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Axiomatic systems By Micah McKee VOCAB: Axiomatic system Postulate/Axiom Theorem Axiomatic system Line segment Ray Point Line Plane An Axiomatic system • “In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems.” An Axiom/postulate • Is true, but can’t be proven A theorem • Is true but must be proven A line segment is LIKE a single string of spaghetti pasta. A line segment is: Part of a line connecting two points. It has definite end points. The word "segment" is important, because a line normally extends in both directions without end. A ray is LIKE a space gun. • A ray is also LIKE a laser beam The point A is considered to be on a member of the ray; Part of a line connecting two points. Some terms are undefinable. Because of this we use KINDA LIKES to help define them. • Point • Line • Plane Point • A point is KINDA LIKE a poppy seed on a bagel A point: An exact location. It has no size, only position. A line is LIKE a line to get into a one direction concert. A geometric line is: A geometrical object that is straight, infinitely long, and infinitely thin. and extends forever in both ways . Plane A plane is KINDA LIKE a huge pizza crust A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space Euclid's Postulates: • 1. A straight line segment can be drawn joining any two points. • 2. Any straight line segment can be extended indefinitely in a straight line. • 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. • 4. All right angles are congruent. • 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. Helpful ways to determine if it Is a axiom or if it is a theorem. • If it is an axiom then it is true but cant be proven. • If it is a theorem it is true but must be proven. • You can read more about it at this source; • http://web.mnstate.edu/peil/geometry/C1AxiomSystem/AxiomaticSystems.htm Examples • Q: What is the difference between a line and a line segment? • Answer: A line is straight (no curves), has no thickness, and extends in both directions without end (infinitely). • A line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points. • So a line extends in both directions forever a line segment is bound by two distinct points. • Q: what are the undefined terms and why are they undefinable? • Answer: The terms in this unit are line segment, ray, point, line, and plane. • Line segment: Part of a line connecting two points. • Ray: Part of a line connecting two points. • Point line and plain are left and they are undefinable. They are undefinable because they can not be put into a category or box. Practice problems • 1. What is the difference between a line and a ray? • 2. What makes a term undefinable? • 3. See fig 1, which example is a ray? Explain your answer. • 4. list the three undefined terms? • 5. See fig 1, which example is a line segment? Fig 1. Practice problems key. • 1. A line extends in both ways forever, a ray has a starting point and only extends in one way forever • 2. The fact the these terms don’t truly exist • 3. (iv) is the ray because it extends forever in one direction and contains two points. • 4. point, line. Plane. • 5. (iii) is the line segment because it is a part of a line containing two points. Source citation • "5 Axioms of Geometry - Google Search." 5 Axioms of Geometry Google Search. N.p., n.d. Web. 03 Dec. 2014. • "Line." Definition of. N.p., n.d. Web. 08 Dec. 2014. THE END