Plane Geometry - Bruce E. Shapiro

Section 50Area in HyperbolicGeometryDefinition 50.1 We say that two polygonal regions R and R ′ are Equivalentby Dissection if they can each be triangulatedR = T 1 ∪ T 2 ∪ · · · ∪ T nwhereR ′ = T ′ 1 ∪ T ′ 2 ∪ · · · ∪ T ′ nT 1∼ = T′1 , . . . , T n∼ = T′nTheorem 50.2 (Fundamental Theorem of Dissection Theory) If Rand R ′ are polygonal regions such that α(R) = α(R ′ ) then R ≡ R ′ inneutral geometry.Proof. (Beyond the scope of this class.)Theorem 50.3 Equivalence by Dissection is reflexive, symmetric, and transitive,i.e., it is an equivalence relation:1. ∀R, R ≡ R2. R 1 ≡ R 2 =⇒ R 2 ≡ R 13. If R 1 ≡ R 2 and R 2 ≡ R 3 then R 1 ≡ R 3Proof. (Beyond the scope of this class.)Theorem 50.4 (Bolyai’s Theorem) If △ABC and △DEF satisfy δ(△ABC) =δ(△DEF ) then △ABC ≡ △DEF .289