2.2 Biconditionals and Definitions September 14, 2009
2­2 Biconditionals and Definitions
Objective
Write biconditionals.
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2.2 Biconditionals and Definitions September 14, 2009
Check Skills You'll Need:
Identify the hypothesis and the conclusion of each conditional statement.
1. If x > 10, then x > 5.
2. If you live in Milwaukee, then you live in Wisconsin.
Write each statement as a conditional.
3. Squares have four sides.
4. All butterflies have wings.
Write the converse of each statement.
5. If the sun shines, then we go on a picnic.
6. If two lines are skew, then they do not intersect.
7. If x = ­3, then x = ­27.
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2.2 Biconditionals and Definitions September 14, 2009
Review
Write the converse of this conditional:
Conditional
If an integer is divisible by 8, then it is divisible by 2.
Converse
If an integer is divisible by 2, then it is divisible by 8.
What is the truth value of the converse? Can you provide a Counterexample?
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2.2 Biconditionals and Definitions September 14, 2009
When a statement and its converse are both true, you can combine them into one statement called a BICONDITIONAL.
A biconditional combines the two statements with the words "if and only if".
EXAMPLE
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2.2 Biconditionals and Definitions September 14, 2009
Writing a Biconditional
Conditional
If two angles have the same measure, then the angles are congruent.
Converse
If two angles are congruent, then they have the same measure.
Are both the Conditional and the Converse true?
Combine them using "if and only if"
Biconditional
Two angles have the same measure if and only if they are congruent.
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2.2 Biconditionals and Definitions September 14, 2009
Writing a Definition as a Biconditional
Definition
Perpendicular lines are two lines that intersect to form right angles.
Conditional
If two lines are perpendicular, then they intersect to form right angles.
Converse
If two lines intersect to form right angles, then they are perpendicular.
Biconditional
Two lines are perpendicular if and only if they intersect to form right angles.
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2.2 Biconditionals and Definitions September 14, 2009
Separating a Biconditional into Parts
Biconditional
A number is divisible by 3 if and only if the sum of its digits is divisible by 3.
Conditional
If a number if divisible by 3, then the sum of its digits is divisible by 3.
Converse
If the sum of a number's digits is divisible by 3, then the number is divisible by 3.
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2.2 Biconditionals and Definitions September 14, 2009
Biconditional Statements
Statement
Example
Biconditional
An angle is a straight angle
if and only if
its measure is 180o.
Symbolic Form
You Read It
p q
p if and only if q
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2.2 Biconditionals and Definitions September 14, 2009
HOMEWORK
p. 90 #1­12, 62­67
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