w Acknowledgments For recognizing the benefits of installing rigid motions into the foundations of geometry.I offer a tribute to Hung-Hsi Wu His persuasive advocacy for introducing rigid motions early in geometry instruction led to the adoption of this approach by the Common Core. preciation to bill barker.who gen in significant improvements. I also want to thank and recognize the editors and technical staff of the AMS,who suddenly had to adapt to working remotely in a pandemic year.Eriko Hironaka was very helpful and supportive throughout the development of this book,and copy editor Arlene O'Sean has been sharp-eyed in making corrections and generous and helpful while getting the book into print. And most of all,I want to thank my wife,Vicki,for e year I pr reading every page of evethouhe cm
xxii Acknowledgments For recognizing the benefits of installing rigid motions into the foundations of geometry, I offer a tribute to Hung-Hsi Wu. His persuasive advocacy for introducing rigid motions early in geometry instruction led to the adoption of this approach by the Common Core. For the book itself, I want to express my great appreciation to Bill Barker, who generously volunteered to read several chapters and then made suggestions that resulted in significant improvements. I also want to thank and recognize the editors and technical staff of the AMS, who suddenly had to adapt to working remotely in a pandemic year. Eriko Hironaka was very helpful and supportive throughout the development of this book, and copy editor Arlene O’Sean has been sharp-eyed in making corrections and generous and helpful while getting the book into print. And most of all, I want to thank my wife, Vicki, for so many things over the years but, in particular, for proofreading every page of this book even though she claims she did not understand any of it
Chapter 1 Congruence and Rigid Motions Congruence is a fundamental concept in geometry.However,it is surprisingly diffi- cult to concoct a mathematically sound general definition in a textbook or in a course based on the most common axioms.The necessary ingredient for a good definition -a transformation called a rigid motion-typically only appears late in the logical development of the subject,rather than early when it is needed.In this book we mod- ify the axiom set for plane geometry so that rigid motions are available at the outset. This provides a powerful new toolkit for geometrical reasoning in the early stages of geometry. Defining Congruence.Textbooks typically focus on congruence of triangles, which is explained reasonably well.But the general concept of congruence is often not defined.In classrooms one often hears the informal mantra of"same size,same shape".This may be helpful as a suggestion of the intended meaning.but it is not a def- inition.For example,somewhat as a joke,we could propose that these first two figures are congruent.They are both rectangles(same shape)and have the same area(same size).No one would really suggest that these are congruent,but it does point out why a real definition should be based on unambiguous words that have clear mathematical meaning. Figure 1.Same Size,Same Shape 五
Chapter 1 Congruence and Rigid Motions Congruence is a fundamental concept in geometry. However, it is surprisingly difficult to concoct a mathematically sound general definition in a textbook or in a course based on the most common axioms. The necessary ingredient for a good definition — a transformation called a rigid motion — typically only appears late in the logical development of the subject, rather than early when it is needed. In this book we modify the axiom set for plane geometry so that rigid motions are available at the outset. This provides a powerful new toolkit for geometrical reasoning in the early stages of geometry. Defining Congruence. Textbooks typically focus on congruence of triangles, which is explained reasonably well. But the general concept of congruence is often not defined. In classrooms one often hears the informal mantra of “same size, same shape”. This may be helpful as a suggestion of the intended meaning, but it is not a definition. For example, somewhat as a joke, we could propose that these first two figures are congruent. They are both rectangles (same shape) and have the same area (same size). No one would really suggest that these are congruent, but it does point out why a real definition should be based on unambiguous words that have clear mathematical meaning. Figure 1. Same Size, Same Shape? 1
1.Congruence and Rigid Motions If one tries to be more precise by using side length and angle measure in a defini tion,such as"corresponding side lengths and angle measures are equal",the definitior still does not explain why the polygons in Figure 2 and the polygonal paths in Figure 3 are not congruent Figure2.Same Side Lengths,Same Angles One can refine the definition of congruence by introducing internal angles and oriented a cify that all distan s ha But this will r ly to etrica om ected objects me abola)as in F To tr or ray or pa ch kind or ngure on case ld he ome a game geometrica whack-a-mole.not a clear definition for a basic concept 八1 Figure3.Same Side Lengths,Same Angles Figure 4.A Disconnected Figure Containingan Arc,Isolated Points,and an Entire Line There is a simple and intuitive,real-world answer to this problem if the two figures are drawn or printed on paper.One simply picks one up and lays it on the other to see whether they can be matched.This is the correspondence needed in the abstract definition as well.But rather than specify the correspondence as point A to D,point B to E,point C to F,as we do for some triangles ABC and DEF,this correspondence moves the whole plane(or the paper in the physical case). movement into a mathematical definition?
2 1. Congruence and Rigid Motions If one tries to be more precise by using side length and angle measure in a definition, such as “corresponding side lengths and angle measures are equal”, the definition still does not explain why the polygons in Figure 2 and the polygonal paths in Figure 3 are not congruent. Figure 2. Same Side Lengths, Same Angles One can refine the definition of congruence by introducing internal angles and oriented angles to cover these examples, or one can specify that all distances between corresponding pairs of vertices are the same. But this will not apply to geometrical objects that are less polygonal and so are harder to deal with. What does one say about congruence when a figure includes a circular arc or two or more disconnected objects, not to mention an unbounded shape such as a line (or ray or parabola) as in Figure 4? To try to cover each kind of figure on a case-by-case basis would become a game of geometrical whack-a-mole, not a clear definition for a basic concept. Figure 3. Same Side Lengths, Same Angles Figure 4. A Disconnected Figure Containing an Arc, Isolated Points, and an Entire Line There is a simple and intuitive, real-world answer to this problem if the two figures are drawn or printed on paper. One simply picks one up and lays it on the other to see whether they can be matched. This is the correspondence needed in the abstract definition as well. But rather than specify the correspondence as point 𝐴 to 𝐷, point 𝐵 to 𝐸, point 𝐶 to 𝐹, as we do for some triangles△𝐴𝐵𝐶 and△𝐷𝐸𝐹, this correspondence moves the whole plane (or the paper in the physical case). This is superposition. The principle of superposition says that two figures are congruent if one can be superimposed on the other. But how can one convert this physical movement into a mathematical definition?
Rigid Motions 3 When the Greeks developed their deductive approach to geometry,they lacked some language and concepts that are central in mathematics today.They did not have the system of real numbers,and they did not have the general concept of function. They clearly had superposition in their thoughts,because it occasionally appears with- out acknowledgment in the reasoning of Euclid.But they treated congrue ce as a for- mal equivalence without featuring the actual correspondence.the superposition,as an object that is part of the picture. Mathematics now does have the language for superposition well worked out.This is the language oftransformations.We can use transformations explicitly in our axioms and our reasoning.Moreover,transformations conform as well or better to the idea of a geometrical tool in our modern world of computer-aided design and rotating photos on phones than do the older tools of straightedge and compass.In addition,transfor- mations become geometrical objects that can be profitably studied in their own right, adding a new layer or richness to the geometry. Rigid Motions The transformations that will model superposition will be called rigid motions.Then two figures will be congruent if there is a rigid motion that maps one to the other. In order to define rigid motions,we need some properties of the plane and the definition of transformation. We will soon introduce axioms for the plane,but at this point we can say the fol- lowing: The plane is a set whose elements are called points. .There is a distance measure:for any two pointsA and B,the distance AB is a nonnegative real number. There is also an angle measure:for any three points,A,B.C,with both B and C distinct from A,there is measure mBAC that is a real number between 0 and 180. A transformation of the plane is a function from the plane into itself that has aere和e urct p阳neo知taA T-(T(A))=A and T(T-(A))=A.In other words,each composition TT-1=T-IT is the identity function I that maps each point A to itself. Definition 1.1.A rigid motion of the plane is a transformation of the plane that preserves distance and angle measure.This means a rigid motion Tis a transformation with these properties: For any points A and B.IlABll =IIT(A)T(B)Il. ,.C.wh oǜ因nq Now that we have defined rigid motions,we can state a general definition of con- gruence
Rigid Motions 3 When the Greeks developed their deductive approach to geometry, they lacked some language and concepts that are central in mathematics today. They did not have the system of real numbers, and they did not have the general concept of function. They clearly had superposition in their thoughts, because it occasionally appears without acknowledgment in the reasoning of Euclid. But they treated congruence as a formal equivalence without featuring the actual correspondence, the superposition, as an object that is part of the picture. Mathematics now does have the language for superposition well worked out. This is the language of transformations. We can use transformations explicitly in our axioms and our reasoning. Moreover, transformations conform as well or better to the idea of a geometrical tool in our modern world of computer-aided design and rotating photos on phones than do the older tools of straightedge and compass. In addition, transformations become geometrical objects that can be profitably studied in their own right, adding a new layer or richness to the geometry. Rigid Motions The transformations that will model superposition will be called rigid motions. Then two figures will be congruent if there is a rigid motion that maps one to the other. In order to define rigid motions, we need some properties of the plane and the definition of transformation. We will soon introduce axioms for the plane, but at this point we can say the following: • The plane is a set whose elements are called points. • There is a distance measure: for any two points 𝐴 and 𝐵, the distance ‖𝐴𝐵‖ is a nonnegative real number. • There is also an angle measure: for any three points, 𝐴, 𝐵, 𝐶, with both 𝐵 and 𝐶 distinct from 𝐴, there is measure 𝑚∠𝐵𝐴𝐶 that is a real number between 0 and 180. A transformation of the plane is a function from the plane into itself that has an inverse. To be more precise, a function 𝑇 from the plane to itself is a transformation if there is another function 𝑇 −1 from the plane to itself so that for any point 𝐴, 𝑇 −1(𝑇(𝐴)) = 𝐴 and 𝑇(𝑇−1(𝐴)) = 𝐴. In other words, each composition 𝑇𝑇−1 = 𝑇−1𝑇 is the identity function 𝐼 that maps each point 𝐴 to itself. Definition 1.1. A rigid motion of the plane is a transformation of the plane that preserves distance and angle measure. This means a rigid motion 𝑇 is a transformation with these properties: • For any points 𝐴 and 𝐵, ‖𝐴𝐵‖ = ‖𝑇(𝐴)𝑇(𝐵)‖. • For any three points, 𝐴, 𝐵, 𝐶, with both 𝐵 and𝐶 distinct from 𝐴, 𝑚∠𝑇(𝐴)𝑇(𝐵)𝑇(𝐶) = 𝑚∠𝐴𝐵𝐶. Now that we have defined rigid motions, we can state a general definition of congruence
1.Congruence and Rigid Motions Aset U in the plane is congruent to a set V if there is a rigid motion T such that T(U)=V Note.For a set U,the notation T(U)means the image of U.the set of all the points that are images T(A)for some point A in U. Rigid Motion vs Isometry.This is a technical point that may be puzzling some who are alr ady f iliar etrical tran nsfo ation ing like this: 山g "O a rigid mo s dis transformation that ance is alled an iso But D always preserves angles.Why do we need this extra condition Readers applying real-world physical intuition may also be thinking that if dis- tances between points are preserved,then angles are not distorted. There is enough truth here to be confusir .An isometry is a transformation tha preserves dista ce.N 330 about angle opment o me e triangle congruer criterion S ngle ide as an ax after a certain number of theorems have been proved t one can prove that an isometry also preserves angle measure and,therefore,is a rigid motion. But in this book we are not assuming side-Angle-Side as an axiom:instead we are going toassume the existence of certain rigd motions at the outset and then prove Side Angle-Side as a theorem.If we only assur med the exist ce of isometries as an axion would not he ssible It is that superposition n.be ny the otion is an isometry,but there is no reaso n to However,to complete the confusion,with our rigid motion axiom,we will eventu- ally prove all the triangle congruence theorems and hence prove that distance preser- vation implies angle preservation;so an isometry will be proved to be a rigid motion. But for an axiom that one assumes without proof to serve as a basis for geometry.we need both conditions in the definition of a rigid motion. ,there are a gs that are true in the world that also eventually turno be true in the abstrac model But these facts do not get built and proved in the abstract model until after some development of the theory. Informal Preview of a Problem Solution Hereis of how this concept of congruen ecan be used for a short clea s geometri w,we will take th ming in advance that m segments to line segments.A segment with endpointsand is denoted ve eady prove uppose triangle ABC is congruent to A'B'C.Let Dbe the mi point of BC and let D'be the midpoint of B'C For points E on AC and E'on A'C assumeCE=CE.Prove that the quadrilateral DCEF is congruent to D'CE'F
4 1. Congruence and Rigid Motions Definition 1.2. A set 𝑈 in the plane is congruent to a set 𝑉 if there is a rigid motion 𝑇 such that 𝑇(𝑈) = 𝑉. Note. For a set 𝑈, the notation 𝑇(𝑈) means the image of 𝑈, the set of all the points that are images 𝑇(𝐴) for some point 𝐴 in 𝑈. Rigid Motion vs Isometry. This is a technical point that may be puzzling some readers. Readers who are already familiar with geometrical transformations may at this point be thinking something like this: “Oh, a rigid motion preserves distance. A transformation that preserves distance is called an isometry. But I thought an isometry always preserves angles. Why do we need this extra condition?” Readers applying real-world physical intuition may also be thinking that if distances between points are preserved, then angles are not distorted. There is enough truth here to be confusing. An isometry is a transformation that preserves distance. Nothing is said about angles. But in an axiomatic development of geometry that assumes the triangle congruence criterion Side-Angle-Side as an axiom, it is true — after a certain number of theorems have been proved — that one can prove that an isometry also preserves angle measure and, therefore, is a rigid motion. But in this book we are not assuming Side-Angle-Side as an axiom; instead we are going to assume the existence of certain rigid motions at the outset and then prove SideAngle-Side as a theorem. If we only assumed the existence of isometries as an axiom, this would not be possible, since we could not reason about angles. It is important that superposition preserve angle measure. So, by definition, before any theorems are proved, every rigid motion is an isometry, but there is no reason to assume that every isometry is a rigid motion. However, to complete the confusion, with our rigid motion axiom, we will eventually prove all the triangle congruence theorems and hence prove that distance preservation implies angle preservation; so an isometry will be proved to be a rigid motion. But for an axiom that one assumes without proof to serve as a basis for geometry, we need both conditions in the definition of a rigid motion. And this confirms the physical intuition about distance preservation and angle. In an axiomatic development of an abstract model of the plane, there are a lot of things that are true in the real world that also eventually turn out to be true in the abstract model. But these facts do not get built and proved in the abstract model until after some development of the theory. Informal Preview of a Problem Solution Here is an example of how this concept of congruence can be used for a short, clear proof that reflects one’s geometric intuition. Since this is a preview, we will take the liberty of assuming in advance that we have already proved rigid motions map line segments to line segments. A segment with endpoints 𝐴 and 𝐵 is denoted 𝐴𝐵. Example 1.3. Suppose triangle △𝐴𝐵𝐶 is congruent to △𝐴′𝐵 ′𝐶 ′ . Let 𝐷 be the midpoint of 𝐵𝐶 and let 𝐷 ′ be the midpoint of 𝐵 ′𝐶 ′ . For points 𝐸 on 𝐴𝐶 and 𝐸 ′ on 𝐴 ′𝐶 ′ , assume ‖𝐶𝐸‖ = ‖𝐶′𝐸 ′‖. Prove that the quadrilateral 𝐷𝐶𝐸𝐹 is congruent to 𝐷 ′𝐶 ′𝐸 ′𝐹 ′