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The SMSG Postulates

By David A. Smith

In an attempt to generate a list of axioms that are more accessible to younger students the School Mathematics Study Group (SMSG) developed a system of axioms that was specifically designed for the use in high school geometry courses. These axioms, which do give rise to all theorems in Euclidean geometry, are not minimal in nature and are meant to move the student almost immediately to more interesting and less intuitively obvious results. The principle is this: if fewer axioms are proposed (such as the Hilbert, Birkhoff, MacLane axioms), then the more elementary (and obvious) results are needed to "start up" the theory of Euclidean geometry. The idea is to get younger students involved in more interesting results in a timely manner. So even though some of the SMSG axioms are redundant, they do achieve the desired effect of almost immediately being able to state significant results. This topic states the SMSG axioms, from which all of Euclidean geometry can be proven.

Comment (Undefined Terms) Point, Line, and Plane.

Postulate (Line Uniqueness) Given any two different points, there is exactly one line which contains both of them.

Postulate (Distance Postulate) To every pair of different points there corresponds a unique positive number.

Postulate (Ruler Postulate) The points of a line can be placed in correspondence with the real numbers in such a way that
    (i) To every point of the line there corresponds exactly one real number.
    (ii) To every real number there corresponds exactly one point of the line.
    (iii) The distance between two points is the absolute value of the difference of the corresponding numbers.

Postulate (Ruler Placement Postulate) Given two points and of a line, the coordinate system can be chosen in such a way that the coordinate of is zero and the corrdinate of is positive.

Postulate (Points Exist) (a) Every plane contains at least three non-collinear points. (b) Space contains at least for non-coplanar points.

Postulate (Points) On Line Lie In Plane  If two points lie in a plane, then the line comtaining these points lies in the same plane.

Postulate (Plane Uniqueness) Any three points are coplanar and any three non-collinear points determine a plane.

Postulate (Plane Inersection)  If two different planes intersect, then their intersection is a line.

Postulate (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
    (i) each of the sets is convex and
    (ii) if is in one set and is in the other, then the segment intersects the line.

Postulate (Space Seperation) The points of space that do not lie in a given plane form two sets such that
    (i) each of the sets is convex
    (ii) if is in one set and is in the other, then the segment intersects the plane.

Postulate (Angle Measurement) To every angle there corresponds a real number between 0 and 180.

Postulate (Angle Construction) Let be a ray on the edge of the half-plane For every number between and there is exactly one ray with in such that

Postulate (Angle Addition) If is a point in the interior of then

Postulate (Supplementary) If two angles form a linear pair, then they are supplementary.

Postulate (Side Angle Side) Given a 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, the the correspondence is a congruence.

Postulate (Parallel Postulate) Through a given external point there is at most one line parallel to a given line.

Postulate (Polygonal Region Number Correspondence) To every polygonal region there corresponds a unique positive number.

Postulate (Area Of Congruent Triangles) If two triangles are congruent, then the triangular regions have the same area.

Postulate  (Summation Of Areas Of Regions) Suppose that the region is the union of two regions and Suppose that and intersect at most in a finite number of segments and points. Then the area of is the sum of the areas of and

Postulate (Area Of A Rectangle) The area of a rectangle is the product of the length of its base of its altitude.

Postulate (Volume Of Rectangular Parallelpiped) The volume of a rectangular parallelpiped is the product of the altitude and the area of the base.

Postulate (Cavalier Principle) Given two solids and a plane. If for every plane which intersects the solids and is parallel to the given plane the intersections have equal areas, then the two solids have the same volume.

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