Pure and Applied UNDERGRADUATE/TEXTS 51 Geometry Transformed Euclidean Plane Geometry Based on Rigid Motions James R.King AMS 我 te for IAS PCMI Park City Mathematics Institute
282 pages on 50 lb • Spine: 9/16 inch • Softcover • Trim Size 7 X 10 51 51 AMSTEXT This series was founded by the highly respected mathematician and educator, Paul J. Sally, Jr. Geometry Transformed Euclidean Plane Geometry Based on Rigid Motions James R. King For additional information and updates on this book, visit www.ams.org/bookpages/amstext-51 AMSTEXT/51 Geometry Transformed James R. King Many paths lead into Euclidean plane geometry. Geometry Transformed offers an expeditious yet rigorous route using axioms based on rigid motions and dilations. Since transformations are available at the outset, interesting theorems can be proved sooner; and proofs can be connected to visual and tactile intuition about symmetry and motion. The reader thus gains valuable experience thinking with transformations, a skill that may be useful in other math courses or applications. For students interested in teaching mathematics at the secondary school level, this approach is particularly useful since geometry in the Common Core State Standards is based on rigid motions. The only prerequisite for this book is a basic understanding of functions. Some previous experience with proofs may be helpful, but students can also learn about proofs by experiencing them in this book—in a context where they can draw and experiment. The eleven chapters are organized in a flexible way to suit a variety of curriculum goals. In addition to a geometrical core that includes finite symmetry groups, there are additional topics on circles and on crystallographic and frieze groups, and a final chapter on affine and Cartesian coordinates. The exercises are a mixture of routine problems, experiments, and proofs. 2-color cover: Pantone 432 C (Gray) and Pantone 1805 C (Red) Photo courtesy of James R. King IAS | PCMI American Mathematical Society Institute for Advanced Study Park City Mathematics Institute
Contents Introduction Advice for Students and Less Experienced Geometers Information for More Experienced Geometer w A Chapter Guide for Instructors and Others xv Acknowledgments Chapter 1.Congruence and Rigid Motions 1 Rigid Motions Informal Preview of a Problem Solution 34 Sameness Properties of Congruence Exercises and Explorations 59 Chapter 2.Axioms for the Plane Incidence Axiom 112 Distance and the Ruler Axiom Protractor Axiom and Angles Plane Separation 2479 Rigid Motions and Lines The Other Axioms Exercises and Explorations 204 Chapter 3.Existence and Properties of Reflections Deducing the Properties of Reflections IsoscelesTriangles and Kites 3379 Circles and Lines
Contents Introduction xi Advice for Students and Less Experienced Geometers xiii Information for More Experienced Geometers xv A Chapter Guide for Instructors and Others xvii Acknowledgments xxi Chapter 1. Congruence and Rigid Motions 1 Rigid Motions 3 Informal Preview of a Problem Solution 4 Sameness Properties of Congruence 5 Exercises and Explorations 9 Chapter 2. Axioms for the Plane 11 Incidence Axiom 12 Distance and the Ruler Axiom 12 Protractor Axiom and Angles 14 Plane Separation 17 Rigid Motions and Lines 19 The Other Axioms 20 Exercises and Explorations 21 Chapter 3. Existence and Properties of Reflections 23 Deducing the Properties of Reflections 23 Isosceles Triangles and Kites 27 Circles and Lines 29 vii
Contents Light,Angles,and Reflections 30 Paper Folding and Tools for Construction 31 Exercises and explorations 32 Chapter 4 Congruence of Triangles 35 Triangle Congruence Tests Applications of Triangle Congruence 3 Properties of Rigid Motions Midpoint Triangle and Angle Sum 04 Exercises and Explorations 42 Chapter 5.Rotation and Orientation Rotations and Double Reflections Rotation-Reflection Relations 50 Symmetry at a Point Orientation of a Plane 5 Orientation-Preserving and Orientation-Reversing Transformations Exercises and Explorations 62 Chapter 6.Half-turns and Inequalities in Triangles 67 Half-turn Properties Inequalities with Angles Circles and Distance to Lines Reflections and the Triangle Inequality 8759 Exercises and Explorations Chapter 7.Parallel Lines and Translations The Euclidean Parallel Postulate Transversals and Parallel Lines Parallelograms Rectangles Midpoint Figures Generalizing Parallelograms 335899%90 Translations as Half-turn Products Products of Translations Direction from Translation 2 Direction and Rotation from Polar Angle 1 Vectors 107 Exercises and Explorations 108
viii Contents Light, Angles, and Reflections 30 Paper Folding and Tools for Construction 31 Exercises and Explorations 32 Chapter 4. Congruence of Triangles 35 Triangle Congruence Tests 36 Applications of Triangle Congruence 39 Properties of Rigid Motions 40 Midpoint Triangle and Angle Sum 41 Exercises and Explorations 42 Chapter 5. Rotation and Orientation 45 Rotations and Double Reflections 46 Rotation-Reflection Relations 50 Symmetry at a Point 51 Orientation of a Plane 55 Orientation-Preserving and Orientation-Reversing Transformations 60 Exercises and Explorations 62 Chapter 6. Half-turns and Inequalities in Triangles 67 Half-turn Properties 67 Inequalities with Angles 68 Circles and Distance to Lines 71 Reflections and the Triangle Inequality 75 Exercises and Explorations 79 Chapter 7. Parallel Lines and Translations 83 The Euclidean Parallel Postulate 83 Transversals and Parallel Lines 85 Parallelograms 88 Rectangles 91 Midpoint Figures 93 Generalizing Parallelograms 96 Translations as Half-turn Products 98 Products of Translations 101 Direction from Translation 102 Direction and Rotation from Polar Angle 104 Vectors 107 Exercises and Explorations 108
Contents Chapter 8.Dilations and Similarity 11 Similarity Theorems for Triangles Right Triangles The Regular Pentagon and Its Ratios Ratios,Signed Ratios,and Scale Factors Transversals of Parallels and Ratios Parallel Segments and Centers of Dilation Construction by Scaling Models Harmonic Division Composition of Dilation Circles.Angles.and Ratios Radical Axis,Intersections,and Triangle Existence Centers of Dilation and the Midpoint Triangle Exercises and Explorations Chapter 9.Area and Its Applications Areas of Triangles and Parallelograms Area Proofs of the Pythagorean Theorem Area and Scaling Area and the circle Affine Relationships and Area Exercises and Explorations Chapter 10.Products and Patterns Products of Rotations Symmetry and 90-Degree Rotations Triangles and 60 Degrees of Rotation Translations and Symmetry Tessellations and Symmetric Wallpaper Designs Translations and Frieze Symmetry Triple Line Reflection Products Exercises and Explorations Chapter 11.Coordinate Geometry Axes and Coordinates Midpoints,Half-turns,and Translations Lines,Dilations,and Equations Euclidean Geometry and Cartesian Coordinates Perpendicular Lines in the Coordinate Plane 25
Contents ix Chapter 8. Dilations and Similarity 113 Similarity Theorems for Triangles 115 Right Triangles 117 The Regular Pentagon and Its Ratios 121 Ratios, Signed Ratios, and Scale Factors 123 Transversals of Parallels and Ratios 125 Parallel Segments and Centers of Dilation 128 Construction by Scaling Models 132 Harmonic Division 134 Composition of Dilations 136 Circles, Angles, and Ratios 139 Radical Axis, Intersections, and Triangle Existence 148 Centers of Dilation and the Midpoint Triangle 152 Exercises and Explorations 155 Chapter 9. Area and Its Applications 161 Areas of Triangles and Parallelograms 161 Area Proofs of the Pythagorean Theorem 165 Area and Scaling 167 Area and the Circle 169 Affine Relationships and Area 172 Exercises and Explorations 175 Chapter 10. Products and Patterns 177 Products of Rotations 178 Symmetry and 90-Degree Rotations 181 Triangles and 60 Degrees of Rotation 188 Translations and Symmetry 192 Tessellations and Symmetric Wallpaper Designs 194 Translations and Frieze Symmetry 200 Triple Line Reflection Products 208 Exercises and Explorations 211 Chapter 11. Coordinate Geometry 217 Axes and Coordinates 217 Midpoints, Half-turns, and Translations 218 Lines, Dilations, and Equations 221 Euclidean Geometry and Cartesian Coordinates 223 Perpendicular Lines in the Coordinate Plane 225
Contents Graphs and Transformations 228 Unit Circle and Rotation Formula 229 Complex Numbers and Transformations of the Plane Barycentric Coordinates Vectors and Affine Transformations Axioms and Models 28 Conclusion Exercises and Explorations Bibliography 253 Index 255
x Contents Graphs and Transformations 228 Unit Circle and Rotation Formula 229 Complex Numbers and Transformations of the Plane 231 Barycentric Coordinates 233 Vectors and Affine Transformations 240 Axioms and Models 245 Conclusion 249 Exercises and Explorations 249 Bibliography 253 Index 255
Introduction This book is an introduction to Euclidean plane geometry with axioms based on rigi motions.The initial spark for thinking about this topic was my reaction to reading the Common Core State Standards for Mathematics (CCSSM),especially this statement: Explain how the criteria for triangle congruence(ASA,SAS,and SSS)follow from the definition of congruence in terms of rigid motions. For almost all geometry textbooks,this statement would not make sense.Even in textbooks with a focus on geometrical transformations,such as Barker and Howe [1]. the Side-Angle-Side criterion for triangle congruence(SAS)is taken as an axiom and is then used to prove the existence and properties of rigid motions. In contrast.a basic assumption of the CCSSM for plane geometry is theexistence of rigid motio s,transformations that pres rve hoth dis and ole me e The are then prove The is intuit on s ap proach e,ea rmally,an ner tha ents the by othe r routes oreover, nis appr ons at an ea rly stage al e traditiona geon etry to Hung-Hsi Wu [19]expounds this pedagogical point of view eloquently and in detai So what would the rigorous mathematics of such an approach look like?What would the axioms be?How would the choice and flavor of topics be changed?And what would be the implications for teachers and students? I was presented a rare opportunity through my association with the IAS/Park City Mathematics Institute(PCMI)2 to ponder and respond to these questions.We in the PCMI program for teachers decided to develop a short online geometry course for teachers reflecting the Common Core approach to geometry.Working with my col- league and co-leader Gabriel Rosenberg.over two summers we collaborated with teach- ers developing ideas for such a course.Then,with Gabe as instructor,we offered the xi
Introduction This book is an introduction to Euclidean plane geometry with axioms based on rigid motions. The initial spark for thinking about this topic was my reaction to reading the Common Core State Standards for Mathematics (CCSSM), especially this statement: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.1 For almost all geometry textbooks, this statement would not make sense. Even in textbooks with a focus on geometrical transformations, such as Barker and Howe [1], the Side-Angle-Side criterion for triangle congruence (SAS) is taken as an axiom and is then used to prove the existence and properties of rigid motions. In contrast, a basic assumption of the CCSSM for plane geometry is the existence of rigid motions, transformations that preserve both distance and angle measure. The triangle congruence criteria are then proved as theorems. The rationale for this approach is that such a path into geometry is intuitive, easily modeled informally, and arrives at interesting theorems sooner than by other routes. Moreover, this approach gives students the valuable tools of geometrical transformations at an early stage along with all the traditional geometry tools. Hung-Hsi Wu [19] expounds this pedagogical point of view eloquently and in detail. So what would the rigorous mathematics of such an approach look like? What would the axioms be? How would the choice and flavor of topics be changed? And what would be the implications for teachers and students? I was presented a rare opportunity through my association with the IAS/Park City Mathematics Institute (PCMI)2 to ponder and respond to these questions. We in the PCMI program for teachers decided to develop a short online geometry course for teachers reflecting the Common Core approach to geometry. Working with my colleague and co-leader Gabriel Rosenberg, over two summers we collaborated with teachers developing ideas for such a course. Then, with Gabe as instructor, we offered the 1CCSSM [14], standard ccss.math.content.hsg.co.b.8 2PCMI is a program of the Institute for Advanced Study in Princeton, New Jersey. xi
i Introduction course several times in a videoconferencing format.The five lessons for that course provided an initial outline for this book. Transformations and Secondary Geometry For about a century,mathematicians and educators have recommended a greater role for transformations in the geometry curriculum,but for the most part,only small steps have taken place. As noted above a difficulty has been that.with the most commonly used sets of avioms the geom try required to define basic transformations,such as reflections,ro- tations,and translations.is developed fairly late in the course.when many opportuni- ties for using these tools have alre ady passed.This means that transformations a more as an advanced topic or an enrichment topic than as a central part of geon The path urged by the Common Core makes these tools available at the very be- ginning in an accessible way.The key to this approach is to assume the existence of transformations that preserve both distance and angle,not just distance.This will be explained further,beginning in Chapter 1. Until recently,a second difficulty with introducing transfo rmations into school ge ometry was that isual and ore abs tract tha figure ea how to raw a idly th tions.This proble em ho n. can s as been ow polygonsmore an fun t years by e ava ity of dynam way that was impossible on paper. With such a dramatic change being proposed in the flow of the geometry curricu lum,there's a concern that teachers are b g asked to teach in a ne w way while text- books are still ized along an earlier model.There have been only a few resou that reflect the C ommon Cor eapproach in useful detail. While this is not a high school textbook,I hope that anyone interested will see in the early chapters ofthis book a way to arrive at familiar territory of congruent triangles while also taking advantage of new tools.This is then followed up by topics such as parallel lines and similarity,intertwined with half-turns,translations,and dilations.I also hope that this book will be helpful to college and university instructors teaching geometry,especially when their students are future teachers who will want and need to understand the Common Core approach. James King
xii Introduction course several times in a videoconferencing format. The five lessons for that course provided an initial outline for this book. Transformations and Secondary Geometry For about a century, mathematicians and educators have recommended a greater role for transformations in the geometry curriculum, but for the most part, only small steps have taken place. As noted above, a difficulty has been that, with the most commonly used sets of axioms, the geometry required to define basic transformations, such as reflections, rotations, and translations, is developed fairly late in the course, when many opportunities for using these tools have already passed. This means that transformations appear more as an advanced topic or an enrichment topic than as a central part of geometry. The path urged by the Common Core makes these tools available at the very beginning in an accessible way. The key to this approach is to assume the existence of transformations that preserve both distance and angle, not just distance. This will be explained further, beginning in Chapter 1. Until recently, a second difficulty with introducing transformations into school geometry was that they seemed less visual and more abstract than figures. It is not clear how to draw a transformation. Pictures can show polygons more vividly than functions. This problem has been alleviated in recent years by the availability of dynamic geometry software that provides movement and interaction with transformations in a way that was impossible on paper. With such a dramatic change being proposed in the flow of the geometry curriculum, there’s a concern that teachers are being asked to teach in a new way while textbooks are still organized along an earlier model. There have been only a few resources that reflect the Common Core approach in useful detail. While this is not a high school textbook, I hope that anyone interested will see in the early chapters of this book a way to arrive at familiar territory of congruent triangles while also taking advantage of new tools. This is then followed up by topics such as parallel lines and similarity, intertwined with half-turns, translations, and dilations. I also hope that this book will be helpful to college and university instructors teaching geometry, especially when their students are future teachers who will want and need to understand the Common Core approach. Most of all, I hope that fellow lovers of geometry will find this an interesting path into their favorite subject. James King
Advice for Students and Less Experienced Geometers The goal of this book istopre sent the beauty and richness of Euclidean geo metho ch en fo d th to begin by assumi ing the entire structure of interrelatior undkybe thetricalled nships from these basi If you have seen this kind of formal mathematical development before,you will now see how such reasoning can be used to develop geometry.If your experience with proofs is less extensive,you will have the opportunity to grow your toolkit for proofs as you work through theorems and learn more about geometry. Whatever your prior mathematical background,here are some suggestions for reading and working with this book. First.geometry isa visual subject.It is veryin portant to look at figures and not just read w than looking is d ou sketch fig curate drawin etter and enjo the etry more any ols is valuable.but it will helpe to in ude inte active repert re With y are you dra g gelements n Since t mation 000 the sc re bility to refle ct,rot e,an translate.Moving figures dynamically will really help you visualize these transforma- tions. Second.while the book is mostly structured as a linear story.building block upon block,you have the freedom to read and think in ways that are most productive for When you encounter a theorem,first really think(and draw)to make sure you understand what the theorem is saying.Then take a stab at an explanation of your 猫
Advice for Students and Less Experienced Geometers The goal of this book is to present the beauty and richness of Euclidean geometry. The method chosen for exploring this mathematics is to begin by assuming a few simple axioms and then deducing the entire structure of interrelationships from these basic building blocks. And a key tool will be the transformations called rigid motions. If you have seen this kind of formal mathematical development before, you will now see how such reasoning can be used to develop geometry. If your experience with proofs is less extensive, you will have the opportunity to grow your toolkit for proofs as you work through theorems and learn more about geometry. Whatever your prior mathematical background, here are some suggestions for reading and working with this book. First, geometry is a visual subject. It is very important to look at figures and not just read words. Even better than looking is drawing. If you sketch figures or construct accurate drawings as you read, you will understand better and enjoy the geometry more. Drawing with any tools is valuable, but it will help enormously to include interactive geometry software in your repertoire. With such software you can draw a figure and then drag elements to create an unlimited number of examples. Since transformations play a big role in this book, the software should include the ability to reflect, rotate, and translate. Moving figures dynamically will really help you visualize these transformations. Second, while the book is mostly structured as a linear story, building block upon block, you have the freedom to read and think in ways that are most productive for you. You are not reading a mystery novel; at the beginning of each chapter, you can look ahead and see what the goals and big ideas seem to be. When you encounter a theorem, first really think (and draw) to make sure you understand what the theorem is saying. Then take a stab at an explanation of your xiii
xiv Advice for Students and Less Experienced Geometers own as to why the theorem is true.When you read the proof,does it reflect your ideas or does it go down a different path?Look ahead to see how the theorem is used. If you get stuck on something.spend a little time trying to figure out the poin of difficulty.but then move on and return later.An idea or theorem or picture you encounter later may illuminate what came before.And if you sleep on it,the answer to the puzzle may become clear the next day:both experience and brain science confirm this. Finally.look at the exercises and explorations.working on problems can provide valuable practice.to be sure.In addition.some interesting examples and special ideas
xiv Advice for Students and Less Experienced Geometers own as to why the theorem is true. When you read the proof, does it reflect your ideas or does it go down a different path? Look ahead to see how the theorem is used. If you get stuck on something, spend a little time trying to figure out the point of difficulty, but then move on and return later. An idea or theorem or picture you encounter later may illuminate what came before. And if you sleep on it, the answer to the puzzle may become clear the next day; both experience and brain science confirm this. Finally, look at the Exercises and Explorations. Working on problems can provide valuable practice, to be sure. In addition, some interesting examples and special ideas are put there for working through or experimentation instead of being presented step by step in the main text
Information for More Experienced Geometers an geo try h d on th ce of rigid aentotpans ons of the plane.A onstrate,by e w th e presence of these tool changes the early experience of geometry.A third goal is to foster an appreciation of how the ways that transformations interact and affect the geometry of the plane. The geometry is based on six axioms.The first four axioms.about lines and angles. are essentially axioms of G.D.Birkhoff [2).often called the ruler and compass axioms. These axioms use the real numbers to arrive quickly at relationships of betweenness and separation at the price of restricting the number field to R. The fifth axiom introduces rigid motions:for each line,it asserts the existence of a rigid motion that fixes the points of the line.In Chapter 3,this rigid motion is quickly proved to be the usual line reflection.Then line reflections are used to prove triangle congruence theorems,to define rotations,to study symmetry and also prove concur- rence theorems and inequalities. The sixth axiom is a similarity is t to the Euclidean allel Post hapter 7. this ax rallel Postul which po The last chapters of the book are devoted to an exploration of area,to considering the structure of symmetric patterns and tessellations on the whole plane,and finally to relating affine and Cartesian coordinates to the geometry that came before. Whe er one is read ing or teaching from the book,I would repeat the injunction in the Advice for Students section to include a lot of drawing,tracing.paper folding. and interactive geometry software to bring figures and transformations to life
Information for More Experienced Geometers A primary goal of this book is to provide a reasonably rigorous development of plane Euclidean geometry from a set of axioms based on the existence of rigid motions of the plane. A secondary goal is to demonstrate, by example, how the presence of these tools changes the early experience of geometry. A third goal is to foster an appreciation of how the ways that transformations interact and affect the geometry of the plane. The geometry is based on six axioms. The first four axioms, about lines and angles, are essentially axioms of G. D. Birkhoff [2], often called the ruler and compass axioms. These axioms use the real numbers to arrive quickly at relationships of betweenness and separation at the price of restricting the number field to ℝ. The fifth axiom introduces rigid motions; for each line, it asserts the existence of a rigid motion that fixes the points of the line. In Chapter 3, this rigid motion is quickly proved to be the usual line reflection. Then line reflections are used to prove triangle congruence theorems, to define rotations, to study symmetry and also prove concurrence theorems and inequalities. The sixth axiom is a similarity axiom, asserting properties of dilations. This axiom is equivalent to the Euclidean Parallel Postulate. In Chapter 7, this axiom is used to prove the Euclidean Parallel Postulate as a theorem, at which point the whole of Euclidean plane geometry becomes available, including the theory of parallels and translations and then similarity. The last chapters of the book are devoted to an exploration of area, to considering the structure of symmetric patterns and tessellations on the whole plane, and finally to relating affine and Cartesian coordinates to the geometry that came before. Whether one is reading or teaching from the book, I would repeat the injunction in the Advice for Students section to include a lot of drawing, tracing, paper folding, and interactive geometry software to bring figures and transformations to life. xv