Plane Geometry - Bruce E. Shapiro
Plane Geometry - Bruce E. Shapiro Plane Geometry - Bruce E. Shapiro
48 SECTION 11. THE UCSMP AXIOMS« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 12Venema’s AxiomsVenema [Venema, 2006] introduces a set of axioms that meld together partsof the axiom systems of Birkhoff, MacLane, the SMSG, and the UCSMP.We present them here for reference, since we will be using this system inthe remainder of the class.Venema’s Undefined TermsThe undefined terms are point, line, distance, half-plane, angle-measure,areaVenema’s Axiom’s of Neutral GeometryExistence says that at least some points exist, and incidents says that everypair of distinct points defines a line.1. Existence Postulate. The collection of all points forms a nonemptyset with more than one (i.e., at least two) points. The set of all points inthe plane is called P.2. Incidence Postulate. Every line is a set of points. For every pair ofdistinct points A, B there is exactly one line l = ←→ AB such that A, B ∈ l.The ruler postulate allows us to associate real numbers – and hence measurements– with distances and line segments.49
- 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
- Page 12 and 13: 8 SECTION 2. NCTMTechnology. Techno
- Page 14 and 15: 10 SECTION 2. NCTM« CC BY-NC-ND 3.
- Page 16 and 17: 12 SECTION 3. CA STANDARDIn 2005 Ca
- Page 18 and 19: 14 SECTION 3. CA STANDARDThe Califo
- Page 20 and 21: 16 SECTION 3. CA STANDARDCalifornia
- Page 22 and 23: 18 SECTION 4. COMMON COREAt all lev
- Page 24 and 25: 20 SECTION 5. LOGIC AND PROOFour th
- Page 26 and 27: 22 SECTION 5. LOGIC AND PROOFRene D
- Page 28 and 29: 24 SECTION 5. LOGIC AND PROOF“I
- Page 30 and 31: 26 SECTION 6. REAL NUMBERSThe inter
- Page 32 and 33: 28 SECTION 6. REAL NUMBERS“The Su
- Page 34 and 35: 30 SECTION 7. EUCLID’S ELEMENTSEu
- Page 36 and 37: 32 SECTION 7. EUCLID’S ELEMENTSFi
- Page 38 and 39: 34 SECTION 8. HILBERT’S AXIOMSiom
- Page 40 and 41: 36 SECTION 8. HILBERT’S AXIOMS«
- Page 42 and 43: 38 SECTION 9. BIRKHOFF/MACLANE AXIO
- Page 44 and 45: 40 SECTION 9. BIRKHOFF/MACLANE AXIO
- Page 46 and 47: 42 SECTION 10. THE SMSG AXIOMSP is
- Page 48 and 49: 44 SECTION 10. THE SMSG AXIOMS« CC
- Page 50 and 51: 46 SECTION 11. THE UCSMP AXIOMSin s
- Page 54 and 55: 50 SECTION 12. VENEMA’S AXIOMS3.
- Page 56 and 57: 52 SECTION 12. VENEMA’S AXIOMS«
- Page 58 and 59: 54 SECTION 13. INCIDENCE GEOMETRYFi
- Page 60 and 61: 56 SECTION 13. INCIDENCE GEOMETRYEx
- Page 62 and 63: 58 SECTION 13. INCIDENCE GEOMETRYFi
- Page 64 and 65: 60 SECTION 13. INCIDENCE GEOMETRY«
- Page 66 and 67: 62 SECTION 14. BETWEENNESSFigure 14
- Page 68 and 69: 64 SECTION 14. BETWEENNESSTheorem 1
- Page 70 and 71: 66 SECTION 14. BETWEENNESS(a) To sh
- Page 72 and 73: 68 SECTION 14. BETWEENNESSThe follo
- Page 74 and 75: 70 SECTION 14. BETWEENNESSExample 1
- Page 76 and 77: 72 SECTION 14. BETWEENNESSorf(A) >
- Page 78 and 79: 74 SECTION 14. BETWEENNESS« CC BY-
- Page 80 and 81: 76 SECTION 15. THE PLANE SEPARATION
- Page 82 and 83: 78 SECTION 15. THE PLANE SEPARATION
- Page 84 and 85: 80 SECTION 16. ANGLESFigure 16.1: T
- Page 86 and 87: 82 SECTION 16. ANGLESCorollary 16.6
- Page 88 and 89: 84 SECTION 16. ANGLESFigure 16.6: I
- Page 90 and 91: 86 SECTION 16. ANGLESD cannot be in
- Page 92 and 93: 88 SECTION 17. THE CROSSBAR THEOREM
- Page 94 and 95: 90 SECTION 17. THE CROSSBAR THEOREM
- Page 96 and 97: 92 SECTION 18. LINEAR PAIRSFigure 1
- Page 98 and 99: 94 SECTION 18. LINEAR PAIRSγ + ∠
- Page 100 and 101: 96 SECTION 19. ANGLE BISECTORSwith
48 SECTION 11. THE UCSMP AXIOMS« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Change language