Plane Geometry - Bruce E. Shapiro
Plane Geometry - Bruce E. Shapiro Plane Geometry - Bruce E. Shapiro
42 SECTION 10. THE SMSG AXIOMSP is zero and the coordinate of Q is positive.Axiom 5 Every plane contains at least three non-collinear points, andspace contains at least four non-coplanar points.Axiom 6 If two points lie in a plane, then the line containing these pointslies in the same plane.Axiom 7 Any three points lie in at least one plane, and any three noncollinearpoints lie in exactly one plane.Axiom 8 If two planes intersect, then that intersection is a line.Edward Griffith Begle (1941-1978), director of SMSG for ten years. Photograph by PaulHalmos, Archives of American Mathematics, Dolph Briscoe Center for American History,University of Texas at Austin.Axiom 9 Plane Separation Postulate. Given a line and a plane containingit, the points of the plane that do not lie on the line form two sets suchthat: (1) each of the sets is convex; and (2) if P is in one set and Q is inthe other, then segment PQ intersects the line.Axiom 10 Space Separation Postulate. The points of space that do not liein 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 PQ intersectsthe plane.Axiom 11 Angle Measurement Postulate. To every angle ∠x there correspondsa real number between 0 ◦ and 180 ◦ .The real number is called the measure of the angle and denoted by m(∠x).Axiom 12 Angle Construction postulate. Let AB be a ray on the edge ofthe half-plane H. For every r between 0 and 180 there is exactly one ray« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 10. THE SMSG AXIOMS 43AP , with P in H such that m(∠P AB) = r.Axiom 13 Angle Addition postulate. If D is a point in the interior of∠BAC, then m∠BAC) = m(∠BAD) + m(∠DAC).Axiom 14 Supplement postulate. If two angles form a linear pair, thenthey are supplementary.Axiom 15 SAS postulate. Given a one-to-one correspondence between twotriangles (or between a triangle and itself). If two sides nd the includedangle of the first triangle are congruent to the corresponding parts of thesecond triangle, then the correspondence is a congruence.Axiom 16 Parallel postulate. Through a given external point there is atmost one line parallel to a given line.Axiom 17 To every polygonal region there corresponds a unique positivereal number called its area.Axiom 18 If two triangles are congruent, then the triangular regions havethe same area.Axiom 19 Suppose that the region R is the union of two regions R 1 andR 2 . If R 1 and R 2 intersect at most in a finite number of segments andpoints, then the area of R is the sum of the areas of R 1 and R 2 .Axiom 20 The area of a rectangle is the product of the length of its andthe length of its altitude.Axiom 21 The volume of a rectangle parallelepiped is equal to the productof the length of its altitude and the area of its base.Axiom 22 Cavalieri’s Principle. Given two solids and a plane. If for everyplane that intersects the solids and is parallel to the given plane the twointersections determine regions that have the same area, then the two solidshave the same volume.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
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SECTION 10. THE SMSG AXIOMS 43AP , with P in H such that m(∠P AB) = r.Axiom 13 Angle Addition postulate. If D is a point in the interior of∠BAC, then m∠BAC) = m(∠BAD) + m(∠DAC).Axiom 14 Supplement postulate. If two angles form a linear pair, thenthey are supplementary.Axiom 15 SAS postulate. Given a one-to-one correspondence between twotriangles (or between a triangle and itself). If two sides nd the includedangle of the first triangle are congruent to the corresponding parts of thesecond triangle, then the correspondence is a congruence.Axiom 16 Parallel postulate. Through a given external point there is atmost one line parallel to a given line.Axiom 17 To every polygonal region there corresponds a unique positivereal number called its area.Axiom 18 If two triangles are congruent, then the triangular regions havethe same area.Axiom 19 Suppose that the region R is the union of two regions R 1 andR 2 . If R 1 and R 2 intersect at most in a finite number of segments andpoints, then the area of R is the sum of the areas of R 1 and R 2 .Axiom 20 The area of a rectangle is the product of the length of its andthe length of its altitude.Axiom 21 The volume of a rectangle parallelepiped is equal to the productof the length of its altitude and the area of its base.Axiom 22 Cavalieri’s Principle. Given two solids and a plane. If for everyplane that intersects the solids and is parallel to the given plane the twointersections determine regions that have the same area, then the two solidshave the same volume.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
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