Section 1.3 Segments and Their Measures Goal 1 Use Segment Postulates Goal 2 Use the Distance Formula

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.

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.

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

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

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.

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.                            

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

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 =

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 =

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.                                 

(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)

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,