Ch. 3 - Lectures 4 - 8

Ch. 3 - Lectures 4 - 8 - Math 3379 Chapter 3, Venema...

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Math 3379 – Chapter 3, Venema Neutral Geometry in the Plane Undefined terms: point, line, distance, half-plane, angle measure, area Axioms: Axiom 1 The Existence Postulate The collection of all points forms a non-empty set. There is more than one point in the set. p. 36 Axiom 2 The Incidence Postulate Every line is a set of points . For every pair of distinct points A and B there is exactly one line l such that A l and B l. p. 36 Axiom 3 The Ruler Postulate For every pair of points P and Q there exists a real number PQ, called the distance from P to Q . For each line l there is a one-to-one correspondence from l to R such that if P and Q are points on the line that correspond to the real numbers x and y , respectively, then PQ = x y - . p. 37 Axiom 4 The Plane Separation Postulate For every line l , the points that do not lie on l form two disjoint, nonempty sets 1 H and 2 H , called half-planes bounded by l , such that the following conditions are satisfied. 1. Each of 1 H and 2 H is convex. 2. If P 1 H and Q 2 H , then PQ intersects l . p. 46 1
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The Protractor Postulate For every angle BAC there is a real number μ( ∠ BAC ), called the measure of BAC , such that the following conditions are satisfied. 1. ≤ ∠ BAC < 180 º for every angle BAC. 2. μ( ∠ BAC ) = 0º if and only if AB AC = uuur uuur . 3. (Angle Construction Postulate) For each real number r , 0 < r < 180, and for each half-plane H bounded by AB uur s there exists a unique ray AE uuur such that E is in H and μ∠ ( BAE ) = r º. 4. (Angle Addition Postulate) If the ray AD uuur is between rays AB uuur and AC uuur , then μ( ∠ BAD ) + μ( ∠ DAC ) = μ( ∠ BAC ). p. 51 Axiom 6 The Side-Angle-Side Postulate (SAS) If ABC and DEF are two triangles such that AB DE 2245 , ABC DEF 2245 ∠ , and BC EF 2245 , then ABC 2245 DEF. p. 64 Axiom 7 The Parallel Postulate p. 66 (Chapter 5 and beyond) Euclidean Parallel Postulate : For every line l and for every point P that does not lie on l , there is exactly one line m such that P lies on m and m is parallel to l . Elliptic Parallel Postulate : For every line l and for every point P that does not lie on l , there is no line m such that P lies on m and m is parallel to l . Hyperbolic Parallel Postulate : For every line l and for every point P that does not lie on l , there are at least two lines m and n such that P lies on both m and n and both m and n are parallel to l . For Chapters 3 and 4 we will work only with Axioms 1 – 6. Starting with Chapter 5 we will make decisions about which parallel postulate we are going to explore. In Chapter 7 we will explore the undefined term area and add axioms about area to our list. 2
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Ch. 3 - Lectures 4 - 8 - Math 3379 Chapter 3, Venema...

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