Plane Geometry - Bruce E. Shapiro

Section 25Angle-Angle-SideTheorem 25.1 (Angle-Angle-Side(AAS) a ) If △ABC and △DEF have∠ABC = ∠DEF , ∠BCA = ∠EF D and AC = DF then △ABC ∼ =△DEF .a Along with Angle-Side-Angle this is Euclid’s Proposition 26.Proof. See figure 25.1. Let P ∈ −→ CB such that CP = F E. Such a pointexists by the ruler postulate.Then by Side Angle Side, △ACP ∼ = △DF E (since AC = DF , β = δ, andCP = F E).There are three possible cases: P = B, C ∗ P ∗ B, and C ∗ B ∗ P .Suppose first that C ∗ P ∗ B. Then ɛ is an exterior angle for △ABP . Henceɛ > α and ɛ > ζ by the Exterior angle theorem (theorem 24.4).Figure 25.1: Angle-Angle-Side with α = γ, β = δ, and AC = DF withC ∗ P ∗ B119