Plane Geometry - Bruce E. Shapiro

SECTION 46. HYPERBOLIC GEOMETRY 267Figure 46.4: In hyperbolic geometry similar triangles are always congruent.Theorem 46.12 The altitude of a Saccheri quadrilateral is shorter thanthe length of its sides.Proof. (Exercise.)Theorem 46.13 The length of the summit of a Saccheri quadrilateral isgreater than the length of its base.Proof. (Exercise.)Theorem 46.14 (AAA in Hyperbolic Geometry) Similar triangles arecongruent:△ABC ∼ △DEF =⇒ △ABC ∼ = △DEFProof. Suppose that △ABC ∼ △DEF but AB ≠ DE, BC ≠ EF andAC ≠ DF . (If any one of the equalities holds the by ASA the triangles arecongruent, so we are in effect assuming that the triangles are not congruent).At least two edges of one triangle are longer than two edges of the secondtriangle. Assume that AB > DE and AC > DF (if not, relabel the verticesaccordingly.)Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.