Plane Geometry - Bruce E. Shapiro

284 SECTION 49. TRIANGLES IN HYPERBOLIC GEOMETRYFigure 49.1: Construction of an isoceles triangle with measure less that ɛ.Proof. Let ɛ > 0 be given.Construct a right triangle △ABC with one interior angle 90 − ɛ as follows:Let A and B be points, and define Q so that ∠ABQ = 90 − ɛ. The dropa perpendicular from A to BQ and call its foot C. Then △ABC is a righttriangle with one interior angle of 90 − ɛ.Thenandσ(△ABC) = 90 + 90 − ɛ + µ(∠ACB)> 180 − ɛδ(△ABC) = 180 − σ(△ABC)< 180 − (180 − ɛ) = ɛTheorem 49.3 For every ɛ > 0 there is a number d > 0 such that if leg oftriangle △ABC has length less than d then δ(△ABC) < ɛ.Proof. Let ɛ > 0 be given.By theorem 49.2 there is a right triangle △EF G with right angle at E suchthat δ(△EF G) < ɛ/2.Let d be the length of the shortest side of △EF G.Let △ABC be any triangle with sides shorter d. Let D be the foot ofthe perpendicdular from C to AB. Then △ADC and △BDC are righttriangles with right angles at D (if D does not satisfy A ∗ D ∗ B relabel thevertices and choose another edge as AB).(See figure 49.2.)By construction we chose AC < d, BC < d, and AB < d.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012