Plane Geometry - Bruce E. Shapiro

194 SECTION 38. THE PYTHAGOREAN THEOREMProof. (Exercise.)Theorem 38.4 (Converse of Pythagorean Theorem) If a 2 + b 2 = c 2then ∠C is a right angle.Proof. We are given △ABC with a 2 + b 2 = c 2 .Construct a right angle at point F on rays −→ −−→F G and F H.Define point E ∈ −−→ F H such that F E = a, and define point D ∈ −→ F G suchthat F D = b. Then △DEF is a right triangle. By the PythagoreanTheoremf 2 = d 2 + e 2 = a 2 + b 2 = c 2This means that f = c and hence by SSS, △ABC ∼ = △DEF .∠C = ∠F = 90.HenceFigure 38.3: Converse of the Pythagorean Theorem.Figure 38.4: Definition of sin θ = BC/AB and cos θ = AC/AB.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012