Discovering Geometry An Investigative Approach

Discovering Geometry An Investigative Approach Practice Your Skills with AnswersTeacher’s Materials Project Editor: Elizabeth DeCarli Project Administrator: Brady Golden Coordinating Writer: Jennifer North Morris C ...

Discovering Geometry An Investigative Approach Practice Your Skills with AnswersTeacher’s Materials Project Editor: Elizabeth DeCarli Project Administrator: Brady Golden Coordinating Writer: Jennifer North Morris Contributors: David Rasmussen, Ralph Bothe, Judy Hicks, Michael Serra Accuracy Checker: Dudley Brooks Production Editor: Holly Rudelitsch Copyeditor: Jill Pellarin Editorial Production Manager: Christine Osborne Production Supervisor: Ann Rothenbuhler Production Coordinator: Jennifer Young Text Designers: Jenny Somerville, Garry Harman Composition, Technical Art, Prepress: ICC Macmillan Inc. Cover Designer: Jill Kongabel Printer: Data Reproductions Textbook Product Manager: James Ryan Executive Editor: Casey FitzSimons Publisher: Steven Rasmussen ©2008 by Key Curriculum Press. All rights reserved. Cover Photo Credits: Background image: Doug Wilson/Westlight/Corbis. Construction site image: Sonda Dawes/The Image Works. All other images: Ken Karp Photography. Limited Reproduction Permission The publisher grants the teacher whose school has adopted Discovering Geometry, and who has received Discovering Geometry: An Investigative Approach, Practice Your Skills with Answers as part of the Teaching Resources package for the book, the right to reproduce material for use in his or her own classroom. Unauthorized copying of Discovering Geometry: An Investigative Approach, Practice Your Skills with Answers constitutes copyright infringement and is a violation of federal law. ®Key Curriculum Press is a registered trademark of Key Curriculum Press. All registered trademarks and trademarks in this book are the property of their respective holders. Key Curriculum Press 1150 65th Street Emeryville, CA 94608 510-595-7000 [email protected] www.keypress.com Printed in the United States of America 10 9 8 7 6 5 4 3 2 13 12 11 10 09 08 ISBN 978-1-55953-894-7Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Chapter 1 Lesson 1.1: Building Blocks of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Lesson 1.2: Poolroom Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Lesson 1.3: What’s a Widget? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Lesson 1.4: Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Lesson 1.5: Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Lesson 1.6: Special Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Lesson 1.7: Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Lesson 1.8: Space Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Lesson 1.9: A Picture Is Worth a Thousand Words . . . . . . . . . . . . . . . . . . . 9 Chapter 2 Lesson 2.1: Inductive Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Lesson 2.2: Finding the nth Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Lesson 2.3: Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Lesson 2.4: Deductive Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Lesson 2.5: Angle Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Lesson 2.6: Special Angles on Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter 3 Lesson 3.1: Duplicating Segments and Angles . . . . . . . . . . . . . . . . . . . . 16 Lesson 3.2: Constructing Perpendicular Bisectors . . . . . . . . . . . . . . . . . . 17 Lesson 3.3: Constructing Perpendiculars to a Line . . . . . . . . . . . . . . . . . 18 Lesson 3.4: Constructing Angle Bisectors . . . . . . . . . . . . . . . . . . . . . . . . . 19 Lesson 3.5: Constructing Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Lesson 3.6: Construction Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Lesson 3.7: Constructing Points of Concurrency . . . . . . . . . . . . . . . . . . . 22 Lesson 3.8: The Centroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Chapter 4 Lesson 4.1: Triangle Sum Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Lesson 4.2: Properties of Isosceles Triangles . . . . . . . . . . . . . . . . . . . . . . 25 Lesson 4.3: Triangle Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Lesson 4.4: Are There Congruence Shortcuts? . . . . . . . . . . . . . . . . . . . . . 27 Lesson 4.5: Are There Other Congruence Shortcuts? . . . . . . . . . . . . . . . 28 Lesson 4.6: Corresponding Parts of Congruent Triangles . . . . . . . . . . . 29 Lesson 4.7: Flowchart Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Lesson 4.8: Proving Special Triangle Conjectures . . . . . . . . . . . . . . . . . . 31 iii Contentsiv Chapter 5 Lesson 5.1: Polygon Sum Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Lesson 5.2: Exterior Angles of a Polygon . . . . . . . . . . . . . . . . . . . . . . . . . 33 Lesson 5.3: Kite and Trapezoid Properties . . . . . . . . . . . . . . . . . . . . . . . . 34 Lesson 5.4: Properties of Midsegments . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Lesson 5.5: Properties of Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Lesson 5.6: Properties of Special Parallelograms . . . . . . . . . . . . . . . . . . 37 Lesson 5.7: Proving Quadrilateral Properties . . . . . . . . . . . . . . . . . . . . . . 38 Chapter 6 Lesson 6.1: Tangent Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Lesson 6.2: Chord Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Lesson 6.3: Arcs and Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Lesson 6.4: Proving Circle Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Lesson 6.5: The Circumference/Diameter Ratio . . . . . . . . . . . . . . . . . . . . 43 Lesson 6.6: Around the World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Lesson 6.7: Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Exploration: Intersecting Secants, Tangents, and Chords . . . . . . . . . . . 46 Chapter 7 Lesson 7.1: Transformations and Symmetry . . . . . . . . . . . . . . . . . . . . . . . 47 Lesson 7.2: Properties of Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Lesson 7.3: Compositions of Transformations . . . . . . . . . . . . . . . . . . . . . 49 Lesson 7.4: Tessellations with Regular Polygons . . . . . . . . . . . . . . . . . . . 50 Lessons 7.5–7.8: Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Chapter 8 Lesson 8.1: Areas of Rectangles and Parallelograms . . . . . . . . . . . . . . . 52 Lesson 8.2: Areas of Triangles, Trapezoids, and Kites . . . . . . . . . . . . . . . 53 Lesson 8.3: Area Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Lesson 8.4: Areas of Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Lesson 8.5: Areas of Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Lesson 8.6: Any Way You Slice It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Lesson 8.7: Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Chapter 9 Lesson 9.1: The Theorem of Pythagoras . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Lesson 9.2: The Converse of the Pythagorean Theorem . . . . . . . . . . . . 60 Lesson 9.3: Two Special Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Lesson 9.4: Story Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Lesson 9.5: Distance in Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . 63 Lesson 9.6: Circles and the Pythagorean Theorem . . . . . . . . . . . . . . . . . 64v Chapter 10 Lesson 10.1: The Geometry of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Lesson 10.2: Volume of Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . 66 Lesson 10.3: Volume of Pyramids and Cones . . . . . . . . . . . . . . . . . . . . . . 67 Lesson 10.4: Volume Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Lesson 10.5: Displacement and Density . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Lesson 10.6: Volume of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Lesson 10.7: Surface Area of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Chapter 11 Lesson 11.1: Similar Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Lesson 11.2: Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Lesson 11.3: Indirect Measurement with Similar Triangles . . . . . . . . . . 74 Lesson 11.4: Corresponding Parts of Similar Triangles . . . . . . . . . . . . . 75 Lesson 11.5: Proportions with Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Lesson 11.6: Proportions with Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Lesson 11.7: Proportional Segments Between Parallel Lines . . . . . . . . 78 Chapter 12 Lesson 12.1: Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Lesson 12.2: Problem Solving with Right Triangles . . . . . . . . . . . . . . . . 80 Lesson 12.3: The Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Lesson 12.4: The Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Lesson 12.5: Problem Solving with Trigonometry . . . . . . . . . . . . . . . . . . 83 Chapter 13 Lesson 13.1: The Premises of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Lesson 13.2: Planning a Geometry Proof . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Lesson 13.3: Triangle Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Lesson 13.4: Quadrilateral Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Lesson 13.5: Indirect Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Lesson 13.6: Circle Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Lesson 13.7: Similarity Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91vi Introduction The author and editors of Discovering Geometry: An Investigative Approach are aware of the importance of students developing geometry skills along with acquiring concepts through investigation. The student book includes many skill-based exercises. These Practice Your Skills worksheets provide problems similar to the introductory exercises in each lesson of Discovering Geometry. Like those exercises, these worksheets allow students to practice and reinforce the important procedures and skills developed in the lessons. Some of these problems provide non contextual skills practice. Others give students an opportunity to apply geometry concepts in fairly simple, straightforward contexts. Some are more complex problems that are broken down into small steps. You might assign the Practice Your Skills worksheet for every lesson, or only for those lessons your students find particularly difficult. Or, you may wish to assign the worksheets on an individual basis, only to those students who need extra help. One worksheet has been provided for nearly every lesson. There are no worksheets for Chapter 0, and the optional tessellation lessons have been combined into two worksheets. To save you the time and expense of copying pages, you can give students the inexpensive Practice Your Skills Student Workbook, which does not have answers. Though the copyright allows you to copy pages from Practice Your Skills with Answers for use with your students, the consumable Practice Your Skills Student Workbook should not be copied. Students, parents, and mentors can also download the student worksheets from www.keymath.com.Lesson 1.1 • Building Blocks of Geometry