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Absolute geometry Other theorems of absolute geometry Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see sections 3.5 and 3.6, pp 166 - 181. These sections have familiar theorems; be sure to look at the proofs in the text. Here’s the theorem list: Scalene inequality If one side of a triangle has greater length than another side, then the angle opposite the longer side has the greater angle measure, and, conversely, the side opposite an angle having the greater measure is the longer side. [Kay, p 166] • You knew this one. • Lets you order sides and angles AC < AB < BC if and only if m∠ B < m∠ C < m∠ A • Proof depends on isosceles triangle theorem, and exterior angle inequality, which in turn depend on SAS postulate. Triangle inequality If A, B, and C are any three distinct points, then AB + BC ≥ AC, with equality only when the points are collinear, and A − B − C. quad [Kay, p 167] • You knew this one too. • Geometric interpretation is that the sum of the lengths of any two legs of a triangle must be greater than the length of the third leg. • Proof depends on the scalene inequality (above), which depends on isosceles triangle theorem, and exterior angle inequality, which depend on SAS. Hinge theorem (SAS inequality) If in 4ABC and 4XY Z we have AB = XY , AC = XZ, but m∠ A > m∠ X, then BC > Y Z, and, conversely, if BC > Y Z, then m∠ A > m∠ X. [Kay, p. 169] • Way to compare between two triangles - side opposite wider angle is longer side, and vice versa. • Extends scalene inequality (which only allows comparisons within one triangle, instead of between two triangles). • I don’t think we’ve worked with this explicitly, but it’s pretty self evident (you’d probably assume it without even thinking about it). • Depends directly on SAS. AAS congruence criterion If under some correspondence of their vertices, two angles and a side opposite in one triangle are congruent to the corresponding two angles and side of a second triangle, then the triangles are congruent. [Kay, p 174] • We’re used to an immediate result of AAS from ASA - if two angles line up, we subtract the third from 180, those match two, and the side falls between two angles. This can’t be used here, because we don’t have angle sum equal to 180 guaranteed. • But it can still be proven (proof by contradiction, plus a construction - very similar to the proof of ASA) that AAS congruence gives congruent triangles. Doesn’t rely on angle sum of 180 at all. • Depends on ASA congruence theorem, which depends on SAS postulate. SSA theorem If under some correspondence of their vertices, two triangles have two pairs of corresponding sides and a pair of corresponding angles congruent, and if the triangles are not congruent under this correspondence, then the remaining pair of angles not included by the congruent sides are supplementary angles. [Kay, p 175] ∠ C and ∠ D are supplementary. m∠ C + m∠ D = 180◦ • SSA correspondence cannot be used to prove triangles congruent. • However, there are only two possible cases. You’ve seen this before - it’s the swinging leg effect. One triangle has an acute angle, the other obtuse. • The SSA theorem says there’s a relationship; if the triangles aren’t congruent (one of the possibilities), then then non-included angles must be supplementary (the other possibility).