Plane Geometry - Bruce E. Shapiro

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102 SECTION 20. THE CONTINUITY AXIOMhence r is in the interior of ∠(BAC).Therefore it intersects BC at some point D such that B ∗ D ∗ C. Since0 < BD < BC, there is some number x such that 0 < x < d and α = f(x).Hence the function is onto.Since the function f(x) is strictly increasing and onto, by theorem 20.1,f(x) is continuous.Now suppose there exist x and y in (0, d) such that f(x) = f(y).By the ruler postulate there must be points D and E on BC such thatx = BD and y = BE.Since f(x) = f(y), we know that µ(∠BAD) = µ(∠BAE). This is onlypossible of −→ AE = −→ AD. Since both rays intersect BC, and since they arethe same ray, their intersection points must be identical, giving D = E ⇒BD = BE ⇒ x = y.Thus f(x) = f(y) ⇒ x = y, which means f is one-to-one. We have alreadyshown that f is onto. Since f is one-to-one and onto, it is invertible.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 21Side-Angle-SideDefinition 21.1 Two triangles △ABC and △DEF are congruent(△ABC ∼ = △DEF ) if there is a correspondence between the vertices ofthe first triangle and the vertices of the second triangle such that the correspondingangles are congruent and corresponding sides are congruent.Figure 21.1: SAS postulate. Distances AB = DE, BC = EF and anglesα = β hence the two triangles are congruent.The Side-Angle-Side Congruence Condition (SAS) is Euclid’s Proposition4; it was taken as an axiom by Hilbert as well as the SMSG (Axiom 15).103
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Foundations of GeometryLecture Note
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ivCONTENTS30 Triangles in Neutral G
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2 SECTION 1. PREFACEare enrolled in
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6 SECTION 2. NCTM3. To guide in the
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8 SECTION 2. NCTMTechnology. Techno
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12 SECTION 3. CA STANDARDIn 2005 Ca
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16 SECTION 3. CA STANDARDCalifornia
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26 SECTION 6. REAL NUMBERSThe inter
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28 SECTION 6. REAL NUMBERS“The Su
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30 SECTION 7. EUCLID’S ELEMENTSEu
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32 SECTION 7. EUCLID’S ELEMENTSFi
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34 SECTION 8. HILBERT’S AXIOMSiom
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36 SECTION 8. HILBERT’S AXIOMS«
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38 SECTION 9. BIRKHOFF/MACLANE AXIO
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40 SECTION 9. BIRKHOFF/MACLANE AXIO
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42 SECTION 10. THE SMSG AXIOMSP is
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44 SECTION 10. THE SMSG AXIOMS« CC
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46 SECTION 11. THE UCSMP AXIOMSin s
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48 SECTION 11. THE UCSMP AXIOMS« C
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50 SECTION 12. VENEMA’S AXIOMS3.
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Page 66 and 67: 62 SECTION 14. BETWEENNESSFigure 14
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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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Page 96 and 97: 92 SECTION 18. LINEAR PAIRSFigure 1
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152 SECTION 31. QUADRILATERALS IN N
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154 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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182 SECTION 36. TRIANGLE CENTERSFig
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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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214 SECTION 40. CIRCLES AND TRIANGL
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240 SECTION 44. ESTIMATING πFigure
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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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306 SECTION 52. ARC LENGTHAs we see
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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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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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