Plane Geometry - Bruce E. Shapiro

82 SECTION 16. ANGLESCorollary 16.6 (The Z-Theorem) Let l be a line and let A, D ∈ l,A ≠ D. If B, E ∉ l are points on opposite sides of l then −→ −−→ AB ∩ DE = ∅.Proof. By corrollary 16.5,∀P ≠ A ∈ −→ AB, P ∈ HB,l∀Q ≠ D ∈ −−→ DE, Q ∈ H E,lSince B and E are on opposite sides of l,H B,l ∩ H E,l = ∅by the plane separation postulate (Axiom 15.2).Thus the only place the rays could intersect is the endpoints (A and D) butthese points are distincy by hypothesis. Hence the rays do not intersect.Corollary 16.7 (Betweenness for Rays is Well Defined) Let D bea point in the interior of angle ∠BAC. Then every point on the ray −→ AD(except for A) is in the interior of ∠BAC.Proof. Let A, B, C be noncollinear and let D be a point in the interior ofangle ∠BAC (figure 16.2).D and C are on the same side of ←→ AB by the definition of interior of anangle.Let P be any point on −→ AD except A.By Corollary 16.5 every point on −→−→AD is on the same side AB as C, i.e.,P ∈ H ←→ C, AB.By the same argument D and B are on the same side of ←→ AB and henceevery point on −→−→AD is on the same side of AC as B, i.e., P ∈ HB, ←→Hence P ∈ H ←→ C, AB∩ H ←→ B, ACHence P is in the interior of ∠BAC (defintion 15.8). Hence every point onthe ray −→ AD is in the interior of the angle.Theorem 16.8 Let A, B, C be distinct, noncollinear points. Let D ∈ ←→ BC.ThenB ∗ D ∗ C ⇐⇒ −→ −→ −→AB ∗ AD ∗ ACAC.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012