Plane Geometry - Bruce E. Shapiro

104 SECTION 21. SIDE-ANGLE-SIDEBirkhoff and MacLane instead chose instead to start with axioms of trianglesimilarity rather than congruence. The UCSMP uses a set of reflectionpostulates instead of SAS.Axiom 21.2 (Side-Angle-Side (SAS) Postulate) If △ABC and△DEF are two triangles such that AB ∼ = DE, ∠ABC ∼ = ∠DEF , andBC ∼ = EF then △ABC ∼ = △DEF .Why can’t we just derive SAS from the other axioms, e.g., like Euclid? Forone thing, Euclid proved his proposition 4 using the concept of placing onefigure on top of another to show congruence (common notion 4). In fact,this should probably have been a separate axiom, and we have not statedany axiom of correspondence.The following example shows that the axioms we have so far are not sufficientto prove SAS.Figure 21.2: SAS failure in taxicab geometry.Example 21.1 Failure of SAS in Taxicab GeometryConsider triangles △ABC and △DEF as shown in figure 21.2.taxicab metric,AB = DE = 2In theAC = DF = 2∠EDF = ∠BAC = 90If SAS held in this model then the two triangles would be congruent. However,EF = 2 ≠ 4 = BCso the two triangles are not congruent.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012