Plane Geometry - Bruce E. Shapiro

Section 18Linear PairsDefinition 18.1 Two angles ∠BAD and ∠DAC are called a linear pair−→ −→if AB and AC are opposite rays.Definition 18.2 Two angles ∠BAD and ∠DAC are called supplementaryangles if ∠BAD + ∠DAC = 180.Figure 18.1: Angles angles ∠BAD and ∠DAC form a linear pairLemma 18.3 (Linear Pair Lemma) If C ∗ A ∗ B and D is in the interiorof ∠BAE then E is in the interior of ∠DAC.Proof. Since D is in the interior of ∠BAE, E and D are on the same sideof line ←→ AB by the definition of interior of an angle.Since ←→ AB = ←→ AC, this means that E and D are on the same side of line ←→ AC,i.e.,E ∈ H ←→ D, ACBy the crossbar theorem, ray −→ AD intersects segment BE (see figure 18.2).91