Plane Geometry - Bruce E. Shapiro

76 SECTION 15. THE PLANE SEPARATION POSTULATEFigure 15.1: Plane Separation Postulate (axiom 15.2). P Q intersects theline because P and Q are in different half planes.Definition 15.3 Let l be a line and A a point not on l. Then we use H A,lto denote the half-plane of l that contains A. When the line is clear fromthe context we will just the notation H A .Definition 15.4 Two points A, B are said to be on the same side of theline l if they are both in the same half-plane. They are said to be onopposite sides of the line if they are in different half planes.In figure 15.1 points P and Q are on opposite sides of l, while points P andF are on the same side of l. In terms of this notation, we can restate theplane separation postulate as follows.Axiom 15.5 (Plane Separation Postulate, Second Form) Let l be aline and let A, B be points not on l. Then A and B are on the same sideof l if and only ifAB ∩ l = ∅,and are on opposite sides of l if and only if.AB ∩ l ≠ ∅Definition 15.6 Two rays −→ −→BA and BC having the same endpoint are oppositerays if BA ≠ BC and−→ −→←→ −→ −→AB = BA ∪ BCor equivalently, A ∗ B ∗ CDefinition 15.7 An angle is the union of two non-opposite rays −→ AB and−→AC having the same endpoint, and is denoted by ∠BAC or ∠CAB. Thepoint A is called the vertex of the angle and the two rays are called thesides of the angle.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012