Lesson 1.2: Congruent Segments and Segment Postulates

Key Concepts

• Lines vs Segments:

  • Lines extend indefinitely and cannot be measured.
  • Segments have endpoints, allowing for measurement.

• Measure of Segments:

  • Use endpoints on a number line or ruler.
  • Ruler Postulate: The distance between two points is the absolute value of the difference of their coordinates.
  • Example: For segment AC with points A at -3 and C at 6:
    • Negative 3 minus 6 equals negative 9
    • Absolute value of -9 is 9
    • Order of subtraction does not matter (|A - C| = |C - A|).

• Points "Between" Other Points:

  • Relevant for collinear points.
  • A point is "between" two other points if all three are collinear.
  • Example: B is between A and C if A, B, and C are collinear.

Segment Addition Postulate

• Definition: If point B is between A and C, then the length of AB + length of BC = length of AC.

• Example Calculations:

  1. Given AB = 15 and BC = 21, find AC:
     • AB + BC = AC
     • 15 + 21 = 36
     • AC = 36
  2. Given AC = 42 and AB = 17, find BC:
     • AB + BC = AC
     • 17 + BC = 42
     • BC = 42 - 17 = 25

• Additional Practice Examples:

  • AB = 12, BC = 19, find AC: 12 + 19 = 31
  • AB = 8, BC = x, AC = 23, find x: 8 + x = 23, x = 15

Example: Algebra with Segment Addition

• Point M between L and N:
  • Segment LM = 7y + 9
  • Segment MN = 3y + 4
  • Segment LN = 143
  • Solve: 7y + 9 + 3y + 4 = 143
  • Combine like terms: 10y + 13 = 143
  • Solve for y: 10y = 130, y = 13

Concept of Congruence

• Definition: Congruent segments have the same length.
• Notation:
  • Equal Lengths: No bar over symbols, e.g., length of AB = length of AD.
  • Congruent Segments: Symbol with equal sign and squiggly line (≅).

Example: Determining Congruence on a Coordinate Grid

• Segment JK (Horizontal):

  • x-coordinates: -3 and 2
  • Length = |(-3) - 2| = 5

• Segment LM (Vertical):

  • y-coordinates: 3 and -2
  • Length = |3 - (-2)| = 5

• Conclusion:

  • JK and LM are congruent as both have a length of 5.

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This concludes the video on using segment postulates to identify congruent segments. Thank you for watching.