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# 2.4 - 2.6 - 2.4 2.5 2.6 Distance Ruler Postulate Segments...

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2.4 Distance, Ruler Postulate, Segments, Rays, and Angles 2.5 Angle Measure and the Protractor Postulate 2.6 Plane Separation, Interior of Angles, Crossbar Theorem Homework assignment 2 1

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2.4 Distance, Ruler Postulate, Segments, Rays, and Angles Being able to measure distance is not a property that comes automatically with a geometry. We did not have distance in any of the non-Euclidean geometries we’ve studied so far. With these axioms we can begin measuring the distance between 2 points and the length of a segment. The Metric Axioms join the Incidence Axioms giving us 9 axioms to work with and will give us more interesting geometries to work with. In particular, adding D4 means that the finite geometries from the preceding part of the chapter no longer model our axioms. D1 Each pair of points A and B is associated with a unique real number called the distance from A to B, denoted AB. (page 78) D2 For all points A and B, AB 0 with equality only when A = B. (page 78) D3 For all points A and B, AB = BA. (page 78) D4 Ruler Postulate (page 83) The points of each line L may be assigned to the entire set of real numbers x, , x < < - called coordinates in such a manner that (1) each point on L is assigned to a unique coordinate (2) no two points are assigned to the same coordinate (3) any two points on L may be assigned to the coordinate zero and a positive real number, respectively (4) if points A and B on L have coordinates a and b, then AB = b a - . D1 says that each pair of points A and B is associated with a unique real number called the distance from A to B, and that adjacent point labels means the distance from the first point to the second point. This means that if you see EF, you’re supposed to know right away that this means “the distance from E to F”. So we can finally talk about distance. We needed an axiom just like this one to continue. D2 confirms something we feel that we already know – distances are positive numbers and that the distance from a point to itself is zero. Since you’ve studied Euclidean geometry some of this seems obvious, but remember in a different geometry these matters might be different than what you’re used to. D3 makes an interesting point: distance is symmetric. The distance from A to B is the same unique positive number as the distance from B to A. This is not a given and there are geometries in which it is not true; we’re not going there, of course, but it is important 2
for you to realize that what seems obvious and true isn’t at all universal. You’ve just been brought up Euclidean and we are building that space but there are lots of different rules out there that work just fine in their own non-Euclidean spaces. The important thing to notice and remember is that this needs to be said in an axiom…it’s not true anywhere but where it’s stated to be an axiom. We will spend a few moments with these first 3 axioms before going on to

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2.4 - 2.6 - 2.4 2.5 2.6 Distance Ruler Postulate Segments...

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