Plane Geometry - Bruce E. Shapiro

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120 SECTION 25. ANGLE-ANGLE-SIDEBy hypothesis α = γ and by construction of CP we have ɛ = γ. Hence α =ɛ, which contradicts the conclusion of the exterior angle theorem (theorem24.4). Hence we must rule out C ∗ P ∗ B.The proof that C ∗ B ∗ P is impossible is completely analogous.P = B, which means △ACB ∼ = △ACP ∼ = △DF EHenceFigure 25.2: Proof of the hypotenuse leg theorem.Definition 25.2 A triangle is called a right triangle if one interior angleis a right angle.Definition 25.3 The side opposite the right angle in a right triangle iscalled the hypotenuse. 1Theorem 25.4 (Hypotenuse-Leg Theorem (ASS for right △)) If△ABC and △DEF are right triangles with right angles at C and F , AB =DE, and BC = EF then △ABC ∼ = △DEF .Proof. Let P be a point on −→ AC such that A ∗ C ∗ P and and CP = F D(ruler postulate) (see figure 25.2).Then by the linear pair theorem (theorem 18.4) ∠BCP = 90.Hence by SAS, △BCP ∼ = △EF D.Hence by congruence P B = ED, and by hypothesis ED = BA henceP B = BA.Thus △ABP is isosceles, and therefore by the isosceles triangle theorem(theorem 21.4) ∠BP C = ∠BACSince △BCP ∼ = △EF D, by congruence ∠BP C = ∠EDF .Hence ∠BAC = ∠EDF . By AAS, this gives △ABC = △DEF .1 From the ancient Greek word hypoteínousa (υπoτɛiνoυσηζ in Euclid, e.g., Proposition47, Book 1).« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 26Side-Side-SideTheorem 26.1 (Side-Side-Side (SSS)) If △ABC and △DEF are trianglessuch that AB = DE, BC = EF , and CA = F D then △ABC ∼ =△DEF .Proof. Given AB = DE, BC = EF and AC = F D for triangles △ABCand △DEF .Figure 26.1: Proof of SSS, case H = A.By the triangle construction theorem there is a point G on the oppositeside of ←→ AB from C such that △DEF ∼ = △ABG.Let H = CG ∩ ←→ AB, which exists by the plane separation postulate becauseC and G are on opposite sides of ←→ AB.There are five possible cases: (1) H = A, (2) H = B, (3) A ∗ H ∗ B, (4)H ∗ A ∗ B, and (5) A ∗ B ∗ H.121
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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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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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52 SECTION 12. VENEMA’S AXIOMS«
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54 SECTION 13. INCIDENCE GEOMETRYFi
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56 SECTION 13. INCIDENCE GEOMETRYEx
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58 SECTION 13. INCIDENCE GEOMETRYFi
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60 SECTION 13. INCIDENCE GEOMETRY«
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62 SECTION 14. BETWEENNESSFigure 14
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64 SECTION 14. BETWEENNESSTheorem 1
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66 SECTION 14. BETWEENNESS(a) To sh
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68 SECTION 14. BETWEENNESSThe follo
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Page 80 and 81: 76 SECTION 15. THE PLANE SEPARATION
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170 SECTION 33. RECTANGLESFigure 33
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174 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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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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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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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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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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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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