1. Definitions and concepts MA418 Spring 2010 McAllister

2. What are ……..? Undefined terms Definitions Postulates and axioms Theorems An axiomatic system

3. Undefined terms Figures or ideas that can be described, but cannot be given precise definitions ◦ Point – location in space. 0-dimension Indicated by a dot, Label with Capital Letters  A ◦ Line – infinite set of points. 1 – dimension Indicated by  Label with 2 points or a lower case letter. ◦ Plane – sheet of points. 2 – dimensions

4. Definitions Figures and objects that are described in terms of undefined terms, previously defined terms, and postulates. We DEFINE space to be the set of all points. We DEFINE a geometric figure to be a subset of space. (p. 39)

5. Postulates (axioms) Statements we assume to be true State relationships among defined and undefined terms. ◦ Every line contains at least two distinct points ◦ Two points are contained in one and only one line ◦ If two points are in a plane, then the line containing these points is also in the plane (p.40)

6. Theorems Deductions that can be made from undefined terms, definitions, postulates, and already proven theorems ◦ The sum of the measures of the angles in a triangle is 180 ⁰. (p. 66) ◦ The sum of the measures of the vertex angles in a polygon with n sides is (n- 2)180 ⁰. (p.68)

7. Axiomatic system An idea structure Begins with a small set of undefined terms and builds on them to produce definitions, postulates and theorems. ◦ Euclidean ◦ Non-Euclidean (pp. 555 – 565)  Taxicab geometry (p. 560)