Protractor Axiom and Angles 15 Figure 2.A Wooden Protractor Measuring from0to180 The ure maps ary angle BAC to a poifive rumber mBAC,wth In addition,when AB AC,the angle measure of this zero angle is mBAC=0. nd AC are oppositera the angle mes oftls t ae For any point A,there exists a protractor at A.This is a mapping 6 from the real sonto the rays with endpoint A.such that 5(t)=5(t+360)for every t,with hatm∠BAC=lb -cl when 5(b)=AB and (c)=AC.provided tha Note.A protractor is also called a polar angle parametrization at A.The degree is the unit of angle measure used here,since this is more convenient for geometry than radians.However,the number 360 could be replaced by another positive number to provide a different choice of unit. Example 2.4.Let's try to connect this angle m nt with ou netric intu ition I kind that gine tha ing a compah the c direc comp ered from ely at o nd360 being the same direction.Suppose this Ifyou are standing at point A,then the rays with endpointA indicate the directions that you can be facing.Suppose directionAB is numbered 40,sort ofa northeast direc. tion,and a dire ction AC i ed 250,sort of s Then w te the ar AC by taking the 2s0210.But this number is greater than 180,so it willnot work as an angle wee measure.However,the number 250-360=-110 also corresponds to the ray AC,so we can try again with this n umber:-110-40=150.This is between 0 and 180,so 150 is the measure of BAC. Polar Angles.Using protractors directly is somewhat awkward:the inverse map from rays to real numbers is often more convenient.For a given protractor 6 at A,we will say the point B(or the ray AB)has polar angle t ift is one of the real numbers for which 5(t)=AB
Protractor Axiom and Angles 15 Figure 2. A Wooden Protractor Measuring from 0 to 180 • The angle measure maps any angle ∠𝐵𝐴𝐶 to a positive number 𝑚∠𝐵𝐴𝐶, with 0 < 𝑚∠𝐵𝐴𝐶 < 180. • In addition, when 𝐴𝐵 = ⃗ 𝐴𝐶⃗, the angle measure of this zero angle is 𝑚∠𝐵𝐴𝐶 = 0. If the rays 𝐴𝐵⃗ and 𝐴𝐶⃗ are opposite rays, the angle measure of this straight angle is 𝑚∠𝐵𝐴𝐶 = 180. • For any point A, there exists a protractor at A. This is a mapping 𝛿 from the real numbers onto the rays with endpoint 𝐴, such that 𝛿(𝑡) = 𝛿(𝑡 + 360) for every 𝑡, with the property that 𝑚∠𝐵𝐴𝐶 = |𝑏 − 𝑐| when 𝛿(𝑏) = 𝐴𝐵⃗ and 𝛿(𝑐) = 𝐴𝐶⃗, provided that |𝑏 − 𝑐| ≤ 180. Note. A protractor is also called a polar angle parametrization at 𝐴. The degree is the unit of angle measure used here, since this is more convenient for geometry than radians. However, the number 360 could be replaced by another positive number to provide a different choice of unit. Example 2.4. Let’s try to connect this angle measurement with our geometric intuition. Imagine that you are holding a compass (the magnetic directional kind, not the kind that draws circles). The directions on the compass face are numbered from 0 to 360, with the directions precisely at 0 and 360 being the same direction. Suppose this 0 direction points rightward (eastward). If you are standing at point 𝐴, then the rays with endpoint 𝐴 indicate the directions that you can be facing. Suppose direction 𝐴𝐵⃗ is numbered 40, sort of a northeast direction, and a direction 𝐴𝐶⃗ is numbered 250, sort of southwesterly. Then we could try to compute the angle measure of ∠𝐵𝐴𝐶 by taking the difference between the numbers: |250 − 40| = 210. But this number is greater than 180, so it will not work as an angle measure. However, the number 250 − 360 = −110 also corresponds to the ray 𝐴𝐶⃗, so we can try again with this number: | − 110 − 40| = 150. This is between 0 and 180, so 150 is the measure of ∠𝐵𝐴𝐶. Polar Angles. Using protractors directly is somewhat awkward; the inverse map from rays to real numbers is often more convenient. For a given protractor 𝛿 at 𝐴, we will say the point 𝐵 (or the ray 𝐴𝐵⃗ ) has polar angle 𝑡 if 𝑡 is one of the real numbers for which 𝛿(𝑡) = 𝐴𝐵⃗
16 2.Axioms for the Plane Two distinc rays never have the same polar angle.By Axiom3,if the polar angles were the same,the angle measure of the rays would be zero and the rays would form a zero angle les of AB and AC satisfy (AB)-(AC)<180.we A choice of polar angle for AB will be denoted by (AB).even though the value of this angle is ambiguous without context,since adding 360 to this number produces another polar angle.To get around this problem in a given situation and to choose a polar angle unambigue ously,we can restrict the protractorto a half-open segment in the real numbers of length 360. For example,given the half-open interval [0,360),which is the set of real numbers t,with ost<360,any protractor at A will define a one-to-one correspondence be- tween this interval and the rays with endpoint A.Therefore,(AB)can be defined as the inverse of the restriction of 6 to this interval,or in other words,(AB)will be the polar angle of B with(AB)<360. Another choice of interval that is sometimes convenient is(-180,+180],the real numbers-180<ts 180.More generally,one can choose unambiguous polar angles 0 either in the interval a s<a+360 or the interval a-180s<a+180,where a is any chosen real number. otractor fun nat A,so are(t+k)and(t+k),since these cha ges erences. This provide convenience for computation:one may assign polar angle 0 to any convenient ray AB Some Angle Terminology.There are a number of terms that specify a relation- ship between two angles or describe the size of an angle. Supplementary Angles:Two angles are supplementary if the sum of their angle measures is 180.The vertices of the two angles do not have to be the same.but if AB and AC are opp osite rays and Dis a point not on AB,then BAD andCAD are supplementary angles of AB and Right Angle ith measure 90.Two lines AB and AC are Acute Angle:An acute angle is an angle with measure less than 90,the measure of a right angle. Obtuse Angle:An obtuse angle is an angle with measure greater than 90,the mea- sure of a right angle. Notice that a supplementary angle of a right angle is also a right angle.Therefore. two perpendicular lines form four right angles
16 2. Axioms for the Plane Two distinct rays never have the same polar angle. By Axiom 3, if the polar angles were the same, the angle measure of the rays would be zero and the rays would form a zero angle. If two values of the polar angles of 𝐴𝐵⃗ and 𝐴𝐶⃗ satisfy |𝜃(𝐴𝐵) − 𝜃( ⃗ 𝐴𝐶)| < 180 ⃗ , we say the polar angles measure ∠𝐵𝐴𝐶. A choice of polar angle for 𝐴𝐵⃗ will be denoted by 𝜃(𝐴𝐵) ⃗ , even though the value of this angle is ambiguous without context, since adding 360 to this number produces another polar angle. To get around this problem in a given situation and to choose a polar angle unambiguously, we can restrict the protractor 𝛿 to a half-open segment in the real numbers of length 360. For example, given the half-open interval [0, 360), which is the set of real numbers 𝑡, with 0 ≤ 𝑡 < 360, any protractor at 𝐴 will define a one-to-one correspondence between this interval and the rays with endpoint 𝐴. Therefore, 𝜃(𝐴𝐵) ⃗ can be defined as the inverse of the restriction of 𝛿 to this interval, or in other words, 𝜃(𝐴𝐵) ⃗ will be the polar angle of 𝐵 with 0 ≤ 𝜃(𝐴𝐵) < 360 ⃗ . Another choice of interval that is sometimes convenient is (−180, +180], the real numbers −180 < 𝑡 ≤ 180. More generally, one can choose unambiguous polar angles 𝜃 either in the interval 𝑎 ≤ 𝜃 < 𝑎 + 360 or the interval 𝑎 − 180 ≤ 𝜃 < 𝑎 + 180, where 𝑎 is any chosen real number. If 𝛿(𝑡)is a protractor function at 𝐴, so are 𝛿(𝑡+𝑘) and 𝛿(−𝑡+𝑘), since these changes of real numbers do not affect the absolute value of the differences. This provides a convenience for computation: one may assign polar angle 0 to any convenient ray 𝐴𝐵⃗ . Some Angle Terminology. There are a number of terms that specify a relationship between two angles or describe the size of an angle. Supplementary Angles: Two angles are supplementary if the sum of their angle measures is 180. The vertices of the two angles do not have to be the same, but if 𝐴𝐵⃗ and 𝐴𝐶⃗ are opposite rays and 𝐷 is a point not on 𝐴𝐵, then ∠𝐵𝐴𝐷 and ∠𝐶𝐴𝐷 are supplementary angles. Vertical Angles: Two angles ∠𝐵𝐴𝐶 and ∠𝐵′𝐴𝐶′ are vertical angles if 𝐴𝐵⃗′ is the opposite ray of 𝐴𝐵⃗ and 𝐴𝐶⃗′ is the opposite ray of 𝐴𝐶⃗. These vertical angles have the same angle measure. Right Angle: A right angle is an angle with measure 90. Two lines 𝐴𝐵 and 𝐴𝐶 are perpendicular if ∠𝐵𝐴𝐶 is a right angle. Acute Angle: An acute angle is an angle with measure less than 90, the measure of a right angle. Obtuse Angle: An obtuse angle is an angle with measure greater than 90, the measure of a right angle. Notice that a supplementary angle of a right angle is also a right angle. Therefore, two perpendicular lines form four right angles
Plane Separation 17 Definition 2.5.The perpendicular bisector ofa segment AB is the line through the midpoint ofAB perpendicular to AB. The perpendicular bisector of a segment always exists,since the midpoint M exists by the Ruler Axiom and a right angle DMA exists by the Protractor Axiom. Plane Separation There is one more axiom that is important for relationships of order and separation.It ensures that our model really acts like a two-sided two-dimensional space. Axiom 4(Plane Separation).For any line m,any point not on the line belongs to one of two disjoint sets called the half-planes of the line. Points P and Q are in different half-planes when PQ intersects m.They are in the same half-plane if PQ does not intersect m. Let m =AB.If a protractor is chosen so that (AB)=0,the points P for which 0<(AP)<180 lie in one half-plane,and the Q for which-180<(AQ)<0 are in the other half-plane. Polar angle =+125 Polar angle-140 Polar angle=-15 Figure 3.Point Pin the Half-plane Opposite the Half-plane of QandR Definition2.6.Given an angleBAC,the interior of theangle is the intersection of the half-plane of AC containing B and the half-plane of AB containing C.For ABAC, the interior of the triangle is the intersection of the interiors of the three angles ∠BAC,∠ACB,∠CBA. .A polygon isa convex polygon if for everyine is a side extended.all the othe olane of the line.The interior of a convex polygon is the inter ection of all these half-planes It follows that in a convex polygon,for any vertex angle,the interior of the polygon is contained in the interior of the angle.Since half-planes are convex,the interior of an angle or the interior of a triangle or other convex polygon is convex also
Plane Separation 17 Definition 2.5. The perpendicular bisector of a segment 𝐴𝐵 is the line through the midpoint of 𝐴𝐵 perpendicular to 𝐴𝐵. The perpendicular bisector of a segment always exists, since the midpoint 𝑀 exists by the Ruler Axiom and a right angle ∠𝐷𝑀𝐴 exists by the Protractor Axiom. Plane Separation There is one more axiom that is important for relationships of order and separation. It ensures that our model really acts like a two-sided two-dimensional space. Axiom 4 (Plane Separation). For any line 𝑚, any point not on the line belongs to one of two disjoint sets called the half-planes of the line. • Points 𝑃 and 𝑄 are in different half-planes when 𝑃𝑄 intersects 𝑚. They are in the same half-plane if 𝑃𝑄 does not intersect 𝑚. • Let 𝑚 = 𝐴𝐵. If a protractor is chosen so that 𝜃(𝐴𝐵) = 0 ⃗ , the points 𝑃 for which 0 < 𝜃(𝐴𝑃) < 180 ⃗ lie in one half-plane, and the 𝑄 for which −180 < 𝜃(𝐴𝑄) < 0 ⃗ are in the other half-plane. Polar angle = -140 Polar angle = -15 Polar angle = +125 A Q B P R Figure 3. Point 𝑃 in the Half-plane Opposite the Half-plane of 𝑄 and 𝑅 Note. We refer to the two half-planes as opposite one another. If two points are in different half-planes, we say they are on opposite sides of the line. Definition 2.6. Given an angle ∠𝐵𝐴𝐶, the interior of the angle is the intersection of the half-plane of 𝐴𝐶 containing 𝐵 and the half-plane of 𝐴𝐵 containing 𝐶. For △𝐵𝐴𝐶, the interior of the triangle is the intersection of the interiors of the three angles ∠𝐵𝐴𝐶, ∠𝐴𝐶𝐵, ∠𝐶𝐵𝐴. Definition 2.7. A polygon 𝑃1𝑃2 . 𝑃𝑛 is a convex polygon if, for every line 𝑃𝑘𝑃𝑘+1 that is a side extended, all the other vertices are in one half-plane of the line. The interior of a convex polygon is the intersection of all these half-planes It follows that in a convex polygon, for any vertex angle, the interior of the polygon is contained in the interior of the angle. Since half-planes are convex, the interior of an angle or the interior of a triangle or other convex polygon is convex also
18 2.Axioms for the Plane h n of the in to dissect the figure into triangles or other convex polygons and then take the interior of the convex polygons,plus common edges,to be the interior.But in general one must ac A Ithis is not jus som and animation,and they can twist into amazingly complicated shapes.One for determining which points are inside a polygon is to intersect the polygon with a line passing thro h sides but not vertices.Then the points of inters e di e po th nis ala mplicated and to show that the interior is well-defined is even more so. B Figure 4.Intersection of CD with Ray Through Interior Point D Theorem 2.8(Addition of Angle Measure).A point D is in the ndme polar anle oD benen the polar n those angles are chosen so that they measure BAC.In this case,mBAD+mDAC= m∠BAC. Proof.Let e(AB)=b,0(AC)=c,(AD)=d,with |b-cl<180.Then we need to prove that D is an interior point if and only if some choice ofd is between band c. Assume without loss of generality that b is the smaller of b and c,so b<c<b+180 D is in the half-plane of AB containing C when for some choice of d,b<d b+180. D is also in the half-plane of AC containing B when d <c.Thus,b<d <c if and only if D is in the interior of the angle. The angle mea ures are m∠BAD=d-b,m∠DAC=c-d,andm∠BAC-c-b This proves the final statement. If D is in the interior of BAC,one also says AD is between AB and AC.All the points of this ray have polar angle d,so they are interior to BAC. Definition 2.9.The angle bisector of BAC is the ray AD for which mBAD m∠DAC=(1/2)m∠BAC Using the notation from the proof of the theorem.the angle bisector exists and has polar angle d =(b +c)/2
18 2. Axioms for the Plane The definition of the interior of a nonconvex polygon in general is surprisingly technical and will not be attempted here, since it would be an inappropriate digression in an introductory work. For a polygon with only a few sides, such as a quadrilateral, it is not hard to dissect the figure into triangles or other convex polygons and then take the interior of the convex polygons, plus common edges, to be the interior. But in general one must account for polygons with hundreds of thousands of sides. And this is not just some sort of “abstract mathematics” problem, for such triangles really appear in computer graphics and animation, and they can twist into amazingly complicated shapes. One algorithm for determining which points are inside a polygon is to intersect the polygon with a line passing through sides but not vertices. Then the points of intersection of sides with the line divide the line into alternating segments of outside points and inside points. But just to prove that the number of intersection points is even, so that this algorithm works, is complicated and to show that the interior is well-defined is even more so. A B C D Figure 4. Intersection of 𝐶𝐷 with Ray Through Interior Point 𝐷 Theorem 2.8 (Addition of Angle Measure). A point 𝐷 is in the interior of ∠𝐵𝐴𝐶 if and only if some polar angle of 𝐴𝐷⃗ is between the polar angles of 𝐴𝐵⃗ and 𝐴𝐶⃗, provided those angles are chosen so that they measure ∠𝐵𝐴𝐶. In this case, 𝑚∠𝐵𝐴𝐷 + 𝑚∠𝐷𝐴𝐶 = 𝑚∠𝐵𝐴𝐶. Proof. Let 𝜃(𝐴𝐵) = 𝑏 ⃗ , 𝜃(𝐴𝐶) = 𝑐 ⃗ , 𝜃(𝐴𝐷) = 𝑑 ⃗ , with |𝑏 − 𝑐| < 180. Then we need to prove that 𝐷 is an interior point if and only if some choice of 𝑑 is between 𝑏 and 𝑐. Assume without loss of generality that 𝑏 is the smaller of 𝑏 and 𝑐, so 𝑏 < 𝑐 < 𝑏+180. 𝐷 is in the half-plane of 𝐴𝐵 containing 𝐶 when for some choice of 𝑑, 𝑏 < 𝑑 < 𝑏 + 180. 𝐷 is also in the half-plane of 𝐴𝐶 containing 𝐵 when 𝑑 < 𝑐. Thus, 𝑏 < 𝑑 < 𝑐 if and only if 𝐷 is in the interior of the angle. The angle measures are 𝑚∠𝐵𝐴𝐷 = 𝑑 − 𝑏, 𝑚∠𝐷𝐴𝐶 = 𝑐 − 𝑑, and 𝑚∠𝐵𝐴𝐶 = 𝑐 − 𝑏. This proves the final statement. □ If 𝐷 is in the interior of ∠𝐵𝐴𝐶, one also says 𝐴𝐷⃗ is between 𝐴𝐵⃗ and 𝐴𝐶⃗. All the points of this ray have polar angle 𝑑, so they are interior to ∠𝐵𝐴𝐶. Definition 2.9. The angle bisector of ∠𝐵𝐴𝐶 is the ray 𝐴𝐷⃗ for which 𝑚∠𝐵𝐴𝐷 = 𝑚∠𝐷𝐴𝐶 = (1/2)𝑚∠𝐵𝐴𝐶. Using the notation from the proof of the theorem, the angle bisector exists and has polar angle 𝑑 = (𝑏 + 𝑐)/2
Rigid Motions and Lines 19 Theorem 2.10(Crossbar).Given any angle BAC: (a)The interior points of BC are interior to BAC. (b)For any point D interior to BAC,the ray AD intersects the segment BC. Proof.Statement(a)istrue since BisonAB,so the interior points of BC are contained in the half-plane of AB containing C.Likewise,the interior points of CB are contained in the half-plane of AC containing B.The set of interior points of BC is the intersection of the interior points of these rays. Toshow(b).continue to use the p olar ang inceb<d<c we see that Band Care on opposite sides of the interior of ZBAC,therefore,at a point of AD Rigid Motions and Lines We will now use our ruler and protractor axioms to prove some essential properties for doing geometry with rigid motions. Theorem 2.11(Line Images).IfT is a rigid motion,the T-image of a line,segment,or ray is, respectively,aline,segment,or ray Proof.IfA,B.C 11 nen,then m.∠BAC =0orm∠BAC 10.Then(T()T(C)also.This proves that the image ofa line is a line Segments and rays are subsets of lines defined by distance relations.a point c on n is in segment AB if and only if llACl +BCll =.This equation still holds true for the image points,so T(C)is in the egment T(A)T(B).Similar reasoning for rays shows that the image of AB is T(A)T(B) -0-0- Figure 5.A and B fixed-All Line Points Fixed This next theorem will play a key role in classifying rigid motions. Theorem 2.12(Two Fixed Points).Ifa rigid motion Tfixes two distinct points Aand B. then it fixes all points of AB. Proof.To sayT fixes a point A means that T(A)=A For a point ConAB,let D=T(C).IfDis distinct from C,then DAll =T(C)T(A)II =IICAll,so A is the midpoint of CD.The same reasoning shows that B is also the mid- point of CD.so A=B.a contradiction.So D must be C. ▣
Rigid Motions and Lines 19 Theorem 2.10 (Crossbar). Given any angle ∠𝐵𝐴𝐶: (a) The interior points of 𝐵𝐶 are interior to ∠𝐵𝐴𝐶. (b) For any point 𝐷 interior to ∠𝐵𝐴𝐶, the ray 𝐴𝐷⃗ intersects the segment 𝐵𝐶. Proof. Statement (a) is true since 𝐵 is on 𝐴𝐵, so the interior points of 𝐵𝐶⃗ are contained in the half-plane of 𝐴𝐵 containing 𝐶. Likewise, the interior points of 𝐶𝐵⃗ are contained in the half-plane of 𝐴𝐶 containing 𝐵. The set of interior points of 𝐵𝐶 is the intersection of the interior points of these rays. To show (b), continue to use the polar angles 𝑏, 𝑐, and 𝑑 as above. Since 𝑏 < 𝑑 < 𝑐, we see that 𝐵 and 𝐶 are on opposite sides of 𝐴𝐷. Thus, 𝐵𝐶 intersects 𝐴𝐷 at a point in the interior of ∠𝐵𝐴𝐶, therefore, at a point of 𝐴𝐷⃗. □ Rigid Motions and Lines We will now use our ruler and protractor axioms to prove some essential properties for doing geometry with rigid motions. Theorem 2.11 (Line Images). If T is a rigid motion, the T-image of a line, segment, or ray is, respectively, a line, segment, or ray. Proof. If 𝐴, 𝐵, 𝐶 are distinct collinear points on line 𝑛, then 𝑚∠𝐵𝐴𝐶 = 0 or 𝑚∠𝐵𝐴𝐶 = 180. Then 𝑚∠𝑇(𝐵)𝑇(𝐴)𝑇(𝐶) also equals 0 or 180. This proves that the image of a line is a line. Segments and rays are subsets of lines defined by distance relations. A point 𝐶 on 𝑛 is in segment 𝐴𝐵 if and only if ‖𝐴𝐶‖ + ‖𝐵𝐶‖ = ‖𝐴𝐵‖. This equation still holds true for the image points, so 𝑇(𝐶) is in the segment 𝑇(𝐴)𝑇(𝐵). Similar reasoning for rays shows that the image of 𝐴𝐵⃗ is ⃖⃖⃖⃖⃖⃖⃖⃖⃗ 𝑇(𝐴)𝑇(𝐵). □ A B Figure 5. 𝐴 and 𝐵 fixed — All Line Points Fixed This next theorem will play a key role in classifying rigid motions. Theorem 2.12 (Two Fixed Points). If a rigid motion T fixes two distinct points 𝐴 and 𝐵, then it fixes all points of 𝐴𝐵. Proof. To say 𝑇 fixes a point 𝐴 means that 𝑇(𝐴) = 𝐴. For a point𝐶 on𝐴𝐵, let 𝐷 = 𝑇(𝐶). If 𝐷 is distinct from 𝐶, then ‖𝐷𝐴‖ = ‖𝑇(𝐶)𝑇(𝐴)‖ = ‖𝐶𝐴‖, so 𝐴 is the midpoint of 𝐶𝐷. The same reasoning shows that 𝐵 is also the midpoint of 𝐶𝐷, so 𝐴 = 𝐵, a contradiction. So 𝐷 must be 𝐶. □