Plane Geometry - Bruce E. Shapiro

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34 SECTION 8. HILBERT’S AXIOMSiom of continuity introduces the continuity of real numbers to geometry,and completeness tells us that everything we can possibly know we canlearn from these axioms.Hilbert’s Axioms of Connection1. Two distinct points A and B always completely determine a straight line a. We writeAB = a or BA = a.2. Any two distinct points of a straight line completely determine that line; that is, ifAB = a and AC = a, where B ≠ C, then also BC = a.3. Three points A, B, C not situated in the same straight line always completely determineaplane α. We write ABC = α.4. Any three points A, B, C of a plane α, which do not lie in the same straightline,completely determine that plane.5. If two points A, B of a straight line a lie in a plane α, then every point of a lies in α.6. If two planes α, β have a point A in common, then they have at least a second pointB in common.7. Upon every straight line there exist at least two points, in every plane at least threepoints not lying in the same straight line, and in space there exist at least four pointsnot lying in a plane.Hilbert’s Axioms of Order1. If A, B, C are points of a straight line and B lies between A and C, then B lies alsobetween C and A.2. If A and C are two points of a straight line, then there exists at least one point Blying between A and C and at least one point D so situated that C lies between Aand D.3. Of any three points situated on a straight line, there is always one and only one whichlies between the other two.4. Any four points A, B, C, D of a straight line can always be so arranged that B shalllie between A and C and also between A and D, and, furthermore, that C shallliebetween A and D and also between B and D.5. Let A, B, C be three points not lying in the same straight line and let a be a straightline lying in the plane ABC and not passing through any of the points A, B, C.Then, if the straight line a passes through a point of the segment AB, it will alsopass through either a point of the segment BC or a point of the segment AC.Hilbert’s Axiom of ParallelsIn a plane α there can be drawn through any point A, lying outside of a straight line a, oneand only one straight line which does not intersect the line a. This straight line is called theparallel to a through the given point A.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 8. HILBERT’S AXIOMS 35Hilbert’s Axioms of Congruence1. If A, B are two points on a straight line a, and if A ′ is a point upon the same oranother straight line a1, then, upon a given side of A ′ on the straight line a ′ , we canalways find one and only one point B ′ so that the segment AB (or BA) is congruentto the segment A ′ B ′ . We indicate this relation by writing AB ∼ = A ′ B ′ Every segmentis congruent to itself; that is, we always have AB ∼ = AB.2. If AB ∼ = A ′ B ′ and also AB ∼ = A ′′ B ′′ , then A ′ B ′ ∼ = A ′′ B ′′3. Let AB and BC be two segments of a straight line a which have no points in commonaside from the point B, and, furthermore, let A ′ B ′ and B ′ C ′ be two segments of thesame or of another straight line a having, likewise, no point other than B ′ in common.Then, if AB ≡ A ′ B ′ and BC ∼ = B ′ C ′ , we have AC ∼ = A ′ C ′ .4. Let an angle (h, k) be given in the plane α and let a straight line a ′ be given in aplane α ′ . Suppose also that, in the plane α ′ , a definite side of the straight line a ′be assigned. Denote by h ′ a half-ray of the straight line a ′ emanating from a pointO ′ of this line. Then in the plane α ′ there is one and only one half-ray k ′ such thatthe angle (h, k), or (k, h), is congruent to the angle (h ′ , k ′ ) and at the same timeall interior points of the angle (h ′ , k ′ ) lie upon the given side of a ′ . We express thisrelation by means of the notation ∠(h, k) ∼ = ∠(h ′ , k ′ ) Every angle is congruent toitself; that is, ∠(h, k) ∼ = ∠(h, k).5. If the ∠(h, k) ∼ = ∠(h ′ , k ′ ) and ∠(h, k) ∼ = ∠(h ′′ , k ′′ ) then ∠(h ′ , k ′ ) ∼ = ∠(h ′′ , k ′′ )6. If, in the two triangles ABC and A ′ B ′ C ′ the congruences ABV A ′ B, AC ∼ = A ′ C ′and ∠BAC ∼ = ∠B ′ A ′ C ′ then ∠ABC ∼ = ∠A ′ B ′ C ′ and ∠ACB ∼ = ∠A ′ C ′ B ′ .Hilbert’s Axiom of Continuity (Archimedian Axiom)Let A 1 A be any point on a straight line AB. Choose A 2 , A 3 , .. so that A 1 is between Aand A 2 , A 2 is between A 1 and A 3 , etc, such thatAA 1 = A 1 A 2 = A 2 A 3 = · · ·Then there always exists a certain point A n such that B lies between A and A n.In more modern terms, ∀ɛ > 0 and ∀x > 0, then ∃m ∈ N such that mɛ > x.Hilbert’s Axioms of CompletenessTo a system of points, straight lines, and planes, it is impossible to add other elements insuch a manner that the system thus generalized shall form a new geometry obeying all ofthe five groups of axioms. In other words, the elements of geometry form a system which isnot susceptible of extension, if we regard the five groups of axioms as valid.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
Page 1: Foundations of GeometryLecture Note
Page 4 and 5: ivCONTENTS30 Triangles in Neutral G
Page 6 and 7: 2 SECTION 1. PREFACEare enrolled in
Page 8 and 9: 4 SECTION 1. PREFACEre-learning it
Page 10 and 11: 6 SECTION 2. NCTM3. To guide in the
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Page 16 and 17: 12 SECTION 3. CA STANDARDIn 2005 Ca
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Page 80 and 81: 76 SECTION 15. THE PLANE SEPARATION
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Page 84 and 85: 80 SECTION 16. ANGLESFigure 16.1: T
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84 SECTION 16. ANGLESFigure 16.6: I
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86 SECTION 16. ANGLESD cannot be in
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88 SECTION 17. THE CROSSBAR THEOREM
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90 SECTION 17. THE CROSSBAR THEOREM
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92 SECTION 18. LINEAR PAIRSFigure 1
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94 SECTION 18. LINEAR PAIRSγ + ∠
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96 SECTION 19. ANGLE BISECTORSwith
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98 SECTION 19. ANGLE BISECTORSAngle
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100 SECTION 20. THE CONTINUITY AXIO
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102 SECTION 20. THE CONTINUITY AXIO
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104 SECTION 21. SIDE-ANGLE-SIDEBirk
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106 SECTION 21. SIDE-ANGLE-SIDEFigu
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108 SECTION 22. NEUTRAL GEOMETRYEll
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110 SECTION 22. NEUTRAL GEOMETRY«
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112 SECTION 23. ANGLE-SIDE-ANGLEFig
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114 SECTION 23. ANGLE-SIDE-ANGLE«
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116 SECTION 24. EXTERIOR ANGLESTheo
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118 SECTION 24. EXTERIOR ANGLES« C
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120 SECTION 25. ANGLE-ANGLE-SIDEBy
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122 SECTION 26. SIDE-SIDE-SIDECase
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124 SECTION 26. SIDE-SIDE-SIDEBy si
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126 SECTION 27. SCALENE AND TRIANGL
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128 SECTION 27. SCALENE AND TRIANGL
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130 SECTION 28. CHARACTERIZATION OF
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132 SECTION 28. CHARACTERIZATION OF
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134 SECTION 29. TRANSVERSALSFigure
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136 SECTION 29. TRANSVERSALSCorolla
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138 SECTION 30. TRIANGLES IN NEUTRA
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146 SECTION 31. QUADRILATERALS IN N
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156 SECTION 32. THE EUCLIDEAN PARAL
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168 SECTION 33. RECTANGLESLemma 33.
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170 SECTION 33. RECTANGLESFigure 33
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172 SECTION 33. RECTANGLES« CC BY-
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174 SECTION 34. THE PARALLEL PROJEC
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176 SECTION 34. THE PARALLEL PROJEC
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178 SECTION 35. SIMILAR TRIANGLESBy
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180 SECTION 35. SIMILAR TRIANGLESCo
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182 SECTION 36. TRIANGLE CENTERSFig
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184 SECTION 36. TRIANGLE CENTERSThe
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186 SECTION 36. TRIANGLE CENTERSCen
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188 SECTION 37. AREADefinition 37.5
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190 SECTION 37. AREAToday’s Lesso
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192 SECTION 38. THE PYTHAGOREAN THE
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202 SECTION 39. CIRCLESFigure 39.1:
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208 SECTION 39. CIRCLESTheorem 39.1
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210 SECTION 39. CIRCLESLet D be the
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212 SECTION 39. CIRCLES« CC BY-NC-
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214 SECTION 40. CIRCLES AND TRIANGL
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228 SECTION 41. EUCLIDEAN CIRCLESBy
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238 SECTION 43. INDIANA BILL 246the
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240 SECTION 44. ESTIMATING πFigure
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244 SECTION 44. ESTIMATING πn π n
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246 SECTION 44. ESTIMATING πTable
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248 SECTION 44. ESTIMATING πnums =
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250 SECTION 44. ESTIMATING π« CC
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252 SECTION 45. EUCLIDEAN CONSTRUCT
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264 SECTION 46. HYPERBOLIC GEOMETRY
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272SECTION 47.PERPENDICULAR LINES I
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274SECTION 47.PERPENDICULAR LINES I
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276 SECTION 48. PARALLEL LINES IN H
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284 SECTION 49. TRIANGLES IN HYPERB
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298 SECTION 51. THE POINCARE DISK M
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306 SECTION 52. ARC LENGTHAs we see
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308 SECTION 52. ARC LENGTHProof. Su
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310 SECTION 52. ARC LENGTHThus the
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312 SECTION 52. ARC LENGTHMeasuring
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314 SECTION 53. SPHERICAL GEOMETRYW
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316 SECTION 53. SPHERICAL GEOMETRYF
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318 SECTION 53. SPHERICAL GEOMETRYP
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320 APPENDIX A. SYMBOLS USEDAppendi
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322 APPENDIX A. SYMBOLS USEDTechnol
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Plane Geometry - Bruce E. Shapiro

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