SMSG Model of Euclidean Geometry

SMSG model for Euclidean Geometry (EG) is an axiomatic system developed by a team of American mathematicians (School Mathematics Study Group-SMSG) in mid-19^th century.  The main characteristics of this model was

• ·         Introducing pedagogical thought (pedagogical orientation) into EG
• ·         Making (Bringing) EG as a discipline of school education

SMSG model for Euclidean Geometry is a re-engineering of Euclidean Geometry to make it more efficient combining the features of H and B (Fusion of H and B to make EG more efficient).
 In SMSG model for Euclidean Geometry, there are three undefined terms and 22 postulates.

• ·  Point
• ·   Line
• ·    Plane

Definitions

In SMSG model, there are number of definitions. Among them, one example is given below.

Triangle: a plane figure enclosed by three lines.

Axioms (Postulates)

In SMSG model, the axioms (postulates) are classified into eight groups. These groups are given below.

Group              I-Incidence-1

II-Distance-2-4 (3)

III-Space relations-5-8 (4)

IV-Separations-9-10 (2)

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V-Angle measure-11-14 (4)

VI-Congruence-15 (1)

VII-Parallelism-16 (1)

VIII-Area and volume -17-22 (6)

Postulates-I: Incidence

1.      (Line Uniqueness) Given any two distinct points there is exactly one line on both.  

Postulates-II: Distance

2.      (Distance Postulate) To every pair of distinct points there corresponds a unique positive number. This number is called the distance between them.  

3.       (Ruler Postulate) The points of a line can be placed in a correspondence with the real numbers such that:

4.      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.
•  (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. 

Postulates-III: Space Relations

5.      (Existence of Points) Every plane contains at least three non-collinear points. Space contains at least four non-coplanar points.

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.  

7.      (Plane Uniqueness) Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane.

8.       (Plane Intersection) If two planes intersect, then intersection is a line.  

Postulates-IV: Separations

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 segment PQ intersects the line.

10.  (Space Separation Postulate) The points of space that do not lie in a plane such that, if P is in one side and Q is in the other, then segment PQ intersects the plane.

Postulates-V: Angle construction (measure)

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

12.   (Angle Construction Postulate) Let, AB be a ray on the edge of the half-plane H. For every r between 0 and 180, there is exactly one ray AP with P in H such that ∠PAB = r.

13.  (Angle Addition Postulate) If D is a point in the interior of ∠BAC, then ∠BAC = ∠BAD + ∠DAC.  

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

Postulates-VIII: Area and Volume

17 .(Area of Polygonal Region) To every polygonal region there corresponds a unique positive real number called the area.

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

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.  

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.  

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.  

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, and the two intersections determine regions that have the same area, then the two solids have the same volume.

Theorem