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.

2 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 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.

3 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 .. 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 .

4 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.

5 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 .. 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.

6 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.

7 75. Lesson : Exterior Angles of a Polygon .. 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.

8 107. 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 .. 127. Lesson : Tessellations with Nonregular Polygons .. 129. Lesson : Tessellations Using Only Translations .. 130. Lesson : Tessellations That Use Rotations.

9 130. 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.

10 155. 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 .. 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.