Plane Geometry - Bruce E. Shapiro

SECTION 18. LINEAR PAIRS 93Figure 18.3: Illustration of symbols in the proof of the linear pair theorem(theorem 18.4).Figure 18.4: Illustration of symbols in the second part of the proof of thelinear pair theorem (theorem 18.4).by the angle addition postulate. Henceα + β = α + γγ = βBut E is in the interior of ∠DAC by lemma 18.3. Hence by the angleaddition postulate,γ + ∠EAC = βHence ∠EAC = 0. By the protractor postulate, this means that −→ AE =−→AC,which contradicts the fact that E is in the interior of ∠DAC. Hencewe must conclude that α + β ≥ 180.Now suppose that α + β > 180. Choose a point F on the same side of ←→ ABas D such that γ = ∠BAF = α + β − 180 (figure 18.4).Since β < 180,α + β − 180 < αBy the betweenness theorem for rays this means that F is in the interior ofangle ∠BAD. By the angle addition postulate∠BAF + ∠F AD = ∠BADRevised: 18 Nov 2012 « CC BY-NC-ND 3.0.