1. Section 1.3 Segments and Their Measures
Goal Use Segment Postulates
Goal Use the Distance Formula

2. Geometry is based on Rules
USING SEGMENT POSTULATES
Geometry is based on Rules
Rules accepted without proof are called Axioms or Postulates.
Rules that must be proved are called Theorems.

3. Postulate 1 Ruler Postulate
USING SEGMENT POSTULATES
Postulate 1 Ruler Postulate
The points on a line can be matched one-to-one with the Real numbers.
The Real number that corresponds to the point is called the coordinate.

4. The Distance between points A and C is written as AC
USING SEGMENT POSTULATES
The Distance between points A and C is written as AC
- It is the Absolute value of the difference of the coordinates of A and C. AC A C
AC = | x2 – x1 |
AC is also called the length of

5. Finding the Distance on a Number Line
USING SEGMENT POSTULATES
Finding the Distance on a Number Line
                                                                
       
Find AB, BA, CB, BC, BD, and DB
Solutions:
AB = 1.5, BA = 1.5, CB = 3, BC = 3, BD = 5, DB = 5

6. Segments can be defined by using the idea of betweenness of points.
USING SEGMENT POSTULATES
Segments can be defined by using the idea of betweenness of points.
DEFINITION OF BETWEENESS - In the figure, Point B is between A and C while Point H is not between A and B.  For B to be between A and C, all three points must be collinear and B must lie on segment AC.

7. Postulate 2 Segment Addition Postulate
USING SEGMENT POSTULATES
Postulate 2 Segment Addition Postulate
If R is between P and Q, then PR + RQ = PQ.  If PR + RQ = PQ, then R is between P and Q.
                           

8. In the diagram, AD = 34, BC = 14, CD = 13 and AB = BE = EC.
USING SEGMENT POSTULATES
In the diagram, AD = 34, BC = 14, CD = 13 and AB = BE = EC.
Find BE, EC, AB, AE, ED, and BD
Solution: BE = 7, EC = 7, AB = 7, AE = 14, ED = 20, and BD = 27

9. USING THE DISTANCE FORMULA
Distance Formula is used to find the distance between points on a coordinate plane.
The Distance Formula
If A(x1, y1) and B(x2, y2) are two points in a coordinate plane, then the distance between A and B is
AB =

10. What is the Distance Formula based on?
USING THE DISTANCE FORMULA
What is the Distance Formula based on?
B(x2, y2)
| y2 – y1 |
| x2 – x1 |
C(x2, y1)
A(x1, y1)
Pythagorean Theorem
c2 = a2 + b2
D =

11. To determine the length of segment CD we would follow these steps.
USING THE DISTANCE FORMULA
    
To determine the length of segment CD we would follow these steps.  
                              

12. (Because the decimal is a rounded answer)
USING THE DISTANCE FORMULA
Find the distance between the points X(2, –5) and Y(–4, –9)
Solution:
Approximately
(Because the decimal is a rounded answer)

13. USING THE DISTANCE FORMULA
Segments that are equal in length are called CONGRUENT SEGMENTS.  Segments of equal length are usually denoted by the same number of dash marks.  In the example to the left, segment AB and segment CD have one dash mark each and thus they are congruent.  The symbol used for congruency is an equals sign with a squiggle over it,