The Crossbar Theorem and Coordinates for Rays

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1 The Crossbar Theorem and Coordinates for Rays Theorem (The Crossbar Theorem: If a point D lies in the interior of pc, then ray Dmeets segment C at some point G interior to C. Note: This is the first of our theorems that isn t pretty straightforward. We will prove it by extending the ray D to a whole line and applying Pasch; but in order to do this we need to have the line go through the interior point of a segment; so we must also extend the side of the angle and create a triangle with C as one side. C E F D G? ~ Use the ruler postulate to find points E and F with E** and F**D. For convenience, call D the line l. Since D is interior to pc we know that D does not lie on line Cor on line = E. Thus line l cannot intersect point C,, or E. If it did, then it would have to be one of these two lines that cannot contain D. Moreover, since E** we know is an interior point of E and l intersects E at an interior point. Thus, Pasch applies to points E, C, and and to line l. We know that l intersects either EC or C. We must show three things to complete the proof:

2 1. Dis the part of l that meets either EC or C ; that is, a Fdoes not meet EC, and b Fdoes not meet C, and 2. D does not meet EC. C E F D G? We tackle each in turn: 1a: Since F**D, F and D are on opposite sides of E. Since D is interior to p, D is on the C-side of E, so C and D are on the same side of E. Thus C and F are on opposite sides of E, and so by the Z Theorem, does not meet. 1b: We just showed that C and F are on opposite sides of E, so another application of the Z Theorem gives us that does not meet. F C F EC

3 2: ecause E**, E and are on opposite sides of C. ecause D is interior to p, D is on the -side of C. Thus, D and are on the same side, and E on the opposite side of C. Since E and D are on opposite sides of C, the Z theorem once again gives us that Ddoes not meet. The only remaining possibility is that ray Dmeets segment C at some point G interior to C. EC The book points out that the Crossbar Theorem and the theorem proved above immediately after the X Theorem can be combined into a single theorem called the Continuity Theorem. Of course, from it you can prove another theorem that was adopted as an axiom in irkhoff s system.

4 ngle Coordinates (Not in the ook efore we move on, I m going to provide another extra theorem that is not in the book, but provides a sort of completion of the material, in the following sense: The Ruler Postulate establishes distance measure and gives a coordinate function for points on a line, from which we can then deal with betweenness. We related betweenness ideas to 1 interiors of segments, 2 being able to sum up distances, 3 inequality of distances, and 4 coordinates provided by the Ruler Postulate. The related ideas are given in this diagram: C C

5 The Protractor Postulate establishes angle measure and some other relationships, from which we also got an idea of betweenness for rays, and related ideas, including 1 interiors of angles, 2 being able to sum up angle measures, and 3 inequality of angle measures. These relationships are shown in the following diagram. However, note that the link to 4 real number coordinates is missing. D D C In fact, as mentioned in the book on pages 69 70, we can create protractor coordinates that fill the same role for angle measure as ruler coordinates did for distance measure. We do that in the pages that follow.

6 Coordinate System for Rays The Protractor Postulate guarantees that, given a line and a half-plane H bounded by, for any real number r with 0< r < 180 we can find a unique ray E with E H such that μ( E = r. H E r I will define a function f with some interesting and useful properties. Given a line and a half-plane H bounded by, define for every X with X H the function f ( X = μ( X We state some properties of f: 1. f is a well-defined function: If X = Y, then X = Y so μ( X = μ( Y so g( X = g( Y 2. The domain of f is the set of all rays with endpoint that lie in the halfplane H bounded by. The range of f is the set of real numbers in the interval 0< r < f is onto since for every real number 0< d < 180the Protractor Postulate guarantees that there is a ray D with μ( D = d, so f ( D = d. 4. f is one-to-one since if f ( X = f ( Y then μ( X = μ( Y = r, and the Protractor Postulate guarantees that there is a unique ray R with μ( R = r. Thus Y = X = R.

7 X and Y μ( X < μ( Y μ( Y < μ( X. We consider these two cases: 5. Now for the payoff: Let be two distinct rays with X and Y in H. Either or Case 1: μ( X < μ( Y. y the etweenness Theorem for Rays, X is between and Y. Thus μ( X + μ( XY = μ( Y, μ( XY = μ( Y μ( X = f ( X f ( Y or. H Y X μ( Y < μ( X Y X μ( Y + μ( YX = μ( X Case 2:. y the etweenness Theorem for Rays, is between and. In this case,, or in other words, μ( XY = μ( YX = μ( X μ( Y = f ( Y f ( X Combining the two cases together, we have μ( XY = f ( X f ( Y Case 1 The nice thing about all of this is: Every ray coming off of point on one side of the line can be assigned a coordinate between 0 and 180 in such a way that the angle formed by any two such rays has measure equal to the absolute value of the difference of their coordinates. The function f we defined above is the coordinate function. This is exactly analogous to how distance between two points on a line can be determined by the absolute value of the difference of the points coordinates. This is even more like how a protractor works. Note that we can extend our coordinate function f to include one more ray, namely, which would be given the coordinate 0. I will leave it to you to prove that this minor extension of the function retains all necessary properties. I will also leave it to you to prove the following statement; it follows almost immediately from the etweenness Theorem for Rays and Property #5 above, and is similar to the proof of the etweenness Theorem for Points: etweenness Theorem for Rays, Strong Version: If a< b< c c< b< a Y X, Y, Z X and Z are rays on one side of with coordinates a, b, c respectively, then is between if and only if either or. This reduces betweenness of rays to ordering of real number coordinates, just as we reduced betweenness of points on a line to ordering of real coordinates in the etweenness Theorem for Points. You can also use this to guarantee the existence of rays with given properties, such as the bisectors of angles or the trisectors of angles, etc.