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  • 1. 2.2 Definitions and Biconditional Statements Objectives: - Recognize and use definitions - Recognize and use biconditional statements
  • 2. A definition . . .
    • Two lines are called perpendicular lines if the intersect to form a right angle.
    • A line perpendicular to a plane is a line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it.
    • The symbol  is read as “is perpendicular to”
  • 3. Look at the picture on p. 79-bottom
    • Using the definitions, you have learned, decide whether each statement is true:
    • Points D, X, and A are collinear.
    • Line AC is perpendicular to line DB
    • Angle AXB is adjacent to Angle CXD
  • 4. Using Biconditional Statements
    • A biconditional statement is a statement that contains the phrase “if and only if.”
    • It is academic Friday only if I have tests in every class.
    • Three lines are coplanar if and only if they lie in the same plane.
  • 5. Rewriting a biconditional
    • A biconditional statement can be rewritten as a conditional statement and its converse.
    • Biconditional: 3 lines are coplanar if and only if
    • they lie in the same plane.
    • Conditional: If 3 lines are coplanar, then they
    • lie in the same plane.
    • Converse: If 3 lines lie in the same plane, then
    • they are coplanar.
  • 6. A biconditional statement
    • Can be either true or false. To be true, both the conditional statement and its converse must be true. This means that a true biconditional statement is true both forward and backward. All definitions can be written as true biconditional statements.
  • 7. X = 3 if and only if x 2 = 9
    • Is this a biconditional statement?
    • Yes, it contains “if and only if”
    • Is the statement true?
    • Yes, because the statement can be rewritten as:
    • Conditional:
    • If x = 3, then x 2 = 9
    • Converse:
    • If x 2 = 9, then x = 3
  • 8. Writing a Biconditional
    • Is the converse true? If it is, combine it with the original to form a biconditional. If not, state a counterexample:
    • If two points lie in a plane, then the line containing them lies in the plane.
    • Converse: If a line containing 2 points lies in a plane, then
    • the points lie in the plane.
    • Biconditional: Two points lie in a plane if and only if
    • the line containing them lies in the plane.
  • 9. Writing a Biconditional
    • Is the converse true? If it is, combine it with the original to form a biconditional. If not, state a counterexample:
    • If a number ends in 0, the the number is divisible by 5.
    • Converse: If a number is divisible by 5, then the number ends in zero.
    • This is false: 15 / 5 = 3
  • 10. Writing a Postulate as a Biconditional
    • Segment Addition Postulate:
      • If B lies between points A and C, then AB + BC = AC
    • Converse:
      • If AB + BC = AC, then
      • B lies between A and C
    • Biconditional:
      • Point B lies between points A and C if and only if
      • AB + BC = AC
  • 11. Do page 82 1 - 12
    • Homework: worksheets