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12 Cards in this Set
- Front
- Back
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What is Playfair's Postulate?
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For every line L and every point P not on L, there exists a unique line M that contains P and is parallel to L.
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What are the undefined terms in Hilbert's axiomatic system for Euclidean geometry?
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Point, Line, Plane, On, Between, Congruence
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State Hilbert's Axioms of Incidence
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I-1. Every two distinct points determine a unique line that contains them.
I-2. There exist at least two points on a given line. There exist at least three points that are not on the same line. I-3. For any three points that do not lie on the same line, there exists a plane that contains these points. For every plane there exists a point that it contains. |
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State Hilbert's Axioms of Order
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II-1. If one point is between two others then all three points must be distinct and collinear.
II-2. For every two distinct points A and B, there is at least one point C on line AB such that B is between A and C (you can extend a line indefinitely) II-3. If three distinct points are collinear then one and only one is between the other two. II-4. If a line cuts one side of a triangle then it must cut one of the other two sides or pass through a vertex. |
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What is the elliptical parallel postulate?
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Given any line L and any point P not on L there is NO line through P that is parallel to L.
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State 4 results in hyperbolic geometry.
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1. There are no rectangles.
2. There exist triangles that cannot be circumscribed. 3. There exist no triangles that are similar but not congruent. 4. The sum of the angles in a triangle is LESS than 180 degrees. 5. Two triangles have the same area if they have the same angle sum. |
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State 2 results in elliptic geometry.
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1. The sum of the angles in any triangle is GREATER than 180 degrees (small triangles are close to 180; large triangles are much larger than 180 but less than 540 degrees)
2. There is a finite limit on the length of an arc, since it cannot exceed the circumference of a great circle. 3. To find the shortest distance between two points, follow the great circle on which both points lie. 4. Given a line L, there exists a point P such that P is equidistant to all points of L and every line connecting P with L is perpendicular to L . |
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What is a model for elliptical geometry?
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The surface of a sphere where lines are "great circles" on the sphere that cut the sphere in two halves, and antipodal points are considered to be the same point.
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What is a model for hyperbolic geometry?
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The Poincare' hyperbolic disk in which a line is represented by an arc of a circle whose ends are perpendicular to the disk's boundary.
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State Hilbert's Axioms of Congruence
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Copy congruent segments; congruent segments are transitive; add segments; construct angles; SAS implies angle congruence
III-1. Given a line, a point on the line and a segment, there exist 2 points on the line that determine with the given point 2 more segments congruent with the given one. (congruent segments can be constructed) III-2. If 2 segments are congruent to a third segment, they are congruent to each other (the congruence of segments is a transitive relation). III-3. The congruence of segments is compatible with the addition of segments. III-4. Given a ray and an angle, there exist 2 angles sharing the given ray congruent to the given angle. III-5. For the SAS congruency case for triangles, another of the corresponding angles is congruent. |
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State Hilbert's Parallel Axiom
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IV-1. Given a line and a point not on the line, there exists at most one line parallel to the given line and containing the point.
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State Hilbert's Axioms of Continuity (completeness)
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V-1. (Archimedes Axiom). Given 2 segments, N copies of the first placed on the ray containing the second will contain the second segment.
V-2. There is no bigger plane in which all previous axioms are satisfied. |
