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Geometry 2.1 and 2.2 Conditional and Biconditional Statements
If-Then Statements
An if-then statement has two parts: a hypothesis (the clause that follows "if") and a
conclusion (the clause that follows "then"). The entire if-then statement is called a
conditional.
The statement "All professional football players are athletes" can be written
as an if-then statement: If a person is a professional football player, then the person
is an athlete. The hypothesis of the statement is "a person is a professional footballplayer" and the conclusion is "the person is an athlete."
The converse of an if-then statement is formed by switching the hypothesis
and conclusion. The converse of the above example is "If a person is an athlete, then the person
is a professional football player." Of course, this is false, because professional baseball players,
soccer stars, and Olympic hopefuls are also athletes. These examples of other types of athletes
are known as counter examples. They show that the "if" part of the statement is true, but the
"then" part is false. To prove that an if-then statement, or conditional, is false, you must give a
counter example.
Rewrite each of the following statements in the if-then form. Underline the hypothesis and circle the
conclusion in each statement.
1. All dogs are mammals.
2. All vertical angles are congruent.
3. Two lines intersect in a point.
Write the converse of each of the following statements. Determine if the converse is true or false.
If false, provide a counterexample.
4. If two angles are vertical angles, then they are congruent.
5. If x2 = 36, then x = 6 or x = -6.
6. If two angles form a linear pair, then they are supplementary.
7. If an angle measures 175°, then it is an obtuse angle.
Biconditional Statements
A biconditional statement is a statement that contains the phrase “if and only if.” Writing a
biconditional is equivalent to writing a conditional statement and its converse. Definitions can
be interpreted as biconditional statements.
Write each of the following statements as a bi-conditional statement.
1. Perpendicular lines intersect to form a right angle.
2. Complementary angles have a sum of 90°.
3. Three collinear points lie on the same line.
Rewrite each biconditional statement as its conditional statement and its converse.
4. Point Y lies between points X and Z if and only if XY + YZ = XZ.
Conditional:
Converse:
5. Two angles are congruent if and only if they have the same measure.
Conditional:
Converse:
6. The car will run if and only if there is gas in the tank.
Conditional:
Converse:
A biconditional statement is a true biconditional if it's conditional statement and its converse are both
true. Which of the above biconditionals are true biconditionals?
Write the converse of each true statement. If the converse is also true, combine the statements to
write a true bi-conditional statement.
7. If you are 15 years old, then you are a teenager.
8. If two angles are supplementary, then the sum of their measures is 180°.
9. If two angles form a linear pair, then they are adjacent.