The UCSMP Postulates
The University of Chicago School Mathematics Project (UCSMP) developed a system of axioms for Euclidean geometry that are still widely used today in most high school geometry textbooks.
These axioms are not minimal; that is, some of the axioms can be proven given some of the other axioms.
The idea is to get younger students involved in more interesting results in a timely manner.
For example, the UCSMP axioms incorporate a transformational approach via the "Reflection Postulate", which asserts that certain transformations exist and have specified properties.
Since understanding geometric relationships in terms of functions of points and angles involves a deeper level of understanding for a rigorous level to be achieved, students benefit by being exposed to a not-so rigorous treatment of transformational result in Euclidean geometry.
This topic states the UCSMP axioms, from which all of Euclidean geometry can be proven.
Comment (Undefined) Terms Point, Line, and Plane
Axiom (Point-Line-Plane) (i) Through any two points there is exactly one line. (ii) Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point corresponding to zero and any other point corresponding to
![the ucsmp postulates _gr_1.gif]](pages/the-ucsmp-postulates/Images/the-ucsmp-postulates_gr_1.gif) (iii) Given a line in a plane, there is at least one point in the plane that is not on the line.
Given a plane in space, there is at least one point in space that is not on the plane. (iv) If two points lie in a plane, the line containing them lies in the plane. (v) Through three noncollinear points, there is exactly one plane. (vi) If two different planes have a point in common, then their intersection is a line.
Axiom (Distance) (i) On a line, there is a unique distance between two points. (ii) If two points on a line have corrdinates
and
the distance between them is
![the ucsmp postulates _gr_4.gif]](pages/the-ucsmp-postulates/Images/the-ucsmp-postulates_gr_4.gif) (iii) If
is on
then
Axiom (Triangle Inequality) The sum of the lengths of two sides of a triangle is greater than the length of the third side.
Axiom (Angle Measure) (i) Every angle has a unique measure from
to
![the ucsmp postulates _gr_9.gif]](pages/the-ucsmp-postulates/Images/the-ucsmp-postulates_gr_9.gif) (ii) Given any ray
and a real number
between
and
there is a unique angle
in each half-plane of
such that
![the ucsmp postulates _gr_16.gif]](pages/the-ucsmp-postulates/Images/the-ucsmp-postulates_gr_16.gif) (iii) If
and
are the same ray, then
![the ucsmp postulates _gr_19.gif]](pages/the-ucsmp-postulates/Images/the-ucsmp-postulates_gr_19.gif) (iv) If
and
are opposite rays, then
![the ucsmp postulates _gr_22.gif]](pages/the-ucsmp-postulates/Images/the-ucsmp-postulates_gr_22.gif) (v) If
(except for point
) is in the interior of
then
Axiom (Corresponding Angle) Suppose two coplanar lines are cut by a transversal.
If two corresponding angles have the same measure, then the lines are parallel.
If the lines are parallel, then the corresponding angles have the same measure.
Axiom (Reflection) Under a reflection: (i) There is a
correspondence between points and their images. (ii) Collinearity is preserved.
If three points
and
lie on the same line then their images
and
are collinear. (iii) Betweenness is preserved.
If
is between
and
, then the image
is between the images
and
![the ucsmp postulates _gr_37.gif]](pages/the-ucsmp-postulates/Images/the-ucsmp-postulates_gr_37.gif) (iv) Distance is preserved.
If
is the image of
then
![the ucsmp postulates _gr_40.gif]](pages/the-ucsmp-postulates/Images/the-ucsmp-postulates_gr_40.gif) (v) Angle measure is preserved.
If
then
![the ucsmp postulates _gr_42.gif]](pages/the-ucsmp-postulates/Images/the-ucsmp-postulates_gr_42.gif) (vi) Orientation is reversed.
A polygon and its image, with vertices taken in corresponding order, have opposite orientations.
Axiom (Area) (i) Given a unit region, every polygonal region has a unique area. (ii) The area of a rectangle with dimensions
and
is
![the ucsmp postulates _gr_45.gif]](pages/the-ucsmp-postulates/Images/the-ucsmp-postulates_gr_45.gif) (iii) Congruent figures have the same area. (iv) The areas of the union of two nonoverlapping regions is the sum of the areas of the regions.
Axiom (Volume) (i) Given a unit cube, every polygonal region has a unique volume. (ii) The volume
of a box with dimensions
and
is found by the formula
![the ucsmp postulates _gr_49.gif]](pages/the-ucsmp-postulates/Images/the-ucsmp-postulates_gr_49.gif) (iii) Congruent figures have the same volumes. (iv) The volume of the union of two nonoverlapping solids is the sum of the volumes of the solids. (v) 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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Cite this as: The Ucsmp Postulates Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/the-ucsmp-postulates.html
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