Plane Geometry - Bruce E. Shapiro

62 SECTION 14. BETWEENNESSFigure 14.1: The Ruler Postulate tells us that there is a 1-1 correspondencebetween the points on a line and the real numbersCorollary 14.9 If l and m are lines, the either (1) l = m; (2)l ‖ m; or (3)l ∩ m contains precisely one point.Definition 14.10 (coordinate) Let l be a line. A 1-1 correspondence 1f : l ↦→ R such that P Q = |f(P )−f(Q)| for every point P, Q ∈ l is called acoordinate function for l and the number f(P ) is called the coordinateof P .We will assume that a coordinate function exists for every line, and use itto define the distance P Q.Axiom 14.11 (Ruler Postulate) For every pair of points P, Q there is anumber P Q called the distance from P to Q. For each line l there is aone-to-one mapping f : l ↦→ R such that if x = f(P ) and y = f(Q) thenP Q = |x − y| is the value of the distance.Definition 14.12 (Betweenness) Let A,B,C are distinct points. We saythat B is between A and C, and write A∗B∗C, if C ∈ ←→ AB and AC+CB =AB.Definition 14.13 The (line) segment (joining A and B) isAB = {A, B} ∪ {P |A ∗ P ∗ B}Definition 14.14 The ray (from A in the direction of B) is−→AB = AB ∪ {P |A ∗ B ∗ P }Definition 14.15 The length of segment AB, denoted by AB, is the distancefrom A to B.Definition 14.16 We call the points A and B the endpoints of AB.Definition 14.17 Two segments AB and BC are said to be congruent ifthey have the same length, and we say AB ∼ = BC, i.e.,AB = BC ⇐⇒ AB ∼ = BC1 Recall from 6.13 that a 1-1 correspondence is a function that is 1-1 and onto.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012