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DG4SE 882 FM qxd 10 13 06 9 33 AM Page ii,Project Editor Compositor and Technical Art. Andres Marti Interactive Composition Corporation,Project Administrator Printer. Brady Golden Von Hoffmann Corp,Consulting Editors, Christian Aviles Scott Elizabeth DeCarli Kendra Lockman Textbook Product Manager. Ladie Malek James Ryan,Mathematics Reviewers Executive Editor. Larry Copes Inner Grove Heights Minnesota Casey FitzSimons. Michael de Villiers Ph D University of Durban Westville. Pinetown South Africa Publisher, David Rasmussen Neil s Harbour Nova Scotia Steven Rasmussen.

Abby Tanenbaum Naperville Illinois,Teacher Reviewers. Rich Crandall Skyline High School Oakland California. Genie Dunn Miami Killian Senior High School Miami, Judy Hicks Ralston Valley High School Arvada Colorado. Susan May University of Texas Austin Texas,Steve Phelps Madeira High School Cincinnati Ohio. Adella Pietrzyk Center Line High School Center Line. Multicultural and Equity Reviewers, David Keiser Montclair State University Upper Montclair. New Jersey, Swapna Mukhopadhyay Ph D San Diego State University.

San Diego California,Accuracy Checkers,Dudley Brooks Marcia Ellen Olmstead. Editorial Production Manager,Christine Osborne,2008 by Michael Serra All rights reserved. Print Production Supervisor, No part of this publication may be reproduced stored in. Ann Rothenbuhler a retrieval system or transmitted in any form or by any. means electronic photocopying recording or otherwise. Production Editor without the prior written permission of the publisher. Angela Chen,The Geometer s Sketchpad Dynamic Geometry and. Production Coordinator Key Curriculum Press are registered trademarks of. Jennifer Young Key Curriculum Press The Discovering Mathematics logo. and Sketchpad are trademarks of Key Curriculum Press. Copyeditor Fathom Dynamic Data is a trademark of,Jill Pellarin KCP Technologies.

All other trademarks are held by their respective owners. Cover Designers,Key Curriculum Press, Jill Kongabel Marilyn Perry Jensen Barnes 1150 65th Street. Emeryville CA 94608,Text Designer editorial keypress com. Marilyn Perry www keypress com,Printed in the United States of America. Illustrators, Juan Alvarez Andy Levine Claudia Newell Bill Pasini 10 9 8 7 6 5 4 3 2 1 11 10 09 08 07. William Rieser Sue Todd Rose Zgodzinski ISBN 978 1 55953 882 4. DG4SE 882 FM qxd 10 13 06 9 33 AM Page iii,Acknowledgments.

F irst to all the teachers who have used Discovering Geometry a sincere thank you. for your wonderful support and encouragement I wish to thank my always. delightful and ever so patient students for their insight humor and hard work. I also wish to thank the many students across the country who have written to me. with their kind words comments and suggestions And thanks to the marketing. and sales staff at Key Curriculum Press for their successful efforts in bringing the. first three editions into so many classrooms, There are three people who have added their touch to earlier editions of Discovering. Geometry Steve Rasmussen was editor on the first edition Dan Bennett was editor. on the second edition and Ladie Malek was editor on the third edition Thank you. Steve Dan and Ladie, While working on this fourth edition of Discovering Geometry I was fortunate to. have the assistance of Andres Marti as project editor Thank you Andres for your. commitment to the pedagogy in Discovering Geometry your cooperative. temperament in the editorial process and your patience with me To the editorial. and production staff and managers at Key the field testers the advisors the. consultants and the reviewers Rich Crandall Genie Dunn Judy Hicks Susan May. Steve Phelps and Adella Pietrzyk I am grateful for your quality work. Michael Serra,DG4SE 882 FM qxd 10 13 06 9 34 AM Page iv. A Note from the Publisher, Y ou probably know from your experience in school that you learn best when you. understand the concepts and are actively engaged in the learning process. Studies confirm that understanding concepts and getting involved are important in. every subject at every grade level This is the foundation that Discovering Geometry. is built on and investigations are at its heart, If you are a student you ll discover many important mathematical principles.

by working together with other students in supportive small groups doing. investigations First you ll explore concepts visually looking for patterns and then. you ll learn to explain why these patterns exist This approach is both challenging. and fun and will make you believe in your ability to succeed at mathematics. Rather than memorizing theorems and formulas you ll learn how to build what you. need from what you already know You ll find that Discovering Geometry is easy to. follow and the design includes many interesting photographs and illustrations that. connect geometry with art architecture science history culture and recreation. If you are a parent you ll appreciate that Discovering Geometry continues to be the. leader in providing a successful discovery based approach You ll know that your. student is actively engaged in the process of learning and constructing his or her. own understanding of concepts developing insight confidence and increasingly. sophisticated mathematical understanding along the way The effectiveness of. Discovering Geometry s approach has been substantiated in thousands of classrooms. by millions of students and has changed the way geometry is taught today If you. go to www keymath com DG you ll find Dynamic Geometry Explorations that. explore important geometry concepts interactively condensed lessons for when. your student is absent downloadable worksheets to help your student practice his. or her skills and resources designed especially for you as a parent. If you are a teacher you ll see that this new edition balances the investigative. approach that is at the heart of the Discovering Mathematics series with an. emphasis on developing students ability to reason deductively You ll be supported. by a deliberate pedagogy based on educational research that carefully develops your. students ability to make sound logical arguments Discovering Geometry introduces. students to reasoning strategies that help them explain their discoveries and enable. them to justify their conjectures through proof There are also more opportunities. to review algebra more ways to use technology especially The Geometer s. Sketchpad and there is an enhanced online textbook and other computer based. resources If you are familiar with earlier editions of Discovering Geometry you ll. still find the original and hallmark features plus improvements based on feedback. from many of your colleagues in geometry classrooms. If you are a student we believe that as you work through this course you ll gain. knowledge for a lifetime If you are a parent we believe you ll enjoy watching your. student develop mathematical power If you are a teacher we believe you ll find. that Discovering Geometry makes a significant positive impact in your classroom. Whether you are learning guiding or teaching we wish you success and urge you. to continue your involvement and interest in mathematics. Steve Rasmussen President,Key Curriculum Press,DG4SE 882 FM qxd 10 13 06 9 34 AM Page v. A Note to Students from the Author xiv,CHAPTER Geometric Art 1. 0 1 Geometry in Nature and in Art 2,0 2 Line Designs 7. 0 3 Circle Designs 10,0 4 Op Art 13,0 5 Knot Designs 16. Project Symbolic Art 19,0 6 Islamic Tile Designs 20.

Project Photo or Video Safari 23,Chapter 0 Review 24. Assessing What You ve Learned 26,CHAPTER Introducing Geometry 27. 1 1 Building Blocks of Geometry 28,Investigation Mathematical Models 32. Project Spiral Designs 35,Using Your Algebra Skills 1 Midpoint 36. 1 2 Poolroom Math 38,Investigation Virtual Pool 41.

1 3 What s a Widget 47,Investigation Defining Angles 49. 1 4 Polygons 54,Investigation Special Polygons 56,1 5 Triangles 59. Investigation Triangles 60,1 6 Special Quadrilaterals 64. Investigation Special Quadrilaterals 64,Project Drawing the Impossible 68. 1 7 Circles 69,Investigation Defining Circle Terms 70.

1 8 Space Geometry 75,Investigation Space Geometry 77. 1 9 A Picture Is Worth a Thousand Words 81,Exploration Geometric Probability I 88. Activity Chances Are 88,Chapter 1 Review 90,Assessing What You ve Learned 94. DG4SE 882 FM qxd 10 13 06 9 34 AM Page vi,CHAPTER Reasoning in Geometry 95. 2 1 Inductive Reasoning 96,Investigation Shape Shifters 98.

2 2 Finding the nth Term 102,Investigation Finding the Rule 102. Project Best Fit Lines 107,2 3 Mathematical Modeling 108. Investigation Party Handshakes 108,2 4 Deductive Reasoning 114. Investigation Overlapping Segments 116,Exploration The Seven Bridges of K nigsberg 120. Activity Traveling Networks 120,2 5 Angle Relationships 122.

Investigation 1 The Linear Pair Conjecture 122,Investigation 2 Vertical Angles Conjecture 123. 2 6 Special Angles on Parallel Lines 128,Investigation 1 Which Angles Are Congruent 128. Investigation 2 Is the Converse True 130,Project Line Designs 134. Using Your Algebra Skills 2 Slope 135,Exploration Patterns in Fractals 137. Activity The Sierpin ski Triangle 138,Chapter 2 Review 140.

Assessing What You ve Learned 142,CHAPTER Using Tools of Geometry 143. 3 1 Duplicating Segments and Angles 144,Investigation 1 Duplicating a Segment 145. Investigation 2 Duplicating an Angle 146,3 2 Constructing Perpendicular Bisectors 149. Investigation 1 Finding the Right Bisector 149, Investigation 2 Constructing the Perpendicular Bisector 150. 3 3 Constructing Perpendiculars to a Line 154,Investigation 1 Finding the Right Line 154.

Investigation 2 Patty Paper Perpendiculars 155,Project Constructing a Tile Design 158. 3 4 Constructing Angle Bisectors 159,Investigation 1 Angle Bisecting by Folding 159. Investigation 2 Angle Bisecting with Compass 160,3 5 Constructing Parallel Lines 163. Investigation Constructing Parallel Lines by Folding 163. Using Your Algebra Skills 3 Slopes of Parallel and. Perpendicular Lines 167,3 6 Construction Problems 170. Exploration Perspective Drawing 174,Activity Boxes in Space 175.

DG4SE 882 FM qxd 10 13 06 9 34 AM Page vii,3 7 Constructing Points of Concurrency 178. Investigation 1 Concurrence 178,Investigation 2 Circumcenter 179. Investigation 3 Incenter 180,3 8 The Centroid 185,Investigation 1 Are Medians Concurrent 185. Investigation 2 Balancing Act 186,Exploration The Euler Line 191. Activity Three Out of Four 191,Project Is There More to the Orthocenter 192.

Chapter 3 Review 193,Mixed Review 196,Assessing What You ve Learned 198. CHAPTER Discovering and Proving Triangle Properties 199. 4 1 Triangle Sum Conjecture 200,Investigation The Triangle Sum 200. 4 2 Properties of Isosceles Triangles 206, Investigation 1 Base Angles in an Isosceles Triangle 207. Investigation 2 Is the Converse True 208,Using Your Algebra Skills 4 Solving Equations 212. 4 3 Triangle Inequalities 215, Investigation 1 What Is the Shortest Path from A to B 216.

Investigation 2 Where Are the Largest and Smallest Angles 217. Investigation 3 Exterior Angles of a Triangle 217,Project Random Triangles 220. 4 4 Are There Congruence Shortcuts 221,Investigation 1 Is SSS a Congruence Shortcut 222. Investigation 2 Is SAS a Congruence Shortcut 223,Investigation 3 Is SSA a Congruence Shortcut 223. 4 5 Are There Other Congruence Shortcuts 227,Investigation 1 Is ASA a Congruence Shortcut 227. Investigation 2 Is SAA a Congruence Shortcut 228,Investigation 3 Is AAA a Congruence Shortcut 228.

4 6 Corresponding Parts of Congruent Triangles 232. Project Polya s Problem 236,4 7 Flowchart Thinking 237. 4 8 Proving Special Triangle Conjectures 243, Investigation The Symmetry Line in an Isosceles Triangle 244. Exploration Napoleon s Theorem 249,Activity Napoleon Triangles 249. Project Lines and Isosceles Triangles 250,Chapter 4 Review 251. Take Another Look 255,Assessing What You ve Learned 256.

DG4SE 882 FM qxd 10 13 06 9 34 AM Page viii, CHAPTER Discovering and Proving Polygon Properties 257. 5 1 Polygon Sum Conjecture 258,Investigation Is There a Polygon Sum Formula 258. 5 2 Exterior Angles of a Polygon 262,Investigation Is There an Exterior Angle Sum 262. Exploration Star Polygons 266,Activity Exploring Star Polygons 266. 5 3 Kite and Trapezoid Properties 268, Investigation 1 What Are Some Properties of Kites 268.

Investigation 2 What Are Some Properties of Trapezoids 270. 5 4 Properties of Midsegments 275, Investigation 1 Triangle Midsegment Properties 275. Investigation 2 Trapezoid Midsegment Properties 276. Project Building an Arch 280,5 5 Properties of Parallelograms 281. Investigation Four Parallelogram Properties 281,Project Drawing Regular Polygons 286. Using Your Algebra Skills 5 Writing Linear Equations 287. 5 6 Properties of Special Parallelograms 291,Investigation 1 What Can You Draw with the. Double Edged Straightedge 291,Investigation 2 Do Rhombus Diagonals Have.

Special Properties 292,Investigation 3 Do Rectangle Diagonals Have. Special Properties 293,5 7 Proving Quadrilateral Properties 298. While working on this fourth edition ofDiscovering Geometry I was fortunate to have the assistance of Andres Marti as Michael Serra Discovering Geometry

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