Discovering Geometry - Flourish

1 DG3SM586_ 8/2/06 6:45 PM Page i Discovering Geometry An Investigative Approach Solutions Manual DG3SM586_ 1/3/07 4:34 PM Page ii Teacher's Materials Project Editor: Joan Lewis Editor: Heather Dever Project Manager: Erin Gray Writers: Abby Tanenbaum, Stacey Miceli Editorial Assistants: Lori Dixon, Karen Rodriguez'G. Accuracy Checker: Dudley Brooks Production Editors: Jacqueline Gamble, Angela Chen Copyeditor: Margaret Moore Editorial Production Manager: Deborah Cogan Production Director: Diana Jean Parks Production Coordinator: Ann Rothenbuhler Text Designers: Jenny Somerville, Garry Harman Art Editor: Jason Luz Composition, Technical Art: Interactive Composition Corporation Art and Design Coordinator: Caroline Ayres Cover Designer: Jill Kongabel Printer, Prepress: Data Reproductions Executive Editor: Casey FitzSimons Publisher: Steven Rasmussen 2003 by Key Curriculum Press. All rights reserved. Cover Photo Credits: Background image: Doug Wilson/Westlight/Corbis.

2 Construction site image: Sonda Dawes/The Image Works. All other images: Ken Karp Photography. Limited Reproduction Permission The publisher grants the teacher who purchases Discovering Geometry : An Investigative Approach, Solutions Manual the right to reproduce material for use in his or her own classroom. Unauthorized copying of Discovering Geometry : An Investigative Approach, Solutions Manual 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. Printed in the United States of America 10 9 8 7 6 09 08 07 ISBN 978-1-55953-586-1. DG3SM586_ 8/2/06 6:45 PM Page iii Introduction Contents Introduction .. vii Chapter 0. Lesson : Geometry in Nature and in Art .. 1. Lesson : Line Designs .. 1. Lesson : Circle Designs.

3 2. Lesson : Op Art .. 4. Lesson : Knot Designs .. 4. Lesson : Islamic Tile Designs .. 5. Chapter 0 Review .. 6. Chapter 1. Lesson : Building Blocks of Geometry .. 7. Using Your Algebra Skills 1: Midpoint .. 8. Lesson : Poolroom Math .. 9. Lesson : What's a Widget? .. 11. Lesson : Polygons .. 13. Lesson : Triangles and Special Quadrilaterals .. 15. Lesson : Circles .. 17. Lesson : A Picture Is Worth a Thousand Words .. 19. Lesson : Space Geometry .. 21. Chapter 1 Review .. 24. Chapter 2. Lesson : Inductive Reasoning .. 25. Lesson : Deductive Reasoning .. 27. Lesson : Finding the nth Term .. 29. Lesson : Mathematical Modeling .. 30. Lesson : Angle Relationships .. 32. Lesson : Special Angles on Parallel Lines .. 34. Using Your Algebra Skills 2: Slope .. 35. Chapter 2 Review .. 36. Chapter 3. Lesson : Duplicating Segments and Angles .. 37. Lesson : Constructing Perpendicular Bisectors .. 39. Lesson : Constructing Perpendiculars to a Line .. 41. Lesson : Constructing Angle Bisectors.

4 43. Lesson : Constructing Parallel Lines .. 45. Using Your Algebra Skills 3: Slopes of Parallel and Perpendicular Lines .. 46. iii DG3SM586_ 8/2/06 6:45 PM Page iv Lesson : Construction Problems .. 47. Exploration: Perspective Drawing .. 50. Lesson : Constructing Points of Concurrency .. 50. Lesson : The Centroid .. 52. Exploration: The Euler Line .. 54. Chapter 3 Review .. 54. Chapter 4. Lesson : Triangle Sum Conjecture .. 56. Lesson : Properties of Special Triangles .. 58. Using Your Algebra Skills 4: Writing Linear Equations .. 61. Lesson : Triangle Inequalities .. 63. Lesson : Are There Congruence Shortcuts? .. 64. Lesson : Are There Other Congruence Shortcuts? .. 66. Lesson : Corresponding Parts of Congruent Triangles .. 68. Lesson : Flowchart Thinking .. 69. Lesson : Proving Isosceles Triangle Conjectures .. 71. Exploration: Napoleon's Theorem .. 72. Chapter 4 Review .. 72. Take Another Look .. 74. Chapter 5. Lesson : Polygon Sum Conjecture .. 75. Lesson : Exterior Angles of a Polygon.

5 78. Lesson : Kite and Trapezoid Properties .. 79. Lesson : Properties of Midsegments .. 81. Lesson : Properties of Parallelograms .. 84. Using Your Algebra Skills 5: Solving Systems of Linear Equations .. 86. Lesson : Properties of Special Parallelograms .. 88. Lesson : Proving Quadrilateral Properties .. 91. Chapter 5 Review .. 93. Take Another Look .. 96. Chapter 6. Lesson : Chord Properties .. 97. Lesson : Tangent Properties .. 99. Lesson : Arcs and Angles .. 102. Lesson : Proving Circle Conjectures .. 105. Using Your Algebra Skills 6: Finding the Circumcenter .. 107. Lesson : The Circumference/Diameter Ratio .. 110. Lesson : Around the World .. 112. Lesson : Arc Length .. 113. Exploration: Cycloids .. 115. Chapter 6 Review .. 116. Take Another Look .. 120. iv DG3SM586_ 8/2/06 6:45 PM Page v Chapter 7. Lesson : Transformations and Symmetry .. 120. Lesson : Properties of Isometries .. 122. Lesson : Compositions of Transformations .. 125. Lesson : Tessellations with Regular Polygons.

6 127. Lesson : Tessellations with Nonregular Polygons .. 129. Lesson : Tessellations Using Only Translations .. 130. Lesson : Tessellations That Use Rotations .. 130. Lesson : Tessellations That Use Glide Reflections .. 131. Using Your Algebra Skills 7: Finding the Orthocenter and Centroid .. 132. Chapter 7 Review .. 134. Chapter 8. Lesson : Areas of Rectangles and Parallelograms .. 136. Lesson : Areas of Triangles, Trapezoids, and Kites .. 138. Lesson : Area Problems .. 142. Lesson : Areas of Regular Polygons .. 144. Lesson : Areas of Circles .. 146. Lesson : Any Way You Slice It .. 147. Exploration: Geometric Probability II .. 149. Lesson : Surface Area .. 149. Exploration: Alternative Area Formulas .. 151. Chapter 8 Review .. 152. Take Another Look .. 155. Chapter 9. Lesson : The Theorem of Pythagoras .. 156. Lesson : The Converse of the Pythagorean Theorem .. 159. Using Your Algebra Skills 8: Radical Expressions .. 161. Lesson : Two Special Right Triangles.

7 162. Lesson : Story Problems .. 165. Lesson : Distance in Coordinate Geometry .. 168. Lesson : Circles and the Pythagorean Theorem .. 171. Chapter 9 Review .. 175. Take Another Look .. 181. Chapter 10. Lesson : The Geometry of Solids .. 182. Lesson : Volume of Prisms and Cylinders .. 183. Lesson : Volume of Pyramids and Cones .. 186. Lesson : Volume Problems .. 189. Lesson : Displacement and Density .. 191. Lesson : Volume of a Sphere .. 194. Lesson : Surface Area of a Sphere .. 197. v DG3SM586_ 8/2/06 6:45 PM Page vi Chapter 10 Review .. 199. Take Another Look .. 203. Chapter 11. Using Your Algebra Skills 9: Proportion and Reasoning .. 203. Lesson : Similar Polygons .. 205. Lesson : Similar Triangles .. 208. Lesson : Indirect Measurement with Similar Triangles .. 212. Lesson : Corresponding Parts of Similar Triangles .. 215. Lesson : Proportions with Area and Volume .. 219. Exploration: Why Elephants Have Big Ears .. 222. Lesson : Proportional Segments Between Parallel Lines.

8 222. Exploration: Two More Forms of Valid Reasoning .. 226. Chapter 11 Review .. 226. Take Another Look .. 228. Chapter 12. Lesson : Trigonometric Ratios .. 229. Lesson : Problem Solving with Right Triangles .. 231. Exploration: Indirect Measurement .. 234. Lesson : The Law of Sines .. 234. Lesson : The Law of Cosines .. 237. Lesson : Problem Solving with Trigonometry .. 241. Exploration: Trigonometric Ratios and the Unit Circle .. 244. Chapter 12 Review .. 245. Take Another Look .. 251. Chapter 13. Lesson : The Premises of Geometry .. 252. Lesson : Planning a Geometry Proof .. 254. Lesson : Triangle Proofs .. 257. Lesson : Quadrilateral Proofs .. 264. Exploration: Proof as Challenge and Discovery .. 272. Lesson : Indirect Proof .. 272. Lesson : Circle Proofs .. 273. Lesson : Similarity Proofs .. 278. Using Your Algebra Skills 10: Coordinate Proof .. 286. Chapter 13 Review .. 291. vi DG3SM586_ 8/2/06 6:45 PM Page vii Introduction The Solutions Manual for Discovering Geometry : An Investigative Approach contains solutions to the exercises at the end of each lesson and to the Extensions, Improving Your Reasoning Skills, Improving Your Visual Thinking Skills, Improving Your Algebra Skills, Projects, and Take Another Look activities.

9 You can find solutions for the Investigations and Explorations in the Teacher's Edition. The solutions in this Solutions Manual are more complete than those offered as annotations in the Teacher's Edition. Although complete solutions for the problems are provided here, keep in mind that often there is more than one method students might use to solve a particular problem. Also, the answers will vary for some problems, depending on assumptions that students make. For problems that could have many different answers, a sample solution is given. In order to save space, the size of constructions has been reduced in the Solutions Manual. Keep this in mind as you check student constructions. Refer to these solutions when your students have difficulty solving a problem and need some assistance in determining a possible approach toward solving it. You might also want to provide a copy of certain solutions for students who have been absent for an extended period of time.

10 Vii DG3SM586_ 8/2/06 6:45 PM Page viii 8/2/06 5:14 PM Page 1. sections parallel to the table are circular (and CHAPTER 0 therefore have both reflectional and rotational symmetry). Cross sections perpendicular to the LESSON table are fairly round for an orange, but not for an apple or a pear. For all three fruits, these perpendi- EXERCISES cular cross sections have approximate reflectional 1. Possible answers: snowflakes and crystals; flowers symmetry, but only the orange has rotational and starfish symmetry. You can slice a banana to get a round 2. Possible answers: to create shapes and patterns, cross section that has both reflectional and to show perspective and proportions, to create rotational symmetry. You can also slice it to create optical illusions a long, skinny cross section that may or may not have reflectional symmetry, depending on the 3. Possible answers: a butterfly, a valentine heart, a shape of the banana. person; bilateral symmetry B. Possible answer: Bilateral symmetry gives humans 4.