Plane Geometry - Bruce E. Shapiro

120 SECTION 25. ANGLE-ANGLE-SIDEBy hypothesis α = γ and by construction of CP we have ɛ = γ. Hence α =ɛ, which contradicts the conclusion of the exterior angle theorem (theorem24.4). Hence we must rule out C ∗ P ∗ B.The proof that C ∗ B ∗ P is impossible is completely analogous.P = B, which means △ACB ∼ = △ACP ∼ = △DF EHenceFigure 25.2: Proof of the hypotenuse leg theorem.Definition 25.2 A triangle is called a right triangle if one interior angleis a right angle.Definition 25.3 The side opposite the right angle in a right triangle iscalled the hypotenuse. 1Theorem 25.4 (Hypotenuse-Leg Theorem (ASS for right △)) If△ABC and △DEF are right triangles with right angles at C and F , AB =DE, and BC = EF then △ABC ∼ = △DEF .Proof. Let P be a point on −→ AC such that A ∗ C ∗ P and and CP = F D(ruler postulate) (see figure 25.2).Then by the linear pair theorem (theorem 18.4) ∠BCP = 90.Hence by SAS, △BCP ∼ = △EF D.Hence by congruence P B = ED, and by hypothesis ED = BA henceP B = BA.Thus △ABP is isosceles, and therefore by the isosceles triangle theorem(theorem 21.4) ∠BP C = ∠BACSince △BCP ∼ = △EF D, by congruence ∠BP C = ∠EDF .Hence ∠BAC = ∠EDF . By AAS, this gives △ABC = △DEF .1 From the ancient Greek word hypoteínousa (υπoτɛiνoυσηζ in Euclid, e.g., Proposition47, Book 1).« CC BY-NC-ND 3.0. Revised: 18 Nov 2012