2 2 Definitions and Biconditional Statements Perpendicular lines
2. 2 Definitions and Biconditional Statements
Perpendicular lines • Perpendicular lines intersect to form four right angles. • Perpendicular Symbol: ⊥
Definitions • All definitions can be interpreted “forward” and “backward” • Ex: – If two lines are perpendicular, then they intersect to form a right angle. – If two lines intersect to form a right angle, then they are perpendicular.
Biconditional Statements • A biconditional statement is a statement that contains the phrase “if and only if. ” • Writing a biconditional statement is the equivalent to writing a conditional statement and its converse.
Examples: Decide if each statement is a biconditional statement: 1. Two lines intersect if and only if their intersection is exactly one point. 2. Points are coplanar if and only if they lie in the same plane. 3. I will go to the movies only if you go.
Examples: Rewriting a biconditional statement • Rewrite as a conditional and its converse: – Three lines are coplanar if and only if they lie in the same plane. – Points are collinear if and only if they lie on the same line.
Example: Analyzing a biconditional statement • Consider the following statement: X=3 if and only if x²=9. a) Is this a biconditional statement? b) Is the statement true?
• Consider the following statement: x²<49 if and only if x<7 a) Is this a biconditional statement? b) Is the statement true?
True Biconditionals • Biconditional statements can be either true or false. • True: both the conditional statement and its converse must be true • False: If either the conditional statement or its converse is false
Example: The following statement is true. Write the converse of each statement and decide whether the converse is true or false. If the converse is true, combine it with the original statement to forma true biconditional. If the converse is false, state a counterexample.
1. If two points lie in a plane, then the line containing them lies in the plane. 2. 2. If a number ends in 0, then the number is divisible by 5.
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