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http://osu.orst.edu/instruct/mth338/garity Read 1.4, 2.6 Incidence Axiom 1. For each two distinct points there exists a unique line on both of them. Incidence Axiom 2. For every line there exist at least two distinct points on it. Incidence Axiom 3. There exist at least three distinct points. Incidence Axiom 4. Not all points lie on the same line. Any geometry that satisfies all four incidence axioms will be called an incidence geometry. 1 Incidence Theorem 1. If two distinct lines intersect, then the intersection is exactly one point. Proof. Incidence Theorem 2. For each point there exist at least two lines containing it. Proof. Incidence Theorem 3. There exist three lines that do not share a common point. Proof 2 If we consider any line and any point P, where P is not on , then three possibilities exist for a parallel axiom. 1. There exist no lines on P that are parallel to , 2. There exists exactly one line on P that is parallel to , or 3. There exists more than one line on P parallel to . 3 THE SMSG POSTULATES FOR EUCLIDEAN GEOMETRY Undefined Terms 1. Point 2. Line 3. Plane Postulate 1. Given any two distinct points, there is exactly one line that contains them. 2. The Distance Postulate. To every pair of distinct points, there corresponds a unique positive number. This number is called the distance between the two points. 4 3. The Ruler Postulate. The points of a line can be placed in a correspondence with the real numbers such that To every point of the line there corresponds exactly one real number, To every real number there corresponds exactly one point of the line, and The distance between two distinct points is the absolute value of the difference of the corresponding real numbers. 5 4. Ruler Placement . Given 2 points P and Q of a line, the coordinate system can be chosen in such a way that the coordinates of P and Q are 0 and 0. 5. (a) Every plane contains at least 3 noncollinear points. (b) Space contains at least 4 noncoplanar points. 6. If 2 points lie in a plane, then the line containing these points lies in the same plane. 7. Any 3 points lie in at least one plane, and any 3 non- colinear points lie in exactly one plane. 6 8. If two planes intersect, then that intersection is a line. 9. Plane Separation. Given a line and a plane containing it, the points of the plane that do not lie on the line form two sets such that each of the sets is convex and if P is in one set and Q is in the other, then segment P Q intersects the line. 10. Space Separation. 11. Angle Measurement. To every angle there corresponds a real number between 0 and 180. 7 l2.Angle Construction. Let AB be a ray on the edge of the half-plane H. For every r between O and 180, there is exactly one ray AP with P in H such that mPAB r. 13. Angle Addition. If D is a point in the interior of BAC, then mBAC mBAD mDAC . l4.Supplements. If two angles form a linear pair,then they are supplementary. 8 15. SAS. Given a 1-1 correspondence between two triangles (or between a triangle and itself): If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence. 16. The Parallel Postulate. Through a given external point there is at most one line parallel to a given line. 17-22. Area and Volume Postulates 9