10 1.Congruence and Rigid Motions 5.(Poduct Inverse)transformations,show that the inverse Prove this with function notation and/or any other way you feel is an effective way to convince others.(Hint:If you put on socks and shoes,in what order do you take them off?) 6.(Proof Notation).ifyou feel it would be helpful write one of the proofs above using me notation (function o r primed)that was not used above.For example.prove that ifTisa rigid motion,then T-preserves angle measure. 7.(Congruence Sets).Suppose F and G are two figures in the plane.Let Ce be the set of figures congruent to F and let Co be the set of figures congruent to G.Prove that if FG.then Cr Cc,but if F is not congruent to G.then Cr and Cc have no elements in common
10 1. Congruence and Rigid Motions 5. (Product Inverse). If 𝑆 and 𝑇 are transformations, show that the inverse of 𝑇𝑆 is 𝑆 −1𝑇 −1. Prove this with function notation and/or any other way you feel is an effective way to convince others. (Hint: If you put on socks and shoes, in what order do you take them off?) 6. (Proof Notation). If you feel it would be helpful, write one of the proofs above using some notation (function or primed) that was not used above. For example, prove that if 𝑇 is a rigid motion, then 𝑇 −1 preserves angle measure. 7. (Congruence Sets). Suppose 𝐹 and 𝐺 are two figures in the plane. Let 𝐶𝐹 be the set of figures congruent to 𝐹 and let 𝐶𝐺 be the set of figures congruent to 𝐺. Prove that if 𝐹 ≅ 𝐺, then 𝐶𝐹 = 𝐶𝐺, but if 𝐹 is not congruent to 𝐺, then 𝐶𝐹 and 𝐶𝐺 have no elements in common
Chapter 2 Axioms for the Plane It is now time to insert a little geometry into our plane.The plane is intended to be an abstract model of the geometry of a flat surface such as a table or a piece of paper, with the difference that our plane will extend without end.So far we have points and some measures(with no details).We have also defined rigid motions,but except for the identity mapping,we have not yet claimed that any rigid motions actually exist.So there are things to do. In this chapter,we state the six axioms for plane geometry that will be used in this book.The first four are closely related to the well-known Ruler and Compass axioms of G.D.Birkhoff 2.3: .Incidence Axiom for points and lines Ruler Axiom concerning distance measure Protractor Axiom concerning angle measure Plane Separation Axiom concerning boundaries and orientation We will explore consequences of these first four axioms in this chapter.From these axioms one can define some elementary but important geometrical concepts and prove a few key theorems. At the end of the chapter,we complete the set of axioms by stating the remainir Reflection Axiom ·Dilation Axiom The Reflection Axiom will add rigid motions to our geometry.Significant geo- metrical connections between distance and angle measure will appear in Chapter 3 as consequences of the Reflection Axiom.This will lead to triangle congruence theo- rems in Chapter 4.In later chapters the importance of the Dilation Axiom will become evident. 五
Chapter 2 Axioms for the Plane It is now time to insert a little geometry into our plane. The plane is intended to be an abstract model of the geometry of a flat surface such as a table or a piece of paper, with the difference that our plane will extend without end. So far we have points and some measures (with no details). We have also defined rigid motions, but except for the identity mapping, we have not yet claimed that any rigid motions actually exist. So there are things to do. In this chapter, we state the six axioms for plane geometry that will be used in this book. The first four are closely related to the well-known Ruler and Compass axioms of G. D. Birkhoff [2], [3]: • Incidence Axiom for points and lines • Ruler Axiom concerning distance measure • Protractor Axiom concerning angle measure • Plane Separation Axiom concerning boundaries and orientation We will explore consequences of these first four axioms in this chapter. From these axioms one can define some elementary but important geometrical concepts and prove a few key theorems. At the end of the chapter, we complete the set of axioms by stating the remaining two axioms: • Reflection Axiom • Dilation Axiom The Reflection Axiom will add rigid motions to our geometry. Significant geometrical connections between distance and angle measure will appear in Chapter 3 as consequences of the Reflection Axiom. This will lead to triangle congruence theorems in Chapter 4. In later chapters the importance of the Dilation Axiom will become evident. 11
12 2.Axioms for the Plane AB=8-川=7 Figure 1.A Ruler Measuring Distance in the Plane Incidence Axiom The first axiom is very simple.The informal version is"two points determine a line". Axiom1(Incidence).The plane consists ofa nonempty set ofelements called points and anonempy set of subsets called lines.Forany two distinct points,there line containing the wo points:it will be denoted AB Asaconsequence of this,two distinct lines cannot intersect in more than one point. This already rules out our model looking like the sphere.We might think of this as the unmarked straightedge axiom:for any two points on a paper.we can draw a line with a straightedge through the two points. Distance and the Ruler Axiom The Ruler Axiom is an existence axiom for a distance measure and for"rulers"that provide distance measurement on a line by means of real numbers.Such a ruler is an abstraction of a marked straightedge. Axiom2(Ruler).The plane has a distance function. .Distance is afunction that maps any two points Aand B in the plane toa nonnega- tive real number ABll. Every line m in the plane has a ruler,a mapping of the real numbers R onto m so that distances are preserved. Another way of de scribinga ruer is that it is an isometry fromtom Explicitly. ifp is a ruler with p(a)=A and p(b)=B.then the distance llABll equals the standard distance between two real numbers,the absolute value la-bl
12 2. Axioms for the Plane Figure 1. A Ruler Measuring Distance in the Plane Incidence Axiom The first axiom is very simple. The informal version is “two points determine a line”. Axiom 1 (Incidence). The plane consists of a nonempty set of elements called points and a nonempty set of subsets called lines. For any two distinct points, there is exactly one line containing the two points; it will be denoted 𝐴𝐵. As a consequence of this, two distinct lines cannot intersect in more than one point. This already rules out our model looking like the sphere. We might think of this as the unmarked straightedge axiom: for any two points on a paper, we can draw a line with a straightedge through the two points. Distance and the Ruler Axiom The Ruler Axiom is an existence axiom for a distance measure and for “rulers” that provide distance measurement on a line by means of real numbers. Such a ruler is an abstraction of a marked straightedge. Axiom 2 (Ruler). The plane has a distance function. • Distance is a function that maps any two points 𝐴 and 𝐵 in the plane to a nonnegative real number ‖𝐴𝐵‖. • Every line 𝑚 in the plane has a ruler, a mapping of the real numbers ℝ onto 𝑚 so that distances are preserved. Another way of describing a ruler is that it is an isometry from ℝ to 𝑚. Explicitly, if 𝜌 is a ruler with 𝜌(𝑎) = 𝐴 and 𝜌(𝑏) = 𝐵, then the distance ‖𝐴𝐵‖ equals the standard distance between two real numbers, the absolute value |𝑎 − 𝑏|
Distance and the Ruler Axiom 13 Remark 2.1.This axiom implies a number of important properties of points,lines, and distance. For any A.IAAll =0.This is true since,for A on a line m with ruler value p(a)= A,the distance AAll la-al=0. Aruler isaone-to-one mapping,for ifpisa ruler on a line m withp(a)=p(b)=A, then la-bl AAll =0,so a =b. For any two distinct points A and B,the distance ABl>0.For a rulerponAB with p(a)=A and p(b)=B for distinct a and b,the distance llABll =la-bl>0. There are an infinite number of points on each line,and the distances on the line are unbounded. If p(x)is a ruler,then so is p(x+k)or p(-x+k)for any real k.Therefore,any point on the line can have coordinate0. preserving transformation of the real numbers before ruler,this position is still a ruler since the distanc still e if o is a ruler for m.let th function Ps be defi 5)Th real numbers s a and b.lle er for m. since fora llp(a+5)p( and this als I (a+5) (b+5=la-lsincep is a ruler Exercise 1 explores these transformations of the real line. The real numbers assigned to each point of m by the inverse map are the coor dinates defined by the ruler.This inverse map will occasionally be designated as a function from points to numbers such as a =x(A)and b =x(B),but usually it will simply be stated that the real numbers a and b correspond to the points A and B in m. These real numbers can be used to prove things about points on a linem. For example,given a point A on m and any distance d,there are two points B and C on m at distance d,with llABll =llACll =d,since for any real number a,there are two numbers b=a+d and c=a-d with b-al lc-al=d. Segments.From the Rule mfor distinct po ints A and B we an define AB the segment with the next few paragraphs,let the numbers a,b,e correspond to the points A.B.ConAB Segment Definition by Ruler:The segment AB is the set of points C of AB such that the numberc belongs to the interval with endpoints a and b:either ascsb orb≤c≤a.Equivalently.c-al+lb-d=lb-al Segment Definition by Distance:From the absc ute value equation above,one sees that the segment AB is the set of points C of AB with llACll +lBCll =IlABll. Midpoint:The midpoint M ofAB is the point on the segment equidistant fromA and B.Hence,2lAMII IABIl
Distance and the Ruler Axiom 13 Remark 2.1. This axiom implies a number of important properties of points, lines, and distance. • For any 𝐴, ‖𝐴𝐴‖ = 0. This is true since, for 𝐴 on a line 𝑚 with ruler value 𝜌(𝑎) = 𝐴, the distance ‖𝐴𝐴‖ = |𝑎 − 𝑎| = 0. • A ruler is a one-to-one mapping, for if 𝜌 is a ruler on a line 𝑚 with 𝜌(𝑎) = 𝜌(𝑏) = 𝐴, then |𝑎 − 𝑏| = ‖𝐴𝐴‖ = 0, so 𝑎 = 𝑏. • For any two distinct points 𝐴 and 𝐵, the distance ‖𝐴𝐵‖ > 0. For a ruler 𝜌 on 𝐴𝐵, with 𝜌(𝑎) = 𝐴 and 𝜌(𝑏) = 𝐵 for distinct 𝑎 and 𝑏, the distance ‖𝐴𝐵‖ = |𝑎 − 𝑏| > 0. • There are an infinite number of points on each line, and the distances on the line are unbounded. • If 𝜌(𝑥) is a ruler, then so is 𝜌(𝑥 + 𝑘) or 𝜌(−𝑥 + 𝑘) for any real 𝑘. Therefore, any point on the line can have coordinate 0. The last item is true because, if we apply any distance-preserving transformation of the real numbers before mapping them by the ruler, this composition is still a ruler since the distance relations are still preserved. For example, if 𝜌 is a ruler for 𝑚, let the function 𝜌5 be defined as 𝜌5 (𝑥) = 𝜌(𝑥 + 5). Then 𝜌5 is also a ruler for 𝑚, since for any real numbers 𝑎 and 𝑏, ‖𝜌5 (𝑎)𝜌5 (𝑏)‖ = ‖𝜌(𝑎 + 5)𝜌(𝑏 + 5)‖, and this equals |(𝑎 + 5) − (𝑏 + 5)| = |𝑎 − 𝑏| since 𝜌 is a ruler. Exercise 1 explores these transformations of the real line. The real numbers assigned to each point of 𝑚 by the inverse map are the coordinates defined by the ruler. This inverse map will occasionally be designated as a function from points to numbers such as 𝑎 = 𝑥(𝐴) and 𝑏 = 𝑥(𝐵), but usually it will simply be stated that the real numbers 𝑎 and 𝑏 correspond to the points 𝐴 and 𝐵 in 𝑚. These real numbers can be used to prove things about points on a line 𝑚. For example, given a point 𝐴 on 𝑚 and any distance 𝑑, there are two points 𝐵 and 𝐶 on 𝑚 at distance 𝑑, with ‖𝐴𝐵‖ = ‖𝐴𝐶‖ = 𝑑, since for any real number 𝑎, there are two numbers 𝑏 = 𝑎 + 𝑑 and 𝑐 = 𝑎 − 𝑑 with |𝑏 − 𝑎| = |𝑐 − 𝑎| = 𝑑. Segments. From the Ruler Axiom, for distinct points 𝐴 and 𝐵 we can define 𝐴𝐵, the segment with endpoints 𝐴 and 𝐵. For the next few paragraphs, let the numbers 𝑎, 𝑏, 𝑐 correspond to the points 𝐴, 𝐵, 𝐶 on 𝐴𝐵. Segment Definition by Ruler: The segment 𝐴𝐵 is the set of points 𝐶 of 𝐴𝐵 such that the number 𝑐 belongs to the interval with endpoints 𝑎 and 𝑏: either 𝑎 ≤ 𝑐 ≤ 𝑏 or 𝑏 ≤ 𝑐 ≤ 𝑎. Equivalently, |𝑐 − 𝑎| + |𝑏 − 𝑐| = |𝑏 − 𝑎|. Segment Definition by Distance: From the absolute value equation above, one sees that the segment 𝐴𝐵 is the set of points 𝐶 of 𝐴𝐵 with ‖𝐴𝐶‖ + ‖𝐵𝐶‖ = ‖𝐴𝐵‖. Segment Interior Point: A point 𝐶 is an interior point of 𝐴𝐵, or is between 𝐴 and 𝐵, if it is in 𝐴𝐵 but is not one of the endpoints. Midpoint: The midpoint 𝑀 of 𝐴𝐵 is the point on the segment equidistant from 𝐴 and 𝐵. Hence, 2‖𝐴𝑀‖ = ‖𝐴𝐵‖
14 2.Axioms for the Plane Convexity:AsetSisconvex if,for any points Aand BinS.the segment AB isalso contained in S. tion of B.We c correspon nd to the points A.B.Con AB. Ray Definition A point C on AB is in AB if A is not between B and C.In other words,c≥aifb>a andc≤aifb<a. Ray Interior Point:The set of all points except the endpoint is the set of interior points of the ray. Opposite Ray:The opposite ray of AB is the ray consisting of endpoint A and the points C on AB not in AB.In other words,ca if b<a andcs a ifb>a. Directed Line:Achoice ofray determinesadirectionororientation ofthe line con- taining it.Two rays in a line define the same direction if their intersection is a ray: this occurs if one ray is contained in the other,so a line has two possible direc- tions. Polygons.Now we can define polygons. Definition 2.2.A polygon RB is the figure consisting of n distinct points,the vertices Pi,and n sides,the segments RB.BB. .BR.such that (1)no interior point ofaside is an intersection point oftwo sides and(2)any three consecutive vertices are noncollinear. The sequ sof three conse ecutive vertices include and PR.B.This n-ih ersecting polygo which beinteresting but are notincudeds closed polygonal paths The most familiar polygons have three sides(triangles).four sides(quadrilaterals). five sides(pentagons),and six sides(hexagons). Protractor Axiom and Angles Next,we will consider how to measure angles.Alas,even in informal geometry this is a bit more complicated than measuring along a line. Definition 2.3.An angle with vertex A consists of two rays AB and AC.denoted BAC,where A,B.C are not collinear. Two opposite rays AB and AC are said to form a straight angle and two equal rays AB and AC are said to form a zero angle even though these are not angles by the definition.This terminology is so well established that we will use it.But to be clear, ifwe say"angleor refer toCwithout explicty includingstraight angles this means an angle by the definition and doe s not include zero or straight angles. Axiom3(Protractor).The plane has an angle measure that maps any two raysAB and AC with common endpoint to a real number m/BAC
14 2. Axioms for the Plane Convexity: A set 𝑆 is convex if, for any points 𝐴 and 𝐵 in 𝑆, the segment 𝐴𝐵 is also contained in 𝑆. Rays. For distinct points 𝐴 and 𝐵, there is a ray 𝐴𝐵⃗ with endpoint 𝐴 in the direction of 𝐵. We continue with the assumption that the numbers 𝑎, 𝑏, 𝑐 correspond to the points 𝐴, 𝐵, 𝐶 on 𝐴𝐵. Ray Definition: A point 𝐶 on 𝐴𝐵 is in 𝐴𝐵⃗ if 𝐴 is not between 𝐵 and 𝐶. In other words, 𝑐 ≥ 𝑎 if 𝑏 > 𝑎 and 𝑐 ≤ 𝑎 if 𝑏 < 𝑎. Ray Interior Point: The set of all points except the endpoint is the set of interior points of the ray. Opposite Ray: The opposite ray of 𝐴𝐵⃗ is the ray consisting of endpoint 𝐴 and the points 𝐶 on 𝐴𝐵 not in 𝐴𝐵⃗ . In other words, 𝑐 ≥ 𝑎 if 𝑏 < 𝑎 and 𝑐 ≤ 𝑎 if 𝑏 > 𝑎. Directed Line: A choice of ray determines a direction or orientation of the line containing it. Two rays in a line define the same direction if their intersection is a ray; this occurs if one ray is contained in the other, so a line has two possible directions. Polygons. Now we can define polygons. Definition 2.2. A polygon 𝑃1𝑃2 . 𝑃𝑛 is the figure consisting of 𝑛 distinct points, the vertices 𝑃𝑖 , and 𝑛 sides, the segments 𝑃1𝑃2 , 𝑃2𝑃3 , . , 𝑃𝑛−1𝑃𝑛, 𝑃𝑛𝑃1 , such that (1) no interior point of a side is an intersection point of two sides and (2) any three consecutive vertices are noncollinear. The sequences of three consecutive vertices include 𝑃𝑛−1, 𝑃𝑛, 𝑃1 and 𝑃𝑛, 𝑃1 , 𝑃2 . This definition excludes self-intersecting polygonal figures and nonclosed polygonal paths, which can also be interesting but are not included as polygons. The most familiar polygons have three sides (triangles), four sides (quadrilaterals), five sides (pentagons), and six sides (hexagons). Protractor Axiom and Angles Next, we will consider how to measure angles. Alas, even in informal geometry this is a bit more complicated than measuring along a line. Definition 2.3. An angle with vertex 𝐴 consists of two rays 𝐴𝐵⃗ and 𝐴𝐶⃗, denoted ∠𝐵𝐴𝐶, where 𝐴, 𝐵, 𝐶 are not collinear. Two opposite rays 𝐴𝐵⃗ and 𝐴𝐶⃗ are said to form a straight angle and two equal rays 𝐴𝐵⃗ and 𝐴𝐶⃗ are said to form a zero angle even though these are not angles by the definition. This terminology is so well established that we will use it. But to be clear, if we say “angle” or refer to ∠𝐵𝐴𝐶 without explictly including zero or straight angles, this means an angle by the definition and does not include zero or straight angles. Axiom 3 (Protractor). The plane has an angle measure that maps any two rays 𝐴𝐵⃗ and 𝐴𝐶⃗ with common endpoint to a real number 𝑚∠𝐵𝐴𝐶