Appendix C - SMSG Axioms for Euclidean Geometry
Everything should be made as simple as possible,
but not simpler.
—Albert Einstein (1879–1955)
The School Mathematics Study Group (SMSG),
developed an axiomatic system designed for use in high school geometry courses,
which was published in 1961.
The axioms are not
independent of each other, but the system does satisfy all the
requirements for Euclidean geometry; that is, all the theorems in Euclidean
geometry can be derived from the system. The lack of independence of the
axiomatic system allows high school students to more quickly study a broader
range of topics without becoming trapped in detailed study of obvious concepts
or difficult proofs. You should compare the similarity and differences between
the SMSG axioms and those by Hilbert and
Birkhoff. Also, you should compare the SMSG axioms
with those found in a high school textbook.
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- Postulate 1. (Line Uniqueness) Given any two distinct points
there is exactly one line that contains them.
- Postulate 2. (Distance Postulate) To every pair of distinct
points there corresponds a unique positive number. This number is called the
distance between the two points.
- Postulate 3. (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.
- The distance between two distinct points is the absolute value of the
difference of the corresponding real numbers.
- Postulate 4. (Ruler Placement Postulate) Given two points
P and Q of a line, the coordinate system can be chosen
in such a way that the coordinate of P is zero and the coordinate
of Q is positive.
- Postulate 5. (Existence of Points)
- Every plane contains at least three non-collinear points.
- Space contains at least four non-coplanar points.
- Postulate 6. (Points on a Line Lie in a Plane) If two points
lie in a plane, then the line containing these points lies in the same plane.
- Postulate 7. (Plane Uniqueness) Any three points lie in at
least one plane, and any three non-collinear points lie in exactly one plane.
- Postulate 8. (Plane Intersection) If two planes intersect,
then that intersection is a line.
- Postulate 9. (Plane Separation Postulate) 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;
- if P is in one set and Q is in the other, then
intersects the line.
- Postulate 10. (Space Separation Postulate) The points of
space that do not lie in a given plane form two sets such that:
- Each of the sets is convex.
- If P is in one set and Q is in the other, then
intersects the plane.
- Postulate 11. (Angle Measurement Postulate) To every angle
there corresponds a real number between 0° and 180°.
- Postulate 12. (Angle Construction Postulate) Let
be a ray on the edge of the half-plane H. For every r
between 0 and 180, there is exactly one
with P in
H such that m∠PAB =
- Postulate 13. (Angle Addition Postulate) If D is
a point in the interior of
m∠BAC = m∠BAD
- Postulate 14. (Supplement Postulate) If two angles form a linear pair, then they
- Postulate 15. (SAS Postulate) Given a one-to-one 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.
- Postulate 16. (Parallel Postulate) Through a given external point there is at most
one line parallel to a given line.
- Postulate 17. (Area of Polygonal Region)
To every polygonal region there corresponds a unique positive real number
called the area.
- Postulate 18. (Area of Congruent
Triangles) If two triangles are congruent, then the triangular regions have the same
- Postulate 19. (Summation of Areas of
Regions) 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. (Area of a Rectangle)
The area of a rectangle is the product of the length of its base and the length
of its altitude.
- Postulate 21. (Volume of Rectangular
Parallelpiped) The volume of a rectangular parallelpiped 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
intersections determine regions that have the same area, then the two solids
have the same volume.
School Mathematics Study Group, Geometry. New Haven:
Yale University Press, 1961.
Mathematics is an interesting intellectual sport but it
should not be allowed to stand in the way of obtaining sensible information
about physical processes.
—Richard W. Hamming, Mathematical Maxims and Minims (1988)