SMSG Postulates - The SMSG Postulates for f Euclidean...

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Unformatted text preview: The SMSG Postulates for f; Euclidean Geometry1 “ Pos Undefined Terms “0‘ “e in 1. Point 1' é‘ . 2. if 2. Lme p. 3. Plane ‘ POS responds Postulates PCS . POSTULA’I‘E 1. Given any two distinct points there is exactly one line that con» edge of ti tains them. I with P in POSTULATE 2. The Distance Postulate. To every pair of distinct points there POS corresponds a unique positive number. This number is called the distance between the 0f :1 B AC two points. Pos POSTULATE 3. The Ruler Postulate. The points of a line can be placed in me“ they correspondence with the real numbers such that P05 1. To every point of the line there corresponds exactly one real number, tween t: 2. To every real number there corresponds exactly one point of the line, angle 0 i and gle, then 3. The distance between two distinct points is the absolute value of the . P08 difference of the corresponding real numbers. ‘3 at 3305‘ Pos POSTULATE 4. The Ruler Placement Pastulate. Given two points P and Q real mm" a line, the coordma' te system can be chosen in such a way that the coordinate of P POE zero and the coordinate of Q is positive. the ' POSTULATE 5. (a) Every plane contains at least three non-collinear poin (b) Space contains at least four non-coplanar points. POSTULATE 6. If two points lie in a plane, then the line containing these points: lies in the same plane. POSTULATE 7. Any three points lie in at least one plane, and any three no collinear points lie in exactly one plane. POSTULATE 8. If two planes intersect, then that intersection is a line. POSTULATE 9. The Plane Separation Postulate. Given a line and a plane com taining it, the points of the plane that do not lie on the line form two sets such that 1Reprinted from School Mathematics Study Group. Geometry, by permission of Yale University Press, New Haven. one line that con— ” stinct points there tance between the :an be placed in a number, f the line, lue of the points P and Q of :oordinate of P is i-collinear points. ‘ Lining these points rd any three non- s a line. > and a plane con— sets such that non of Yale nasasy¢as§~gawyms 1. each of the sets is convex and 2. if P is in one set and Q is in the other, then segment P Q intersects the line. POSTULATE 10. The Space Separation Postulate. The points of space that do not lie in a given plane form two sets such that 1. each of the sets is convex, and 2. if P is in one set and Q is in the other, then segment w intersects the plane. POSTULATE 1 l . The Angle Measurement Postulate. To every angle there cor- responds a real number between 0° and 180°. POSTULATE 12. The Angle Construction Postulate. Let in be a ray on the edge of the half-plane H . For every r between 0 and 180 there is exactly one ray AP, with P in H such mathPAB = r. POSTULATE 13. The Angle Addition Postulate. If D is a point in the interior of LBAC, then mABAC = mZBAD + mZDAC. POSTULATE 14. The Supplement Postulate. 'If two angles form a linear pair, then they are supplementary. POSTULATE 15. The SAS Postulate. Given a one-to-one correspondence be- tween 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 trian— gle, then the correSpondence is a congruence. POSTULATE 16. The Parallel Postulate. Through a given external point there is at most one line parallel to a given line. POSTULATE 17. To every polygonal region there corresponds a unique positive real number called its area. POSTULATE 18. If two triangles are congruent, then the triangular regions have the same area. POSTULATE 19. Suppose that the region R is the union of two regions R1 and R2. If R1 and R2 intersect at most in a finite number of segments and points, then the area of R is the sum of the areas of R1 and R2. POSTULATE 20. The area of a rectangle is the product of the length of its base and the length of its altitude. POSTULATE 21. The volume of a rectangular parallelepiped is equal to the product of the length of its altitude and the area of its base. POSTULATE 22. Cavalieri’s Principle. Given two solids and a plane, if for every plane that intersects the solids and is parallel to the given plane the two m ; sections determine regions that have the same area, then the two solids have the volume. ...
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This note was uploaded on 11/02/2010 for the course MATH modern geo taught by Professor Shirley during the Spring '10 term at University of Texas.

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SMSG Postulates - The SMSG Postulates for f Euclidean...

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