Absolute geometry Congruent triangles - SAS, ASA, SSS Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see section 3.3, pp 139-150. The problems are all from section 3.3. Proving SSS from SAS SSS Theorem If, under some correspondence between their vertices, two triangles have the three sides of one congruent to the corresponding three sides of the other, then the triangles are congruent under that correspondence. [Kay, p 141] picture conclusions justifications Suppose you have two triangles which Hypothesis (Given) satisfy the hypothesis of SSS: three sides of one congruent to the corre- sponding three sides of the other. The goal is to show that having SSS (thing you want to prove) always leads to having SAS (the postulate); i.e. having the corresponding parts congruent that are marked below: always leads to having the additional marked angles congruent: which proves the triangles are congruent by SAS. The previously proven ASA theorem may also be used as justifi- cation in the proof. The trick in this one is to construct a copy XYZ onto ABC, and show that the copy is congruent to ABC. This is a slightly simplied version - you can use the betweenness relations apparent in the figures. Also assume that the figures are oriented so that the angles at the base are acute (i.e., if this were an obtuse triangle, I’d orient it with the obtuse angle at the “top”). picture conclusions justifications Given Copy ∠ X and ∠ Z onto segment AB, as shown with the vertices at A and C. ∼ ADC = XYZ This gives AD = XY and therefore AD = AB,andalso (At this point you can ignore XYZ for a while.) Construct segment BD. picture conclusions justifications What can you con- clude about DAB? Likewise ... (Assuming betweenness as it appears - you could throw in a couple extra steps and the Crossbar theorem, but just go straight to ...) Angle Addition Postulate Conclusion: So, traingles having an SSS correspondence automatically have an SAS correspondence..