1
2
Section 2.2 (Day 2) - Biconditional Statements
Biconditional statement- can be used when a conditional
statement and its converse are both true. Contains the phrase "if
and only if"
Symbol:
Consider the following example:
Original: If two segments are congruent, then they have equal lengths. True or False?
Converse: If two segments have equal length, then they are congruent. True or False?
BicondiBonal: Two segments are congruent if and only if they have equal lengths.
3
Steps for writing a biconditional:
1
st
– Determine if the conditional statement is true.
2nd – Write the converse and determine if the converse is true.
3
rd – If the conditional statement and its converse are both true,
then you can write a biconditional statement (an “if and only if” statement)
4
Example 1: Write the definition of perpendicular lines as a biconditional
statement.
Defini&on: If two lines intersect to form a right angle, then they are perpendicular.
Is the conditional statement true?
Converse:
Is the converse true?
Biconditional:
5
With your group, create a statement that could be wriEen as a bicondiBonal statement. Remember the converse must also be true! Ex: If you are at Washington-Lee High School,
then you are at 1301 N. Stafford St.
Then, think of a statement that could not be wriEen as a bicondiBonal statement.
Ex: If you are a Redskins fan, then you will always
be disappointed.
Be prepared to share your statement with the class!
6
You try...
Example 2: Write the definition of a right angle as a biconditional statement.
Defini&on: If the measure of an angle is , then it is a right angle.
Is the conditional statement true?
Converse:
Is the converse true?
Biconditional:
7
Example 3: Rewrite the following statements as a biconditional statement.
If Mary is in theater class, then she will be in the fall play.
If Mary is in the fall play, then she must be taking theater class.
8
Counterexamples:
counterexample: An example that shows a condiBonal is false.
Example 4: Is the following condi:onal true or false? Explain why. If false, provide a counterexample.
If you are an athlete, then you are a hockey player.
Example 5: Is the following condi:onal true or false? Explain why. If false, provide a counterexample.
If two angles are congruent, then they are verBcal angles.
Example 6: Is the following condi:onal true or false? Explain why. If false, provide a counterexample.
If two angles are supplementary, then they are a linear pair.
9