? Artists use transformations to create decorative patterns. (See Example 4.) The Alhambra, a 13th-century palace in Grenada, Spain, is famous for the geometric patterns that cover its walls and floors. To create a variety of designs, the builders based the patterns on several different transformations. A transformation is a change in the position, size, or shape of a figure. The original figure is called the preimage. The resulting figure is called the image. A transformation maps the preimage to the image. Arrow notation (→) is used to describe a transformation, and primes (′) are used to label the image. Transformations REFLECTION ROTATION TRANSLATION A reflection (or flip) is a transformation across a line, called the line of reflection. Each point and its image are the same distance from the line of reflection. A rotation (or turn) is a transformation about a point P, called the center of rotation. Each point and its image are the same distance from P. A translation (or slide) is a transformation in which all the points of a figure move the same distance in the same direction. E X A M P L E 1 Identifying Transformations Identify the transformation. Then use arrow notation to describe the transformation. A The transformation cannot be a translation because each point and its image are not in the same position. The transformation is a reflection. △EFG → △E′F′G′ 50 50 Chapter 1 Foundations for Geometry ������������������������������������������������������������������������������������� Identify the transformation. Then use arrow notation to describe the transformation. B The transformation cannot be a reflection because each point and its image are not the same distance from a line of reflection. The transformation is a 90° rotation. RSTU → R′S′T′U′ Identify each transformation. Then use arrow notation to describe the transformation. 1a. 1b. |
E X A M P L E 2 Drawing and Identifying Transformations A figure has vertices at A (-1, 4), B (-1, 1), and C (3, 1). After a transformation, the image of the figure has vertices at A′ (-1, -4), B′ (-1, -1), and C′ (3, -1). Draw the preimage and image. Then identify the transformation. Plot the points. Then use a straightedge to connect the vertices. The transformation is a reflection across the x-axis because each point and its image are the same distance from the x-axis. 2. A figure has vertices at E (2, 0), F (2, -1), G (5, -1), and H (5, 0). After a transformation, the image of the figure has vertices at E′ (0, 2), F′ (1, 2), G′ (1, 5), and H′ (0, 5). Draw the preimage and image. Then identify the transformation. To find coordinates for the image of a figure in a translation, add a to the x-coordinates of the preimage and add b to the y-coordinates of the preimage. Translations can also be described by a rule such as (x, y) → (x + a, y + b Translations in the Coordinate Plane Find the coordinates for the image of △ABC after the translation (x, y) → (x + 3, y - 4). Draw the image. Step 1 Find the coordinates of △ABC. The vertices of △ABC are A (-1, 1), B (-3, 3), and C (-4, 0). 1- 7 Transformations in the Coordinate Plane 51 51 �������������������������������������������������������������� Step 2 Apply the rule to find the vertices of the image. A′ (-1 + 3, 1 - 4) = A′ (2, -3) B′ (-3 + 3, 3 - 4) = B′ (0, -1) C′ (-4 + 3, 0 - 4) = C′ (-1, -4) Step 3 Plot the points. Then finish drawing the image by using a straightedge to connect the vertices. 3. Find the coordinates for the image of JK |
LM after the translation (x, y) → (x - 2, y + 4). Draw the image. E X A M P L E 4 Art History Application The pattern shown is similar to a pattern on a wall of the Alhambra. Write a rule for the translation of square 1 to square 2. Step 1 Choose 2 points Choose a point A on the preimage and a corresponding point A′ on the image. A has coordinates (3, 1), and A′ has coordinates (1, 3). Step 2 Translate To translate A to A′, 2 units are subtracted from the x-coordinate and 2 units are added to the y-coordinate. Therefore, the translation rule is (x, y) → (x - 2, y + 2). 4. Use the diagram to write a rule for the translation of square 1 to square 3. THINK AND DISCUSS 1. Explain how to recognize a reflection when given a figure and its image. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, sketch an example of each transformation. 52 52 Chapter 1 Foundations for Geometry �����������������������������geo7sec01l07002a2A‘A13yx�������������������������������������������� 1-7 Exercises Exercises KEYWORD: MG7 1-7 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Given the transformation △XYZ → △X′Y′Z′, name the preimage and image of the transformation. 2. The types of transformations of geometric figures in the coordinate plane can be described as a slide, a flip, or a turn. What are the other names used to identify these transformations Identify each transformation. Then use arrow notation to describe the transformation. p. 50 3. 4. 51 5. A figure has vertices at A (-3, 2), B (-1, -1), and C (-4, -2). After a transformation, the image of the figure has vertices at A′ (3, 2), B′ (1, -1), and C′ (4, -2). Draw the preimage and image. Then identify the transformation. 51. 52 6. Multi-Step The coordinates of the vertices of △DEF are D (2 |
, 3), E (1, 1), and F (4, 0). Find the coordinates for the image of △DEF after the translation (x, y) → (x - 3, y - 2). Draw the preimage and image. 7. Animation In an animated film, a simple scene can be created by translating a figure against a still background. Write a rule for the translation that maps the rocket from position 1 to position 2. PRACTICE AND PROBLEM SOLVING Identify each transformation. Then use arrow notation to describe the transformation. 8. 9. Independent Practice For See Exercises Example 8–9 10 11 12 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S5 Application Practice p. S28 10. A figure has vertices at J (-2, 3), K (0, 3), L (0, 1), and M (-2, 1). After a transformation, the image of the figure has vertices at J′ (2, 1), K′ (4, 1), L′ (4, -1), and M′ (2, -1). Draw the preimage and image. Then identify the transformation. 1- 7 Transformations in the Coordinate Plane 53 53 ���������������������x– 44y4 – 412������������������������ 11. Multi-Step The coordinates of the vertices of rectangle ABCD are A (-4, 1), B (1, 1), C (1, -2), and D (-4, -2). Find the coordinates for the image of rectangle ABCD after the translation (x, y) → (x + 3, y - 2). Draw the preimage and the image. 12. Travel Write a rule for the translation that maps the descent of the hot air balloon. Which transformation is suggested by each of the following? 13. mountain range and its image on a lake 14. straight line path of a band marching down a street 15. wings of a butterfly Given points F (3, 5), G (-1, 4), and H (5, 0), draw △FGH and its reflection across each of the following lines. 16. the x-axis 17. the y-axis 18. Find the vertices of one of the triangles on the graph. Then use arrow notation to write a rule for translating the other three |
triangles. A transformation maps A onto B and C onto D. 19. Name the image of A. 20. Name the preimage of B. 21. Name the image of C. 22. Name the preimage of D. 23. Find the coordinates for the image of △RST with vertices R (1, -4), S (-1, -1), and T (-5, 1) after the translation (x, y) → (x - 2, y - 8). 24. Critical Thinking Consider the translations (x, y) → (x + 5, y + 3) and (x, y) → (x + 10, y + 5). Compare the two translations. Graph each figure and its image after the given translation. _ MN with endpoints M (2, 8) and N (-3, 4) after the translation (x, y) → (x + 2, y - 5) _ KL with endpoints K (-1, 1) and L (3, -4) after the translation (x, y) → (x - 4, y + 3) 25. 26. 27. Write About It Given a triangle in the coordinate plane, explain how to draw its image after the translation (x, y) → (x + 1, y + 1). 28. This problem will prepare you for the Multi-Step TAKS Prep on page 58. Greg wants to rearrange the triangular pattern of colored stones on his patio. What combination of transformations could he use to transform △CAE to the image on the coordinate plane? 54 54 Chapter 1 Foundations for Geometry xyge07sec01/07004a������������������������������ 29. Which type of transformation maps △XYZ to △X′Y′Z′? Reflection Rotation Translation Not here 30. △DEF has vertices at D (-4, 2), E (-3, -3), and F (1, 4). Which of these points is a vertex of the image of △DEF after the translation (x, y) → (x - 2, y + 1)? (-2, 1) (3, 3) (-5, -2) (-6, -1) 31. Consider the translation (1, 4) → (-2, 3). What number was added to the x-coordinate? -3 -1 1 7 32. Consider |
the translation (-5, -7) → (-2, -1). What number was added to the y-coordinate? -3 3 6 8 CHALLENGE AND EXTEND 33. △RST with vertices R (-2, -2), S (-3, 1), and T (1, 1) is translated by (x, y) → (x - 1, y + 3). Then the image, △R′S′T ′, is translated by (x, y) → (x + 4, y - 1), resulting in △R "S"T ". a. Find the coordinates for the vertices of △R "S"T ". b. Write a rule for a single translation that maps △RST to △R"S"T ". 34. Find the angle through which the minute hand of a clock rotates over a period of 12 minutes. 35. A triangle has vertices A (1, 0), B (5, 0), and C (2, 2). The triangle is rotated 90° counterclockwise about the origin. Draw and label the image of the triangle. Determine the coordinates for the reflection image of any point A (x, y) across the given line. 36. x-axis 37. y-axis SPIRAL REVIEW Use factoring to find the zeros of each function. (Previous course) 39. y = x 2 + 3x - 18 38. y = x 2 + 12x + 35 40. y = x 2 - 18x + 81 41. y = x 2 - 3x + 2 Given m∠A = 76.1°, find the measure of each of the following. (Lesson 1-4) 42. supplement of ∠A 43. complement of ∠A Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. (Lesson 1-6) 44. (2, 3) and (4, 6) 46. (-3, 7) and (-6, -2) 45. (-1, 4) and (0, 8) 47. (5, 1) and (-1, 3) 1- 7 Transformations in the Coordinate Plane 55 55 ��������� 1-7 Explore Transformations A transformation is a movement of a figure from its original position (preimage) to a new |
position (image). In this lab, you will use geometry software to perform transformations and explore their properties. Use with Lesson 1-7 TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.2.B KEYWORD: MG7 Lab1 Activity 1 1 Construct a triangle using the segment tool. Use the text tool to label the vertices A, B, and C. 2 Select points A and B in that order. Choose Mark Vector from the Transform menu. 3 Select △ABC by clicking on all three segments of the triangle. 4 Choose Translate from the Transform menu, using Marked as the translation vector. What do you notice about the relationship between your preimage and its image? 5 What happens when you drag a vertex or a side of △ABC? Try This For Problems 1 and 2 choose New Sketch from the File menu. 1. Construct a triangle and a segment outside the triangle. Mark this segment as a translation vector as you did in Step 2 of Activity 1. Use Step 4 of Activity 1 to translate the triangle. What happens when you drag an endpoint of the new segment? 2. Instead of translating by a marked vector, use Rectangular as the translation vector and translate by a horizontal distance of 1 cm and a vertical distance of 2 cm. Compare this method with the marked vector method. What happens when you drag a side or vertex of the triangle? 3. Select the angles and sides of the preimage and image triangles. Use the tools in the Measure menu to measure length, angle measure, perimeter, and area. What do you think is true about these two figures? 56 56 Chapter 1 Foundations for Geometry Activity 2 1 Construct a triangle. Label the vertices G, H, and I. 2 Select point H and choose Mark Center from the Transform menu. 3 Select ∠GHI by selecting points G, H, and I in that order. Choose Mark Angle from the Transform menu. 4 Select the entire triangle △GHI by dragging a selection box around the figure. 5 Choose Rotate from the Transform menu, using Marked Angle as the angle of rotation. 6 What happens when you drag a vertex or a side of △GHI? Try This For Problems 4–6 choose New Sketch from the File menu. 4. Instead of selecting an angle of the triangle as the rotation angle, draw a new angle outside of the triangle. Mark this angle. |
Mark ∠GHI as Center and rotate the triangle. What happens when you drag one of the points that form the rotation angle? 5. Construct △QRS, a new rotation angle, and a point P not on the triangle. Mark P as the center and mark the angle. Rotate the triangle. What happens when you drag P outside, inside, or on the preimage triangle? 6. Instead of rotating by a marked angle, use Fixed Angle as the rotation method and rotate by a fixed angle measure of 30°. Compare this method with the marked angle method. 7. Using the fixed angle method of rotation, can you find an angle measure that will result in an image figure that exactly covers the preimage figure? 1- 7 Technology Lab 57 57 SECTION 1B Coordinate and Transformation Tools Pave the Way Julia wants to use L-shaped paving stones to pave a patio. Two stones will cover a 12 in. by 18 in. rectangle. 1. She drew diagram ABCDEF to represent the patio. Find the area and perimeter of the patio. How many paving stones would Julia need to purchase to pave the patio? If each stone costs $2.25, what is the total cost of the stones for the patio? Describe how you calculated your answer. 2. Julia plans to place a fountain at the ̶̶ AF. How far is the fountain midpoint of from B, C, E, and F? Round to the nearest tenth. 3. Julia used a pair of paving stones to create another pattern for the patio. Describe the transformation she used to create the pattern. If she uses just one transformation, how many other patterns can she create using two stones? Draw all the possible combinations. Describe the transformation used to create each pattern. 58 58 Chapter 1 Foundations for Geometry ���������������������������������������������� SECTION 1B Quiz for Lessons 1-5 Through 1-7 1-5 Using Formulas in Geometry Find the perimeter and area of each figure. 1. 3. 2. 4. 5. Find the circumference and area of a circle with a radius of 6 m. Use the π key on your calculator and round to the nearest tenth. 1-6 Midpoint and Distance in the Coordinate Plane 6. Find the coordinates for the midpoint of ̶̶ XY with endpoints X (-4, 6) and Y (3, 8). 7. J is the midpoint of ̶̶ HK, H has |
coordinates (6, -2), and J has coordinates (9, 3). Find the coordinates of K. 8. Using the Distance Formula, find QR and ST to the nearest tenth. Then determine if ̶̶ QR ≅ ̶̶ ST. 9. Using the Distance Formula and the Pythagorean Theorem, find the distance, to the nearest tenth, from F (4, 3) to G (-3, -2). 1-7 Transformations in the Coordinate Plane Identify the transformation. Then use arrow notation to describe the transformation. 10. 11. 12. A graphic designer used the translation (x, y) → (x - 3, y + 2) to transform square HJKL. Find the coordinates and graph the image of square HJKL. 13. A figure has vertices at X (1, 1), Y (3, 1), and Z (3, 4). After a transformation, the image of the figure has vertices at X′ (-1, -1), Y′ (-3, -1), and Z′ (-3, -4). Graph the preimage and image. Then identify the transformation. Ready to Go On? 5959 �������������������������������������������������������������������������������������������������� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary acute angle.................. 21 diameter.................... 37 plane........................ 6 adjacent angles.............. 28 distance..................... 13 point........................ 6 angle....................... 20 endpoint..................... 7 postulate..................... 7 |
angle bisector............... 23 exterior of an angle.......... 20 preimage.................... 50 area........................ 36 height...................... 36 radius....................... 37 base........................ 36 hypotenuse................. 45 ray.......................... 7 between..................... 14 image....................... 50 reflection................... 50 bisect....................... 15 interior of an angle.......... 20 right angle.................. 21 circumference............... 37 leg.......................... 45 rotation..................... 50 collinear..................... 6 length...................... 13 segment..................... 7 complementary angles....... 29 line...................... |
.... 6 segment bisector............ 16 congruent angles............ 22 linear pair................... 28 straight angle................ 21 congruent segments......... 13 measure.................... 20 supplementary angles........ 29 construction................ 14 midpoint.................... 15 transformation.............. 50 coordinate.................. 13 obtuse angle................. 21 translation.................. 50 coordinate plane............ 43 opposite rays................. 7 undefined term............... 6 coplanar..................... 6 perimeter................... 36 vertex....................... 20 degree...................... 20 pi.......................... 37 vertical angles............... 30 Complete the sentences below with vocabulary words from the list above. 1. A(n)? divides an angle into two congruent angles. ̶̶̶̶̶̶ 2.? are two angles whose measures have a sum of 90°. ̶̶̶̶̶̶ 3. The length of the longest side of a right triangle is called |
the?. ̶̶̶̶̶̶ 1-1 Understanding Points, Lines, and Planes (pp. 6–11) TEKS G.1.A, G.7.A E X A M P L E S ■ Name the common endpoint of SR and ST. ST are opposite rays with common SR and endpoint S. 60 60 Chapter 1 Foundations for Geometry EXERCISES Name each of the following. 4. four coplanar points 5. line containing B and C 6. plane that contains A, G, and E ������������ ■ Draw and label three coplanar lines intersecting in one point. Draw and label each of the following. 7. line containing P and Q 8. pair of opposite rays both containing C 9. CD intersecting plane P at B 1-2 Measuring and Constructing Segments (pp. 13–19) TEKS G.2.A, G.2.B, G.3.B, G.7.C E X A M P L E S ■ Find the length of XY = ⎜-2 - 1⎟ = ⎜-3⎟ = 3 ̶ XY. ■ S is between R and T. Find RT. RT = RS + ST 3x + 2 = 5x - 6 + 2x 3x + 2 = 7x - 6 x = 2 RT = 3 (2) + 2 = 8 EXERCISES Find each length. 10. JL 11. HK 12. Y is between X and Z, XY = 13.8, and XZ = 21.4. Find YZ. 13. Q is between P and R. Find PR. 14. U is the midpoint of ̶ TV, TU = 3x + 4, and UV = 5x - 2. Find TU, UV, and TV. 15. E is the midpoint of ̶ DF, DE = 9x, and EF = 4x + 10. Find DE, EF, and DF. 1-3 Measuring and Constructing Angles (pp. 20–27) TEKS G.1.A, G.1.B, G.2.A, G.2.B, G.3.B E X |
A M P L E S EXERCISES ■ Classify each angle as acute, right, or obtuse. 16. Classify each angle as acute, right, or obtuse. ∠ABC acute; ∠CBD acute; ∠ABD obtuse; ∠DBE acute; ∠CBE obtuse ■ ̶ KM bisects ∠JKL, m∠JKM = (3x + 4) °, and m∠MKL = (6x - 5) °. Find m∠JKL. 3x + 4 = 6x - 5 Def. of ∠ bisector 3x + 9 = 6x Add 5 to both sides. 9 = 3x Subtract 3x from both sides. x = 3 Divide both sides by 3. 17. m∠HJL = 116°. Find m∠HJK. 18. NP bisects ∠MNQ, m∠MNP = (6x - 12) °, and m∠PNQ = (4x + 8) °. Find m∠MNQ. m∠JKL = 3x + 4 + 6x - 5 = 9x -1 = 9 (3) - 1 = 26° Study Guide: Review 61 61 ���������������������������������������������������������������������������������������������������������������� 1-4 Pairs of Angles (pp. 28–33) TEKS G.1.A, G.2.B E X A M P L E S EXERCISES ■ Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. ∠1 and ∠2 are only adjacent. ∠2 and ∠4 are not adjacent. ∠2 and ∠3 are adjacent and form a linear pair. ∠1 and ∠4 are adjacent and form a linear pair. ■ Find the measure of the complement and supplement of each angle. 90 - 67.3 = 22.7° 180 - 67.3 = 112.7° 90 - (3x - 8) = (98 - 3x) ° 180 - (3x - 8) = (188 - 3x) ° Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. |
19. ∠1 and ∠2 20. ∠3 and ∠4 21. ∠2 and ∠5 Find the measure of the complement and supplement of each angle. 22. 23. 24. An angle measures 5 degrees more than 4 times its complement. Find the measure of the angle. 1-5 Using Formulas in Geometry (pp. 36–41) TEKS G.1.A, G.1.B, G.8.A E X A M P L E S EXERCISES ■ Find the perimeter and area of the triangle. P = 2x + 3x + 5 + 10 = 5x + 15 A = 1 _ (3x + 5) (2x) 2 Find the perimeter and area of each figure. 25. 26. = 3 x 2 + 5x 27. 28. ■ Find the circumference and area of the circle to the nearest tenth. C = 2π r = 2π (11) = 22π ≈ 69.1 cm A = π r 2 = π (11) 2 = 121π ≈ 380.1 cm 2 Find the circumference and area of each circle to the nearest tenth. 29. 30. 31. The area of a triangle is 102 m 2. The base of the triangle is 17 m. What is the height of the triangle? 62 62 Chapter 1 Foundations for Geometry �������������������������������������������������������������������������������������������� 1-6 Midpoint and Distance in the Coordinate Plane (pp. 43–49) TEKS G.1.A, G.7.A, E X A M P L E S EXERCISES G.7.C, G.8.C ̶ AB. Find the missing coordinates Y is the midpoint of of each point. 32. A (3, 2) ; B (-1, 4) ; Y ( 33. A (5, 0) ; B ( 34 (-2, 3) ) ; B (-4, 4) ; Y (-2, 3) Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. 35. X (-2, 4) and Y (6, 1) 36. H (0, 3) and K (-2, -4) 37. L (-4, 2) and M (3, -2) ■ X is |
the midpoint of ̶ CD. C has coordinates (-4, 1), and X has coordinates (3, -2). Find the coordinates of D. (3, -2) = ( ) 1 + y -4 + x _ _, 2 2 -2 = 3 = 6 = -4 + x -4 = 1 + y 10 = x -5 = y The coordinates of D are (10, -5). ■ Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from (1, 6) to (4, 2). c2 = a 2 + b 2 d = √ 4 - (1) 2 + 2 - (6) 2 3 2 + (-4) 2 = √ = √ 9 + 16 = √ 25 = 5.0 = 3 2 + 4 2 = 9 + 16 = 25 c = √ 25 = 5.0 1-7 Transformations in the Coordinate Plane (pp. 50–55) TEKS G.1.A, G.5.C E X A M P L E S EXERCISES ■ Identify the transformation. Then use arrow notation to describe the transformation. Identify each transformation. Then use arrow notation to describe the transformation. The transformation is a reflection. △ABC → △A′B′C′ ■ The coordinates of the vertices of rectangle HJKL are H (2, -1), J (5, -1), K (5, -3), and L (2, -3). Find the coordinates of the image of rectangle HJKL after the translation (x, y) → (x - 4, y + 1). H′ = (2 - 4, -1 + 1) = H′ (-2, 0) J′ = (5 - 4, -1 + 1) = J′ (1, 0) K′ = (5 - 4, -3 + 1) = K′ (1, -2) L′ = (2 - 4, -3 + 1) = L′ (-2, -2) 38. 39. 40. The coordinates |
for the vertices of △XYZ are X (-5, -4), Y (-3, -1), and Z (-2, -2). Find the coordinates for the image of △XYZ after the translation (x, y) → (x + 4, y + 5). Study Guide: Review 63 63 ��������������������������������� 1. Draw and label plane N containing two lines that intersect at B. Use the figure to name each of the following. 2. four noncoplanar points 3. line containing B and E 4. The coordinate of A is -3, and the coordinate of B is 0.5. Find AB. 5. E, F, and G represent mile markers along a straight highway. Find EF. 6. J is the midpoint of ̶ HK. Find HJ, JK, and HK. Classify each angle by its measure. 7. m∠LMP = 70° 8. m∠QMN = 90° 9. m∠PMN = 125° 10. TV bisects ∠RTS. If the m∠RTV = (16x - 6) ° and m∠VTS = (13x + 9) °, what is the m∠RTV? 11. An angle’s measure is 5 degrees less than 3 times the measure of its supplement. Find the measure of the angle and its supplement. Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 12. ∠2 and ∠3 13. ∠4 and ∠5 14. ∠1 and ∠4 15. Find the perimeter and area of a rectangle with b = 8 ft and h = 4 ft. Find the circumference and area of each circle to the nearest tenth. 16. r = 15 m 17. d = 25 ft 18. d = 2.8 cm 19. Find the midpoint of the segment with endpoints (-4, 6) and (3, 2). 20. M is the midpoint of ̶ LN. M has coordinates (-5, 1), and L has coordinates (2, 4). Find the coordinates of N. 21. Given A (-5, 1), B (-1, 3), C (1, 4), and D (4, 1), is ̶ AB ≅ |
̶ CD? Explain. Identify each transformation. Then use arrow notation to describe the transformation. 22. 23. 24. A designer used the translation (x, y) → (x + 3, y - 3) to transform a triangular-shaped pin ABC. Find the coordinates and draw the image of △ABC. 64 64 Chapter 1 Foundations for Geometry �������������������������������������������������������������������������� FOCUS ON SAT The SAT has three sections: Math, Critical Reading, and Writing. Your SAT scores show how you compare with other students. It can be used by colleges to determine admission and to award merit-based financial aid. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. On SAT multiple-choice questions, you receive one point for each correct answer, but you lose a fraction of a point for each incorrect response. Guess only when you can eliminate at least one of the answer choices. 1. Points D, E, F, and G are on a line, in that order. If DE = 2, FG = 5, and DF = 6, what is the value of EG (DG)? (A) 13 (B) 18 (C) 19 (D) 42 (E) 99 4. What is the area of the square? (A) 16 (B) 25 (C) 32 (D) 36 (E) 41 2. QS bisects ∠PQR, m∠PQR = (4x + 2) °, and m∠SQR = (3x - 6) °. What is the value of x? 5. If ∠BFD and ∠AFC are right angles and m∠CFD = 72°, what is the value of x? (A) 1 (B) 4 (C) 7 (D) 10 (E) 19 Note: Figure not drawn to scale. 3. A rectangular garden is enclosed by a brick border. The total length of bricks used to enclose the garden is 42 meters. If the length of the garden is twice the width, what is the area of the garden? (A) 18 (B) 36 (C) 72 (D) 90 (E) 108 (A) 7 meters (B) 14 meters (C) 42 meters (D) 42 square meters ( |
E) 98 square meters College Entrance Exam Practice 65 65 ���������������� Multiple Choice: Work Backward When you do not know how to solve a multiple-choice test item, use the answer choices and work the question backward. Plug in the answer choices to see which choice makes the question true. T is the midpoint of ̶ RC, RT = 12x - 8, and TC = 28. What is the value of x? -4 2 3 28 Since T is the midpoint of ̶ RC, then RT = RC, or 12x - 8 = 28. Find what value of x makes the left side of the equation equal 28. Try choice A: If x = -4, then 12x - 8 = 12 (-4) - 8 = -56. This choice is not correct because length is always a positive number. Try choice B: If x = 2, then 12x - 8 = 12 (2) - 8 = 16. Since 16 ≠ 28, choice B is not the answer. Try choice C: If x = 3, then 12x - 8 = 12 (3) - 8 = 28. Since 28 = 28, the correct answer is C, 3. Joel used 6400 feet of fencing to make a rectangular horse pen. The width of the pen is 4 times as long as the length. What is the length of the horse pen? 25 feet 480 feet 640 feet 1600 feet Use the formula P = 2ℓ + 2w. P = 6400 and w = 4ℓ. You can work backward to determine which answer choice is the most reasonable. Try choice J: Use mental math. If ℓ = 1600, then 4ℓ = 6400. This choice is not reasonable because the perimeter of the pen would then be far greater than 6400 feet. Try choice F: Use mental math. If ℓ = 25, then 4ℓ = 100. This choice is incorrect because the perimeter of the pen is 6400 ft, which is far greater than 2 (25) + 2 (100). Try choice H: If ℓ = 640, then 4ℓ = 2560. When you substitute these values into the perimeter formula, it makes a true statement. The correct answer is H, 640 ft. 66 66 Chapter 1 Foundations for Geometry ��������������� Read each test item and answer the questions that follow. �� �� �� �� ��� � Item A The |
measure of an angle is 3 times as great as that of its complement. Which value is the measure of the smaller angle? 22.5° 27.5° 63.5° 67.5° 1. Are there any definitions that you can use to solve this problem? If so, what are they? 2. Describe how to work backward to find the correct answer. When you work a test question backward start with choice C. The choices are usually listed in order from least to greatest. If choice C is incorrect because it is too low, you do not need to plug in the smaller numbers. Item D △QRS has vertices at Q (3, 5), R (3, 9), and S (7, 5). Which of these points is a vertex of the image of △QRS after the translation (x, y) → (x - 7, y - 6)? Item B In a town’s annual relay marathon race, the second runner of each team starts at mile marker 4 and runs to the halfway point of the 26-mile marathon. At that point the second runner passes the relay baton to the third runner of the team. How many total miles does the second runner of each team run? 4 miles 6.5 miles 9 miles 13 miles (-4, 3) (0, 0) (4, 1) (4, -3) 7. Explain how to use mental math to find an answer that is NOT reasonable. 8. Describe, by working backward, how you can determine the correct answer. 3. Which answer choice should you plug in first? Why? 4. Describe, by working backward, how you know that choices F and G are not correct. Item E TS bisects ∠PTR. If m∠PTS = (9x + 2) ° and m∠STR = (x + 18) °, what is the value of x? Item C Consider the translation (-2, 8) → (8, -4). What number was added to the x-coordinate? -12 -6 4 10 -10 0 2 20 5. Which answer choice should you plug in first? Why? 6. Explain how to work the test question backward to determine the correct answer. 9. Explain how to use mental math to find an answer that is NOT reasonable. 10. Describe how to use the answer choices to work backward to find which answer |
is reasonable. TAKS Tackler 67 67 ������������������ KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTER 1 Multiple Choice Use the diagram for Items 1–3. Use the diagram for Items 8–10. 1. Which points are collinear? A, B, and C B, C, and D A, B, and E B, D, and E 2. What is another name for plane R? Plane C Plane AB Plane ACE Plane BDE 3. Use your protractor to find the approximate measure of ∠ABD. 123° 117° 77° 63° 4. S is between R and T. The distance between R and T is 4 times the distance between S and T. If RS = 18, what is RT? 24 22.5 14.4 6 5. A ray bisects a straight angle into two congruent angles. Which term describes each of the congruent angles that are formed? Acute Obtuse Right Straight 6. Which expression states that ̶ AB is congruent ̶ CD? to AB ≅ CD AB = CD ̶ AB = ̶ AB ≅ ̶ CD ̶ CD 7. The measure of an angle is 35°. What is the measure of its complement? 35° 45° 55° 145° 68 68 Chapter 1 Foundations for Geometry 8. Which of these angles is adjacent to ∠MQN? ∠QMN ∠NPQ ∠QNP ∠PQN 9. What is the area of △NQP? 3.7 square meters 7.4 square meters 6.8 square meters 13.6 square meters 10. Which of the following pairs of angles are complementary? ∠MNQ and ∠QNP ∠NQP and ∠QPN ∠MNP and ∠QNP ∠QMN and ∠NPQ 11. K is the midpoint of ̶ JL. J has coordinates (2, -1), and K has coordinates (-4, 3). What are the coordinates of L? (3, -2) (1, -1) (-1, 1) (-10, 7) 12. A circle with a diameter of 10 inches has a circumference equal to the perimeter of a square. To the nearest tenth, what is the length of each side of the square? 2.5 inches 3.9 inches 5.6 inches |
7.9 inches 13. The map coordinates of a campground are (1, 4), and the coordinates of a fishing pier are (4, 7). Each unit on the map represents 1 kilometer. If Alejandro walks in a straight line from the campground to the pier, how many kilometers, to the nearest tenth, will he walk? 3.5 kilometers 6.0 kilometers 4.2 kilometers 12.1 kilometers ������������������ ���� ���� ���� For many types of geometry problems, it may be helpful to draw a diagram and label it with the information given in the problem. This method is a good way of organizing the information and helping you decide how to solve the problem. 14. m∠R is 57°. What is the measure of its supplement? 33° 43° 123° 133° 15. What rule would you use to translate a triangle STANDARDIZED TEST PREP Short Response 23. △ABC has vertices A (-2, 0), B (0, 0), and C (0, 3). The image of △ABC has vertices A′(1, -4), B′ (3, -4), and C′ (3, -1). a. Draw △ABC and its image △A′B′C′ on a coordinate plane. b. Write a rule for the transformation of △ABC using arrow notation. 24. You are given the measure of ∠4. You also know the following angles are supplementary: ∠1 and ∠2, ∠2 and ∠3, and ∠1 and ∠4. 4 units to the right? (x, y) → (x + 4, y) (x, y) → (x - 4, y) (x, y) → (x, y + 4) (x, y) → (x, y - 4) 16. If ̶ WZ bisects ∠XWY, which of the following statements is true? m∠XWZ > m∠YWZ m∠XWZ < m∠YWZ m∠XWZ = m∠YWZ m∠XWZ ≅ m∠YWZ 17. The x- and y-axes separate the coordinate plane into four regions, called quadrants. If (c, d) is a point that is not on the axes, |
such that c < 0 and d < 0, which quadrant would contain point (c, d)? I III II IV Gridded Response 18. The measure of ∠1 is 4 times the measure of its supplement. What is the measure, in degrees, of ∠1? 19. The exits for Market St. and Finch St. are 3.5 miles apart on a straight highway. The exit for King St. is at the midpoint between these two exits. How many miles apart are the King St. and Finch St. exits? 20. R has coordinates (-4, 9). S has coordinates (4, -6). What is RS? 21. If ∠A is a supplement of ∠B and is a right angle, then what is m∠B in degrees? 22. ∠C and ∠D are complementary. m∠C is 4 times m∠D. What is m∠C? Explain how you can determine the measures of ∠1, ∠2, and ∠3. 25. Marian is making a circular tablecloth from a rectangular piece of fabric that measures 6 yards by 4 yards. What is the area of the largest circular piece that can be cut from the fabric? Leave your answer in terms of π. Show your work or explain in words how you found your answer. Extended Response 26. Demara is creating a design using a computer illustration program. She begins by drawing the rectangle shown on the coordinate grid. a. Demara translates rectangle PQRS using the rule (x, y) → (x - 4, y - 6). On a copy of the coordinate grid, draw this translation and label each vertex. b. Describe one way that Demara could have moved rectangle PQRS to the same position in part a using a reflection and then a translation. c. On the same coordinate grid, Demara reflects rectangle PQRS across the x-axis. She draws a figure with vertices at (1, -3), (3, -3), (3, -5), and (1, -5). Did Demara reflect rectangle PQRS correctly? Explain your answer. Cumulative Assessment, Chapter 1 69 69 ����������������� Geometric Reasoning 2A Inductive and Deductive Reasoning 2-1 Using Inductive Reasoning to Make Conjectures 2-2 Conditional Statements 2-3 Using Deductive Reasoning to Verify |
Conjectures Lab Solve Logic Puzzles 2-4 Biconditional Statements and Definitions 2B Mathematical Proof 2-5 Algebraic Proof 2-6 Geometric Proof Lab Design Plans for Proofs 2-7 Flowchart and Paragraph Proofs Ext Introduction to Symbolic Logic KEYWORD: MG7 ChProj A corn maze from the 7A Ranch near Hondo 70 70 Chapter 2 Vocabulary Match each term on the left with a definition on the right. 1. angle A. a straight path that has no thickness and extends forever 2. line 3. midpoint 4. plane 5. segment B. a figure formed by two rays with a common endpoint C. a flat surface that has no thickness and extends forever D. a part of a line between two points E. names a location and has no size F. a point that divides a segment into two congruent segments Angle Relationships Select the best description for each labeled angle pair. 6. 7. 8. linear pair or vertical angles adjacent angles or vertical angles supplementary angles or complementary angles Classify Real Numbers Tell if each number is a natural number, a whole number, an integer, or a rational number. Give all the names that apply. 9. 6 11. –3 12. 5.2 10. –0.8 13. 3_ 8 Points, Lines, and Planes Name each of the following. 15. a point 16. a line 17. a ray 18. a segment 19. a plane 14. 0 � � � � � � � Solve One-Step Equations Solve. 20. 8 + x = 5 23. p - 7 = 9 21. 6y = -12 24. z_ 5 = 5 22. 9 = 6s 25. 8.4 = -1.2r Geometric Reasoning 71 71 ������ Key Vocabulary/Vocabulario conjecture conjetura counterexample contraejemplo deductive reasoning razonamiento deductivo inductive reasoning razonamiento inductivo polygon proof polígono demostración quadrilateral cuadrilátero theorem triangle teorema triángulo Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word counterexample is made up of two words: counter and example. In this case, counter |
means “against.” What is a counterexample to the statement “All numbers are positive”? 2. The root of the word inductive is ducere, which means “to lead.” When you are inducted into a club, you are “led into” membership. When you use inductive reasoning in math, you start with specific examples. What do you think inductive reasoning leads you to? 3. In Greek, the word poly means “many,” and the word gon means “angle.” How can you use these meanings to understand the term polygon? Geometry TEKS G.1.A Geometric structure* develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems Les. 2-1 Les. 2-2 Les. 2-3 2-3 Geo. Lab Les. 2-4 Les. 2-5 Les. 2-6 2-6 Geo. Lab Les. 2-7 Ext. ★ ★ ★ G.2.B Geometric structure* make conjectures... and determine ★ ★ ★ the validity of the conjectures,... G.3.A Geometric structure* determine the validity of a conditional statement, its converse, inverse, and contrapositive ★ ★ G.3.B Geometric structure* construct and justify statements ★ ★ ★ ★ ★ about geometric figures and their properties G.3.C Geometric structure* use logical reasoning to prove ★ ★ ★ ★ ★ ★ ★ statements are true and find counterexamples to disprove statements that are false G.3.D Geometric structure* use inductive reasoning to ★ formulate a conjecture G.3.E Geometric structure* use deductive reasoning to prove a ★ ★ ★ ★ ★ statement G.4.A Geometric structure* select an appropriate representation... in order to solve problems G.5.B Geometric patterns* use numeric and geometric patterns ★ to make generalizations about geometric properties... ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 72 72 Chapter 2 Reading Strategy: Read and Interpret a Diagram A diagram is an informational tool. To correctly read a diagram, you must know what you can and cannot assume based on what you see in it. ✔ Collinear points ✔ Betweenness of points ✔ Coplanar points ✘ Measures of segments ✘ Measures of angles ✘ Congru |
ent segments ✔ Straight angles and lines ✘ Congruent angles ✔ Adjacent angles ✔ Linear pairs of angles ✔ Vertical angles ✘ Right angles If a diagram includes labeled information, such as an angle measure or a right angle mark, treat this information as given. ✔ Points A, B, and C are collinear. ✘ ∠CBD is acute. ✔ Points A, B, C, and D are coplanar. ✔ B is between A and C. ✔ AC is a line. ✔ ∠ABD and ∠CBD are adjacent angles. ✔ ∠ABD and ∠CBD form a linear pair. ✘ ∠ABD is obtuse. ̶̶ BC ̶̶ AB ≅ ✘ Try This List what you can and cannot assume from each diagram. 1. 2. Geometric Reasoning 73 73 ������������� 2-1 Using Inductive Reasoning to Make Conjectures TEKS G.3.D Geometric structure: use inductive reasoning to formulate a conjecture. Also G.2.B, G.5.B Objectives Use inductive reasoning to identify patterns and make conjectures. Find counterexamples to disprove conjectures. Who uses this? Biologists use inductive reasoning to develop theories about migration patterns. Vocabulary inductive reasoning conjecture counterexample Biologists studying the migration patterns of California gray whales developed two theories about the whales’ route across Monterey Bay. The whales either swam directly across the bay or followed the shoreline. E X A M P L E 1 Identifying a Pattern Find the next item in each pattern. A Monday, Wednesday, Friday, … Alternating days of the week make up the pattern. The next day is Sunday. B 3, 6, 9, 12, 15, … Multiples of 3 make up the pattern. The next multiple is 18. C ←, ↖, ↑, … In this pattern, the figure rotates 45° clockwise each time. The next figure is ↗. 1. Find the next item in the pattern 0.4, 0.04, 0.004, … When several examples form a pattern and you assume the pattern will continue, you are applying inductive reasoning. Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. You may use inductive reasoning to draw a conclusion from a pattern. A |
statement you believe to be true based on inductive reasoning is called a conjecture. E X A M P L E 2 Making a Conjecture Complete each conjecture. A The product of an even number and an odd number is?. ̶̶̶ List some examples and look for a pattern. (2) (3) = 6 The product of an even number and an odd number is even. (4) (3) = 12 (2) (5) = 10 (4) (5) = 20 74 74 Chapter 2 Geometric Reasoning Complete each conjecture. B The number of segments formed by n collinear points is?. ̶̶̶ Draw a segment. Mark points on the segment, and count the number of individual segments formed. Be sure to include overlapping segments. Points Segments = 10 The number of segments formed by n collinear points is the sum of the whole numbers less than n. 2. Complete the conjecture: The product of two odd numbers is?. ̶̶̶ E X A M P L E 3 Biology Application To learn about the migration behavior of California gray whales, biologists observed whales along two routes. For seven days they counted the numbers of whales seen along each route. Make a conjecture based on the data. Numbers of Whales Each Day Direct Route Shore Route More whales were seen along the shore route each day. The data supports the conjecture that most California gray whales migrate along the shoreline. 3. Make a conjecture about the lengths of male and female whales based on the data. Average Whale Lengths Length of Female (ft) Length of Male (ft) 49 47 51 45 50 44 48 46 51 48 47 48 To show that a conjecture is always true, you must prove it. To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample. A counterexample can be a drawing, a statement, or a number. Inductive Reasoning 1. Look for a pattern 2. Make a conjecture. 3. Prove the conjecture or find a counterexample. 2- 1 Using Inductive Reasoning to Make Conjectures 75 75 E X A M P L E 4 Finding a Counterexample Show that each conjecture is false by finding a counterexample. A For all positive numbers n, 1 _ n ≤ n. Pick positive values for n and substitute them into the equation to see if the conjecture holds. Let n |
= 1. Since 1 _ n = 1 and 1 ≤ 1, the conjecture holds. Let n = 2. Since 1 _ n = 1 _ and 1 _ ≤ 2, the conjecture holds. 2 2. Since 1 _ n = 1 _ Let and 2 ≰ 1 _ 2, the conjecture is false. n = 1 _ 2 is a counterexample. B For any three points in a plane, there are three different lines that contain two of the points. Draw three collinear points. If the three points are collinear, the conjecture is false. C The temperature in Abilene, Texas, never exceeds 100°F during the spring months (March, April, and May). Monthly High Temperatures (°F) in Abilene, Texas Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 88 89 97 99 107 109 110 107 106 103 92 89 The temperature in May was 107°F, so the conjecture is false. Show that each conjecture is false by finding a counterexample. 4a. For any real number x, x 2 ≥ x. 4b. Supplementary angles are adjacent. 4c. The radius of every planet in the solar system is less than 50,000 km. Planets’ Diameters (km) Mercury Venus Earth Mars Jupiter Saturn Uranus Nepture Pluto 4880 12,100 12,800 6790 143,000 121,000 51,100 49,500 2300 THINK AND DISCUSS 1. Can you prove a conjecture by giving one example in which the conjecture is true? Explain your reasoning. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe the steps of the inductive reasoning process. 76 76 Chapter 2 Geometric Reasoning ������������������������������������ 2-1 Exercises Exercises KEYWORD: MG7 2-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain why a conjecture may be true or false. 74 Find the next item in each pattern., 3 _, 2 _ 3. 1 _ 5 4 3 2. March, May, July, …, … 4 Complete each conjecture. p. 74 5. The product of two even numbers is?. ̶̶̶ 6. A rule in terms of n for the sum of the first n odd positive integers is?. ̶̶̶. Biology A laboratory culture contains 150 bacteria. After twenty minutes, the |
p. 75 culture contains 300 bacteria. After one hour, the culture contains 1200 bacteria. Make a conjecture about the rate at which the bacteria increases. 76 Show that each conjecture is false by finding a counterexample. 8. Kennedy is the youngest U.S. president to be inaugurated. 9. Three points on a plane always form a triangle. 10. For any real number x, if x 2 ≥ 1, then x ≥ 1. President Washington T. Roosevelt Truman Kennedy Clinton Age at Inauguration 57 42 60 43 46 Independent Practice For See Exercises Example 11–13 14–15 16 17–19 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S6 Application Practice p. S29 PRACTICE AND PROBLEM SOLVING Find the next item in each pattern. 11. 8 A.M., 11 A.M., 2 P.M., … 12. 75, 64, 53, … 13. △, □,, … Complete each conjecture. 14. A rule in terms of n for the sum of the first n even positive integers is?. ̶̶̶ 15. The number of nonoverlapping segments formed by n collinear points is?. ̶̶̶ 16. Industrial Arts About 5% of the students at Lubbock High School usually participate in the robotics competition. There are 526 students in the school this year. Make a conjecture about the number of students who will participate in the robotics competition this year. Show that each conjecture is false by finding a counterexample. 17. If 1 - y > 0, then 0 < y < 1. 18. For any real number x, x 3 ≥ x 2. 19. Every pair of supplementary angles includes one obtuse angle. Make a conjecture about each pattern. Write the next two items., 1 _, 1 _ 21. 1 _ 4 8 2 20. 2, 4, 16, …, … 22. –3, 6, –9, 12, … 23. Draw a square of dots. Make a conjecture about the number of dots needed to increase the size of the square from n × n to (n + 1) × (n + 1). 2- 1 Using Inductive Reasoning to Make Conjectures 77 77 ����� Math History Goldbach first stated his conjecture in a letter to Leonhard Euler in 1742. Euler, a Swiss mathematician who published over 800 |
papers, replied, “I consider [the conjecture] a theorem which is quite true, although I cannot demonstrate it.” Determine if each conjecture is true. If not, write or draw a counterexample. 24. Points X, Y, and Z are coplanar. 25. If n is an integer, then –n is positive. 26. In a triangle with one right angle, two of the sides are congruent. 27. If BD bisects ∠ABC, then m∠ABD = m∠CBD. 28. Estimation The Westside High School band is selling coupon books to raise money for a trip. The table shows the amount of money raised for the first four days of the sale. If the pattern continues, estimate the amount of money raised during the sixth day. Day Money Raised ($) 1 2 3 4 146.25 195.75 246.25 295.50 29. Write each fraction in the pattern 1 _ 11 description of the fraction pattern and the resulting decimal pattern., 2 _ 11, 3 _ 11, … as a repeating decimal. Then write a 30. Math History Remember that a prime number is a whole number greater than 1 that has exactly two factors, itself and 1. Goldbach’s conjecture states that every even number greater than 2 can be written as the sum of two primes. For example, 4 = 2 + 2. Write the next five even numbers as the sum of two primes. 31. The pattern 1, 1, 2, 3, 5, 8, 13, 21, … is known as the Fibonacci sequence. Find the next three terms in the sequence and write a conjecture for the pattern. 32. Look at a monthly calendar and pick any three squares in a row—across, down, or diagonal. Make a conjecture about the number in the middle. 33. Make a conjecture about the value of 2n - 1 when n is an integer. 34. Critical Thinking The turnaround date for migrating gray whales occurs when the number of northbound whales exceeds the number of southbound whales. Make a conjecture about the turnaround date, based on the table below. What factors might affect the validity of your conjecture in the future? Migration Direction of Gray Whales Feb. 16 Feb. 17 Feb. 18 Feb. 19 Feb. 20 Feb. 21 Feb. 22 Southbound Northbound 35. Write About It Explain why a true conjecture |
about even numbers does not necessarily hold for all numbers. Give an example to support your answer. 36. This problem will prepare you for the Multi-Step TAKS Prep on page 102. a. For how many hours did the Mock Turtle do lessons on the third day? b. On what day did the Mock Turtle do 1 hour of lessons? “And how many hours a day did you do lessons?” said Alice, in a hurry to change the subject. “Ten hours the first day,” said the Mock Turtle: “nine the next, and so on.” 78 78 Chapter 2 Geometric Reasoning ������������������ 37. Which of the following conjectures is false? If x is odd, then x + 1 is even. The sum of two odd numbers is even. The difference of two even numbers is positive. If x is positive, then –x is negative. 38. A student conjectures that if x is a prime number, then x + 1 is not prime. Which of the following is a counterexample? x = 11 x = 6 x = 3 x = 2 39. The class of 2004 holds a reunion each year. In 2005, 87.5% of the 120 graduates attended. In 2006, 90 students went, and in 2007, 75 students went. About how many students do you predict will go to the reunion in 2010? 12 15 24 30 CHALLENGE AND EXTEND 40. Multi-Step Make a table of values for the rule x 2 + x + 11 when x is an integer from 1 to 8. Make a conjecture about the type of number generated by the rule. Continue your table. What value of x generates a counterexample? 41. Political Science Presidential elections are held every four years. U.S. senators are elected to 6-year terms, but only 1 __ 3 of the Senate is up for election every two years. If 1 __ 3 of the Senate is elected during a presidential election year, how many years must pass before these same senate seats are up for election during another presidential election year? 42. Physical Fitness Rob is training for the President’s Challenge physical fitness program. During his first week of training, Rob does 15 sit-ups each day. He will add 20 sit-ups to his daily routine each week. His goal is to reach 150 sit-ups per day. a. Make a table of the number of sit-ups Rob does each week from week 1 |
through week 10. b. During which week will Rob reach his goal? c. Write a conjecture for the number of sit-ups Rob does during week n. ̶̶ AB and is the ̶̶ BC. Compare m∠CAB and m∠CBA ̶̶ AB. Then construct point C so that it is not on ̶̶ AC and 43. Construction Draw same distance from A and B. Construct and compare AC and CB. Make a conjecture. SPIRAL REVIEW Determine if the given point is a solution to y = 3x - 5. (Previous course) 44. (1, 8) 45. (-2, -11) 46. (3, 4) 47. (-3.5, 0.5) Find the perimeter or circumference of each of the following. Leave answers in terms of x. (Lesson 1-5) 48. a square whose area is x 2 49. a rectangle with dimensions x and 4x - 3 50. a triangle with side lengths of x + 2 51. a circle whose area is 9π x 2 A triangle has vertices (-1, -1), (0, 1), and (4, 0). Find the coordinates for the vertices of the image of the triangle after each transformation. (Lesson 1-7) 52. (x, y) → (x, y + 2) 53. (x, y) → (x + 4, y - 1) 2- 1 Using Inductive Reasoning to Make Conjectures 79 79 Venn Diagrams Number Theory Recall that in a Venn diagram, ovals are used to represent each set. The ovals can overlap if the sets share common elements. The real number system contains an infinite number of subsets. The following chart shows some of them. Other examples of subsets are even numbers, multiples of 3, and numbers less than 6. See Skills Bank pages S53 and S81 Set Description Natural numbers The counting numbers Examples 1, 2, 3, 4, 5, … Whole numbers The set of natural numbers and 0 0, 1, 2, 3, 4, … Integers The set of whole numbers and their opposites …, -2, -1, 0, 1, 2, … Rational numbers The set of numbers that can be written as a ratio of integers - 3 _ 4, 5, -2, 0.5, 0 Irrational numbers The set of numbers that cannot |
be written as a ratio of integers π, √ 10, 8 + √ 2 Example Draw a Venn diagram to show the relationship between the set of even numbers and the set of natural numbers. The set of even numbers includes all numbers that are divisible by 2. This includes natural numbers such as 2, 4, and 6. But even numbers such as –4 and –10 are not natural numbers. So the set of even numbers includes some, but not all, elements in the set of natural numbers. Similarly, the set of natural numbers includes some, but not all, even numbers. Draw a rectangle to represent all real numbers. Draw overlapping ovals to represent the sets of even and natural numbers. You may write individual elements in each region. Try This TAKS Grades 9–11 Obj. 1, 10 Draw a Venn diagram to show the relationship between the given sets. 1. natural numbers, whole numbers whole numbers 2. odd numbers, 3. irrational numbers, integers 80 80 Chapter 2 Geometric Reasoning ���������������������������������������������������� 2-2 Conditional Statements TEKS G.3.A Geometric structure: determine the validity of a conditional statement, its converse, inverse, and contrapositive. Also G.3.C Objectives Identify, write, and analyze the truth value of conditional statements. Why learn this? To identify a species of butterfly, you must know what characteristics one butterfly species has that another does not. Write the inverse, converse, and contrapositive of a conditional statement. Vocabulary conditional statement hypothesis conclusion truth value negation converse inverse contrapositive logically equivalent statements It is thought that the viceroy butterfly mimics the bad-tasting monarch butterfly to avoid being eaten by birds. By comparing the appearance of the two butterfly species, you can make the following conjecture: If a butterfly has a curved black line on its hind wing, then it is a viceroy. Conditional Statements DEFINITION SYMBOLS VENN DIAGRAM A conditional statement is a statement that can be written in the form “if p, then q.” The hypothesis is the part p of a conditional statement following the word if. p → q The conclusion is the part q of a conditional statement following the word then. By phrasing a conjecture as an if-then statement, you can quickly identify its hypothesis and conclusion. E X A M P L E 1 Identifying |
the Parts of a Conditional Statement “If p, then q” can also be written as “if p, q,” “q, if p,” “p implies q,” and “p only if q.” Identify the hypothesis and conclusion of each conditional. A If a butterfly has a curved black line on its hind wing, then it is a viceroy. Hypothesis: A butterfly has a curved black line on its hind wing. Conclusion: The butterfly is a Viceroy. B A number is an integer if it is a natural number. Hypothesis: A number is a natural number. Conclusion: The number is an integer. 1. Identify the hypothesis and conclusion of the statement “A number is divisible by 3 if it is divisible by 6.” Many sentences without the words if and then can be written as conditionals. To do so, identify the sentence’s hypothesis and conclusion by figuring out which part of the statement depends on the other. 2- 2 Conditional Statements 81 81 ��� E X A M P L E 2 Writing a Conditional Statement Write a conditional statement from each of the following. A The midpoint M of a segment bisects the segment. The midpoint M of a segment bisects the segment. Conditional: If M is the midpoint of a segment, and conclusion. Identify the hypothesis then M bisects the segment. B The inner oval represents the hypothesis, and the outer oval represents the conclusion. Conditional: If an animal is a tarantula, then it is a spider. 2. Write a conditional statement from the sentence “Two angles that are complementary are acute.” A conditional statement has a truth value of either true (T) or false (F). It is false only when the hypothesis is true and the conclusion is false. Consider the conditional “If I get paid, I will take you to the movie.” If I don’t get paid, I haven’t broken my promise. So the statement is still true. To show that a conditional statement is false, you need to find only one counterexample where the hypothesis is true and the conclusion is false. E X A M P L E 3 Analyzing the Truth Value of a Conditional Statement Determine if each conditional is true. If false, give a counterexample. A If you live in El |
Paso, then you live in Texas. When the hypothesis is true, the conclusion is also true because El Paso is in Texas. So the conditional is true. B If an angle is obtuse, then it has a measure of 100°. You can draw an obtuse angle whose measure is not 100°. In this case, the hypothesis is true, but the conclusion is false. Since you can find a counterexample, the conditional is false. C If an odd number is divisible by 2, then 8 is a perfect square. An odd number is never divisible by 2, so the hypothesis is false. The number 8 is not a perfect square, so the conclusion is false. However, the conditional is true because the hypothesis is false. 3. Determine if the conditional “If a number is odd, then it is divisible by 3” is true. If false, give a counterexample. If the hypothesis is false, the conditional statement is true, regardless of the truth value of the conclusion. The negation of statement p is “not p,” written as ∼p. The negation of the statement “M is the midpoint of The negation of a true statement is false, and the negation of a false statement is true. Negations are used to write related conditional statements. ̶̶ AB ” is “M is not the midpoint of ̶̶ AB.” 82 82 Chapter 2 Geometric Reasoning ����������������� Related Conditionals DEFINITION SYMBOLS A conditional is a statement that can be written in the form “If p, then q.” The converse is the statement formed by exchanging the hypothesis and conclusion. The inverse is the statement formed by negating the hypothesis and the conclusion. The contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion. p → q q → p ∼p → ∼q ∼q → ∼p E X A M P L E 4 Biology Application Moth The logical equivalence of a conditional and its contrapositive is known as the Law of Contrapositive. Write the converse, inverse, and contrapositive of the conditional statement. Use the photos to find the truth value of each. If an insect is a butterfly, then it has four wings. If an insect is a butterfly, then it has four wings. Converse: If an insect has four wings, then it is a butterfly. A moth also |
is an insect with four wings. So the converse is false. Inverse: If an insect is not a butterfly, then it does not have four wings. A moth is not a butterfly, but it has four wings. So the inverse is false. Contrapositive: If an insect does not have four wings, then it is not Butterfly a butterfly. Butterflies must have four wings. So the contrapositive is true. 4. Write the converse, inverse, and contrapositive of the conditional statement “If an animal is a cat, then it has four paws.” Find the truth value of each. In the example above, the conditional statement and its contrapositive are both true, and the converse and inverse are both false. Related conditional statements that have the same truth value are called logically equivalent statements. A conditional and its contrapositive are logically equivalent, and so are the converse and inverse. Statement Example Truth Value Conditional If m∠A = 95°, then ∠A is obtuse. Converse Inverse If ∠A is obtuse, then m∠A = 95°. If m∠A ≠ 95°, then ∠A is not obtuse. Contrapositive If ∠A is not obtuse, then m∠A ≠ 95°. T F F T However, the converse of a true conditional is not necessarily false. All four related conditionals can be true, or all four can be false, depending on the statement. 2- 2 Conditional Statements 83 83 THINK AND DISCUSS 1. If a conditional statement is false, what are the truth values of its hypothesis and conclusion? 2. What is the truth value of a conditional whose hypothesis is false? 3. Can a conditional statement and its converse be logically equivalent? Support your answer with an example. 4. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the definition and give an example. 2-2 Exercises Exercises KEYWORD: MG7 2-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. The? of a conditional statement is formed by exchanging the hypothesis ̶̶̶̶ and conclusion. (converse, inverse, or contrapositive) 2. A conditional and its contrapositive are value. (logically equivalent or converses)? because they have the same truth |
̶̶̶̶ Identify the hypothesis and conclusion of each conditional. p. 81 3. If a person is at least 16 years old, then the person can drive a car. 4. A figure is a parallelogram if it is a rectangle. 5. The statement a - b < a implies that b is a positive number Write a conditional statement from each of the following. p. 82 6. Eighteen-year-olds are eligible to vote. 2 7 when 0 < a < b. 8 Determine if each conditional is true. If false, give a counterexample. p. 82 9. If three points form the vertices of a triangle, then they lie in the same plane. 10. If x > y, then ⎜x⎟ > ⎜y⎟. 11. If the season is spring, then the month is March 12. Travel Write the converse, inverse, and contrapositive of the following conditional p. 83 statement. Find the truth value of each. If Brielle drives at exactly 30 mi/h, then she travels 10 mi in 20 min. 84 84 Chapter 2 Geometric Reasoning �������������������������������������������������������������������������� Independent Practice For See Exercises Example 13–15 16–18 19–21 22–23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S6 Application Practice p. S29 PRACTICE AND PROBLEM SOLVING Identify the hypothesis and conclusion of each conditional. 13. If an animal is a tabby, then it is a cat. 14. Four angles are formed if two lines intersect. 15. If 8 ounces of cereal cost $2.99, then 16 ounces of cereal cost $5.98. Write a conditional statement from each sentence. 16. You should monitor the heart rate of a patient who is ill. 17. After three strikes, the batter is out. 18. Congruent segments have equal measures. Determine if each conditional is true. If false, give a counterexample. 19. If you subtract -2 from -6, then the result is -4. 20. If two planes intersect, then they intersect in exactly one point. 21. If a cat is a bird, then today is Friday. Write the converse, inverse, and contrapositive of each conditional statement. Find the truth value of each. 22. |
Probability If the probability of an event is 0.1, then the event is unlikely to occur. 23. Meteorology If freezing rain is falling, then the air temperature is 32°F or less. (Hint: The freezing point of water is 32°F.) Find the truth value of each statement. 24. E lies in plane R. 25. CD lies in plane F. 26. C, E, and D are coplanar. 27. Plane F contains ED. 28. B and E are collinear. 29. BC contains F and R. Draw a Venn diagram. 30. All integers are rational numbers. 31. All natural numbers are real. 32. All rectangles are quadrilaterals. 33. Plane is an undefined term. Write a conditional statement from each Venn diagram. 34. 35. 36. 37. This problem will prepare you for the Multi-Step TAKS Prep on page 102. a. Identify the hypothesis and conclusion in the Duchess’s statement. b. Rewrite the Duchess’s claim as a conditional statement. “Tut, tut, child!” said the Duchess. “Everything’s got a moral, if only you can find it.” And she squeezed herself up closer to Alice’s side as she spoke. 2- 2 Conditional Statements 85 85 �������������������������������������������������� Find a counterexample to show that the converse of each conditional is false. 38. If x = -5, then x 2 = 25. 39. If two angles are vertical angles, then they are congruent. 40. If two angles are adjacent, then they share a vertex. 41. If you use sunscreen, then you will not get sunburned. Geology Geology Mohs’ scale is used to identify minerals. A mineral with a higher number is harder than a mineral with a lower number. Mohs’ Scale Hardness Mineral Diamond is four times as hard as the next mineral on Mohs’ scale, corundum (ruby and sapphire). Use the table and the statements below for Exercises 42–47. Write each conditional and find its truth value. p: calcite q: not apatite r: a hardness of 3 s: a hardness |
less than 5 42. p → r 45. q → p 43. s → q 46. r → q 44. q → s 47. p → s 48. Critical Thinking Consider the conditional “If two angles are congruent, then they have the same measure.” Write the converse, inverse, and contrapositive and find the truth value of each. Use the related conditionals to draw a Venn diagram that represents the relationship between congruent angles and their measures 10 Talc Gypsum Calcite Fluorite Apatite Orthoclase Quartz Topaz Corundum Diamond 49. Write About It When is a conditional statement false? Explain why a true conditional statement can have a hypothesis that is false. 50. What is the inverse of “If it is Saturday, then it is the weekend”? If it is the weekend, then it is Saturday. If it is not Saturday, then it is the weekend. If it is not Saturday, then it is not the weekend. If it is not the weekend, then it is not Saturday. 51. Let a represent “Two lines are parallel to the same line,” and let b represent “The two lines are parallel.” Which symbolic statement represents the conditional “If two lines are NOT parallel, then they are parallel to the same line”? a → b b → a ∼b → a b → ∼a 52. Which statement is a counterexample for the conditional statement “If f (x) = √ 25 - x 2, then f (x) is positive”? 53. Which statement has the same truth value as its converse? If a triangle has a right angle, its side lengths are 3 centimeters, 4 centimeters, and 5 centimeters. If an angle measures 104°, then the angle is obtuse. If a number is an integer, then it is a natural number. If an angle measures 90°, then it is an acute angle. 86 86 Chapter 2 Geometric Reasoning CHALLENGE AND EXTEND For each Venn diagram, write two statements beginning with Some, All, or No. 54. 55. 56. Given: If a figure is a square, then it is a rectangle. Figure A is not a rectangle. Conclusion: Figure A is not a square. a. Draw a Venn diagram to represent the given conditional statement. Use the Venn diagram to |
explain why the conclusion is valid. b. Write the contrapositive of the given conditional. How can you use the contrapositive to justify the conclusion? 57. Multi-Step How many true conditionals can you write using the statements below? r : n is a natural number. q: n is a whole number. p: n is an integer. SPIRAL REVIEW Write a rule to describe each relationship. (Previous course) 58. x -8 y -5 4 7 7 9 10 12 59. x -3 -1 y -5 -1 0 1 4 9 60. x -2 0 y -9 -4 4 6 6 11 Determine whether each statement is true or false. If false, explain why. (Lesson 1-4) 61. If two angles are complementary and congruent, then the measure of each is 45°. 62. A pair of acute angles can be supplementary. 63. A linear pair of angles is also a pair of supplementary angles. Find the next item in each pattern. (Lesson 2-1) 64. 1, 13, 131, 1313, …, 2 _, 2 _ 65. 2, 2 _ 27 9 3, … 66. x: What high school math classes did you take? A: I took three years of math: Pre-Algebra, Algebra, and Geometry. KEYWORD: MG7 Career Q: What training do you need to be a desktop publisher? A: Most of my training was done on the job. The computer science and typing classes I took in high school have been helpful. Q: How do you use math? A: Part of my job is to make sure all the text, charts, and photographs are formatted to fit the layout of each page. I have to manipulate things by comparing ratios, calculating areas, and using estimation. Stephanie Poulin Desktop Publisher Daily Reporter Q: What future plans do you have? A: My goal is to start my own business as a freelance graphic artist. 2- 2 Conditional Statements 87 87 ������������������������� 2-3 Using Deductive Reasoning to Verify Conjectures TEKS G.3.E Geometric structure: use deductive reasoning to prove a statement. Also G.2.B, G.3.B, G.3.C Objective Apply the Law of Detachment and the Law of Syllogism in logical reasoning. Vocabulary deductive reasoning Why learn this? You can use inductive |
and deductive reasoning to decide whether a common myth is accurate. You learned in Lesson 2-1 that one counterexample is enough to disprove a conjecture. But to prove that a conjecture is true, you must use deductive reasoning. Deductive reasoning is the process of using logic to draw conclusions from given facts, definitions, and properties. E X A M P L E 1 Media Application Urban legends and modern myths spread quickly through the media. Many Web sites and television shows are dedicated to confirming or disproving such myths. Is each conclusion a result of inductive or deductive reasoning? A There is a myth that toilets and sinks drain in opposite directions in the Southern and Northern Hemispheres. However, if you were to observe sinks draining in the two hemispheres, you would see that this myth is false. Since the conclusion is based on a pattern of observation, it is a result of inductive reasoning. B There is a myth that you should not touch a baby bird that has fallen from its nest because the mother bird will disown the baby if she detects human scent. However, biologists have shown that birds cannot detect human scent. Therefore, the myth cannot be true. The conclusion is based on logical reasoning from scientific research. It is a result of deductive reasoning. 1. There is a myth that an eelskin wallet will demagnetize credit cards because the skin of the electric eels used to make the wallet holds an electric charge. However, eelskin products are not made from electric eels. Therefore, the myth cannot be true. Is this conclusion a result of inductive or deductive reasoning? In deductive reasoning, if the given facts are true and you apply the correct logic, then the conclusion must be true. The Law of Detachment is one valid form of deductive reasoning. 88 88 Chapter 2 Geometric Reasoning Law of Detachment If p → q is a true statement and p is true, then q is true. E X A M P L E 2 Verifying Conjectures by Using the Law of Detachment Determine if each conjecture is valid by the Law of Detachment. A Given: If two segments are congruent, then they have the same length. ̶̶ XY. ̶̶ AB ≅ Conjecture: AB = XY Identify the hypothesis and conclusion in the given conditional. If two segments are congruent, then they have the same length. The given statement conditional. By the Law of Detachment |
AB = XY. The conjecture is valid. ̶̶ XY matches the hypothesis of a true ̶̶ AB ≅ B Given: If you are tardy 3 times, you must go to detention. Shea is in detention. Conjecture: Shea was tardy at least 3 times. Identify the hypothesis and conclusion in the given conditional. If you are tardy 3 times, you must go to detention. The given statement “Shea is in detention” matches the conclusion of a true conditional. But this does not mean the hypothesis is true. Shea could be in detention for another reason. The conjecture is not valid. 2. Determine if the conjecture is valid by the Law of Detachment. Given: If a student passes his classes, the student is eligible to play sports. Ramon passed his classes. Conjecture: Ramon is eligible to play sports. Another valid form of deductive reasoning is the Law of Syllogism. It allows you to draw conclusions from two conditional statements when the conclusion of one is the hypothesis of the other. Law of Syllogism If p → q and q → r are true statements, then p → r is a true statement. E X A M P L E 3 Verifying Conjectures by Using the Law of Syllogism Determine if each conjecture is valid by the Law of Syllogism. A Given: If m∠ A < 90°, then ∠ A is acute. If ∠ A is acute, then it is not a right angle. Conjecture: If m∠ A < 90°, then it is not a right angle. Let p, q, and r represent the following. p: The measure of an angle is less than 90°. q: The angle is acute. r : The angle is not a right angle. You are given that p → q and q → r. Since q is the conclusion of the first conditional and the hypothesis of the second conditional, you can conclude that p → r. The conjecture is valid by the Law of Syllogism. 2- 3 Using Deductive Reasoning to Verify Conjectures 89 89 It is possible to arrive at a true conclusion by applying invalid logical reasoning, as in Example 3B. Determine if each conjecture is valid by the Law of Syllogism. B Given: If a number is divisible by 4, then it is divisible by 2. If a number is even, then it is |
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