Plane Geometry - Bruce E. Shapiro

92 SECTION 18. LINEAR PAIRSFigure 18.2: Illustration of proof of Linear Pair Lemma (Lemma 18.3).Therefore E and B are on different sides of ←→ AD by the plane separationpostulate.Since C ∗ A ∗ B then B and C are on opposite sides of ←→ AD.Therefore C and E are on the same side of ←→ AD, i.e.HenceE ∈ H C,←→ ADE ∈ H ←→ D, AC∩ H ←→ C, ADThen by definition E is in the interior of angle ∠CAD.Theorem 18.4 (Linear Pair Theorem) If ∠BAD and angle ∠DAC fora linear pair then∠BAD + ∠DAC = 180 (18.1)Proof. Suppose that ∠BAD and ∠DAC form a linear pair. Then −→ AB and−→AC are opposite.Let α = ∠BAD and let β = ∠DAC (figure 18.3).α + β = 180.We must show thatThere are three possible cases: α + β > 180, α + β < 180, and α + β = 180.Suppose that α + β < 180. By the angle construction postulate there is apoint E on the same side of ←→ AB as D such that ∠BAE = α + β < 180.Define γ = ∠DAE.By the betweenness theorem for rays, D is in the interior of angle ∠BAE.Hence∠BAE = ∠BAD + ∠DAE« CC BY-NC-ND 3.0. Revised: 18 Nov 2012