Plane Geometry - Bruce E. Shapiro

96 SECTION 19. ANGLE BISECTORSwith D on the same side of ←→ AB as C.By the Betweenness theorem for rays,By the angle addition postulateThis proves existence.−→AB ∗ −→ −→ AD ∗ AC∠BAC = ∠BAD + ∠DAC= β + α= 1 2 ∠BAC + α⇒ α = 1 2 ∠BAC = β−→ −→Figure 19.2: Uniqueness of Angle bisectors. AD and AE are both bisectorsof ∠BAC. To prove uniqueness, they must be the same ray.To prove uniqueness, let −→ AE be another angle bisector of ∠BAC (see figure19.2.)If we do the same construction as before we find that ∠BAE = ∠BAD;by the uniqueness part of the angle construction postulate, this means that−→AE = −→ AD.Definition 19.3 Two lines l and m are perpendicular (l ⊥ m) if thereexists a point A on both l and m, a point B ∈ l, and a point C ∈ m suchthat ∠BAC is a right angle.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012