Understanding Segments and Rays: Key Postulates and Definitions

1. Segments and Rays Lesson 2-2 UNIT 2: Segments and Rays

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

3. Postulates Examples: • If two planes intersect, • then the intersecting is a line. • If two points lie in a plane, • then the line containing the two • points lie in the same plane. UNIT 2: Segments and Rays

4. 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 │ UNIT 2: Segments and Rays

5. Ruler Postulate : Example 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 │ PK = | 3 --2 | = 5 Remember : Distance is always positive UNIT 2: Segments and Rays

6. Is Alex between Ty and Josh? Yes! In order for a point to be between 2 others, all 3 points MUST BE collinear!! Ty Alex Josh No, but why not? How about now? UNIT 2: Segments and Rays

7. Between Definition: X is between A and B if AX + XB = AB. AX + XB = AB AX + XB > AB UNIT 2: Segments and Rays

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

9. 12 AC + CB = AB x + 2x = 12 3x = 12 x = 4 The Segment Addition Postulate Postulate: If C is between A and B, then AC + CB = AB. If AC = x , CB = 2x and AB = 12, then, find x, AC and CB. Example: 2x x Step 1: Draw a figure Step 2: Label fig. with given info. Step 3: Write an equation x = 4 AC = 4 CB = 8 Step 4: Solve and find all the answers UNIT 2: Segments and Rays

10. 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 Definition: Segments with equal lengths. (congruent symbol: ) Congruent segments can be marked with dashes. Correct notation: Incorrect notation: UNIT 2: Segments and Rays

11. Midpoint Definition: A point that divides a segment into two congruent segments Formulas: 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 . UNIT 2: Segments and Rays

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

13. Reminders: • Pythagorean Theorem – a2+b2=c2 a & b are the lengths of the legs of a right triangle and c is the length of the hypotenuse. • Distance formula – (x1,y1) & (x2,y2) are the 2 points. UNIT 2: Segments and Rays

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

15. RA : RA and all points Y such that A is between R and Y. ( the symbol RA is read as “ray RA” ) Ray Definition: How to sketch: How to name: UNIT 2: Segments and Rays

16. Opposite Rays Definition: If A is between X and Y, AX and AY are opposite rays. ( Opposite rays must have the same “endpoint” ) opposite rays not opposite rays UNIT 2: Segments and Rays