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Segments and Rays in Geometry

Learn about the postulates and definitions related to segments, rays, ruler postulate, distances, between points, segment addition postulate, congruent segments, midpoint, segment bisector, rays, and opposite rays.

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Segments and Rays in Geometry

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  1. Segments and Rays Lesson 1-2 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. • A line contains at least two points. • Through any three points, there is • exactly one plane. • A plane contains at least three points. Lesson 1-2: 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. Lesson 1-2: 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 │ Lesson 1-2: 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 Lesson 1-2: Segments and Rays

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

  7. 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. Lesson 1-2: Segments and Rays

  8. 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 Lesson 1-2: Segments and Rays

  9. 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: Lesson 1-2: Segments and Rays

  10. 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 . Lesson 1-2: Segments and Rays

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

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

  13. 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: Lesson 1-2: Segments and Rays

  14. 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 Lesson 1-2: Segments and Rays

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