1. 1.3 Early Definitions & Postulates
Four Parts of a Mathematical System*
Undefined terms 
 vocabulary
Defined terms 
Axioms or postulates 
 principles
Theorems 
* Examples are Algebra, Geometry, Calculus
9/22/2019

2. Characteristics of a Good Definition
A good definition must have certain characteristics.
It names the term being defined.
It places the term into a set or category.
It distinguishes itself from other terms in that category.
It is reversible.
Ex: Definition: A line segment is the part of a line that consists of two points, known as endpoints and all the points between them.
Reverse: The part of a line between and including two points is a line segment.
9/22/2019

3. The reverse of a definition will prove useful in our later work when attempting to establish geometric properties of lines, segments, angles, & figures.
Ex. A midpoint of a segment is defined as a point M that divides a segment into two segments of equal length. How can we prove that a point M is the midpoint of segment ?
We must appeal to the reverse of the definition of a midpoint: a point that divides a segment into 2 segments of equal length is the midpoint of the segment. In other words, we must show that AM  MB.
Once that is accomplished, we can then conclude that point M is the midpoint of segment AB.
9/22/2019

4. Symbols for Lines/Line Segments
9/22/2019

5. Initial Postulates Basic assumptions that are accepted as true.
Postulates that can be proven are called theorems.
Postulate 1: Through two distinct points there is exactly one line. (distinct means different)
Postulate 2: (Ruler Postulate) The measure of any line segment is a unique positive number.( unique: one and only; exactly one; one and no more than one) __
Postulate 3: (Segment-Addition Postulate). If X is a point of AB and X is between A and B then AX + XB = AB. (p. 24 Ex 2)
9/22/2019

6. Other definitions
Distance : The distance between two points A and B is the length of the line segment AB that joins the two points.
Congruent (  )segments: Two segments that have the same length. Ex.3 and 4 p. 25
 __
Ray: Ray AB, denoted by →, is the union of AB and all 
points X on AB such that B is between A and X.
Fig Opposite e rays are two rays with a common endpoint whose union is a straight line. Fig 1.39 p. a p. 26
Parallel lines: Lines that lie in the same plane but do not intersect.
9/22/2019

7. More Postulates
Postulate 4: If two lines intersect, they intersect at a point.
Postulate 5: Through three non-collinear points, there is exactly one plane. Fig 1.42 p. 27
Postulate 6: If two distinct planes intersect, then their intersection is a line. Fig 1.44 p. 27
Postulate 7: Given two distinct points in a plane, the line containing these points also lies in the plane.
Theorems are postulates which can be proven.
Theorem 1.3.1: The midpoint of a line segment is unique.
Note: Numbering system for Theorems in this book is Chapter.Section.Order
9/22/2019