1. Lesson: Segments and Rays 1 Geometry Segments and Rays

2. Lesson 1-2: Segments and Rays 2 Postulates Definition: An assumption that needs no explanation. Examples: Through any two points there is exactly one line. Through any three points, there is exactly one plane. A line contains at least two points. A plane contains at least three points.

3. Lesson 1-2: Segments and Rays 3 The Ruler Postulate The Ruler Postulate: Points on a line can be paired with the real numbers in such a way that: Any two chosen points can be paired with 0 and 1. The distance between any two points on a number line is the absolute value of the difference of the real numbers corresponding to the points. Formula: Take the absolute value of the difference of the two coordinates a and b: │a – b │

4. Lesson 1-2: Segments and Rays 4 Ruler Postulate : Example PK =| 3 - -2 | = 5 Remember : Distance is always positive Find the distance between P and K. Note: The coordinates are the numbers on the ruler or number line! The capital letters are the names of the points. Therefore, the coordinates of points P and K are 3 and -2 respectively. Substituting the coordinates in the formula │a – b │

5. Measuring Segment Lengths What is ST? What is SV? What is UV? What is TV?

6. Measuring Segment Lengths ST = | -4 – 8 | = | -12| = 12 SV = |-4 – 14 | = | -18| = 18 UV = | 10 – 14| = | -4| = 4 TV = |8 – 14| = | -6 | = 6

7. Postulate: Segment Addition Postulate If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC.

8. Lesson 1-2: Segments and Rays 8 Between Definition: X is between A and B if AX + XB = AB. AX + XB = ABAX + XB > AB

9. Lesson 1-2: Segments and Rays 9 Segment Part of a line that consists of two points called the endpoints and all points between them. How to sketch: How to name: Definition: AB (without a symbol) means the length of the segment or the distance between points A and B.

10. Lesson 1-2: Segments and Rays 10 The Segment Addition Postulate If C is between A and B, then AC + CB = AB. Postulate: Example: If AC = x, CB = 2x and AB = 12, then, find x, AC and CB. AC + CB = AB x + 2x = 12 3x = 12 x = 4 2x x 12 x = 4 AC = 4 CB = 8 Step 1: Draw a figure Step 2: Label fig. with given info. Step 3: Write an equation Step 4: Solve and find all the answers

11. Lesson 1-2: Segments and Rays 11 Congruent Segments Definition: If numbers are equal the objects are congruent. AB: the segment AB ( an object ) AB: the distance from A to B ( a number ) Congruent segments can be marked with dashes. Correct notation: Incorrect notation: Segments with equal lengths. (congruent symbol: )

12. Lesson 1-2: Segments and Rays 12 Midpoint A point that divides a segment into two congruent segments Definition: On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is. In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and is. Formulas:

13. Lesson 1-2: Segments and Rays 13 Midpoint on Number Line - Example Find the coordinate of the midpoint of the segment PK. Now find the midpoint on the number line.

14. Lesson 1-2: Segments and Rays 14 Segment Bisector Any segment, line or plane that divides a segment into two congruent parts is called segment bisector. Definition: