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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
    (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
    (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
    (ii) Given any ray and a real number between and there is a unique angle in each half-plane of such that
    (iii) If and are the same ray, then
    (iv) If and are opposite rays, then
    (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
    (iv) Distance is preserved. If is the image of then
    (v) Angle measure is preserved. If then
    (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
    (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
    (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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