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.

The SAS hypothesis: Under the correspondence ABC ↔ XYZ, let two sides and the included angle of ABC be congruent, ∼ respectively, to the corresponding two sides and included angle of XYZ.Thatis,forexample,AB = ∼ ∼ XY, BC = YZ,and∠ ABC = ∠ XYZ.

And the SAS postulate: If the SAS hypothesis holds for two triangles under some correspondence between their vertices, then the triangles are congruent.

Key points: • SAS must be assumed as an axiom; you cannot establish it from previous axioms.

• Geometry with SAS congruence as an axiom is absolute geometry. Both Euclidean and hyperbolic (a non- Euclidean) geometry are extensions of absolute geometry.

• Absolute geometry lacks a parallel postulate of any sort. However, it does include the idea that parallel lines exist. One of the non-Euclidean geometries, spherical or elliptical geometry, does not even allow for the existence of parallel lines ... and is not an absolute geometry. No guarantee of SAS there, unless we explicitly assume it again.

• You’ve encountered a geometry that lacks SAS congruence - Taxicab geometry. So Taxicab geometry is not an absolute geometry either.

• Once we have assumed the SAS postulate for congruence, we can prove ASA and SSS. You are asked to do some fill-in-the-blank proofs of these as exercises.

• There are three other key theorems that come out of the SAS postulate: (1) the isosceles triangle theorem, (2) the perpendicular bisector theorem, (3) existence of a perpendicular. Theorem list on the next page; please see text for proof.

• Absolute geometry with congruent triangles is starting to look very much like Euclidean geometry. You have to be very careful to base results on things already established, and not bring in any prior knowledge of Euclidean geometry. Be sure you can justify any calculation you do by citing a theorem/axiom which has been explicitly established. Isosceles triangle theorem: [Kay, p 142] A triangle is isosceles if and only if its base angles are congruent.

Note that the definition of isosceles is two sides equal. Equal base angles is not part of the definition - it has to be proven as a theorem, and is only true in absolute geometry. (In Taxicab geometry, you saw isosceles triangles with unequal base angles ... so it can’t be part of the definition of isosceles!)

Also note that there’s a neat and simple proof of this (Euclid’s is complicated) - an isosceles triangle is congruent to a mirror image of itself, which lets you match up base angles and opposite sides as corresponding parts.

Perpendicular bisector theorem: [Kay, p 144] The set of all points equidistant from two distinct points A and B is the perpendicular bisector of the segment AB.

This alomst reads like a definition, but it isn’t - perpendicular and bisector are already defined. The theorem tells you that if you take all points equidistant from A and B, you will get a line that (1) intersects AB at right angles, and (2) intersects AB at its midpoint. It also suggests how to construct a perpendicular bisector, using the compass to draw circles of equal radius centered at A and B.

Existence of a perpendicular from an external point: [Kay, p 147] Let the line l and a point A not on l be given. Then, there exists a unique line m perpendicular to l and passing through A.

And another construction theorem - an existence theorem is generally an assertion that the thing existing can be constructed somehow ... and a description of how to construct it (and that the construction gives you what you want) is proof that it exists.