from LEEs axiomatic geometrySolutionIn accordance with the h

(from LEE\'s axiomatic geometry)

Solution

In accordance with the hypothesis, we let AB be a segment and CD a ray. Note that C and D are necessarily distinct. Using the Ruler Placement Theorem (See exercise 5.6) and/or emailed attachment), we can put a coordinate function f on the line l = CD, such that the coordinate of C (i.e., f(C)) is zero and the coordinate of D (i.e., f(D)) is > 0. Using this coordinate function, the ray CD is precisely the set of points with coordinates greater than or equal to zero! (CD = {P|f(P) 0}). (Think: D determines the positive direction). Select E CD such that f(E) = AB (reminder: AB R and AB > 0). Since f is a coordinate function, the point E is unique. Furthermore, CE = |f(C) f(E)| = |0 f(E)| = f(E) = AB. On account of CE = f(E) = AB, we conclude CD = AB.