Chapter 2.2 Notes: Analyze
Conditional Statements
Goal: You will write definitions as conditional
statements.
• A conditional statement is a logical statement that
has two parts, a hypothesis and a conclusion.
• When a conditional statement is written in if-then
form, the “if” part contains the hypothesis and the
“then” part contains the conclusion.
i.e. If it is raining, then there are clouds in the sky.
Ex.1: Rewrite the conditional statement in if-then
form.
a. All birds have feathers.
If an animal is a bird, then it has feathers.
b. Two angles are supplementary if they are a linear
pair.
If two angles are supplementary angles, then
they are a linear pair.
OR
If two angles form a linear pair, then they are
supplementary angles.
Ex.2: Rewrite the conditional statement in if-then
form.
a. All whales are mammals.
If an animal is a whale, then it is a mammal.
b. Three points are collinear if there is a line
containing them.
If three points are collinear, then there is a line
containing them.
OR
If there is a line containing three points, then
the three points are collinear.
Ex.3: Rewrite the conditional statement in if-then
form.
a. All 90o angles are right angles.
If an angle measures 90o, then it is a right angle.
b. Tourists at the Alamo are in Texas.
If tourists are at the Alamo, then they are in
Texas.
• Negation:
• The negation of a statement is the opposite of the
original statement.
Ex.4: Statement 1: The ball is red.
Negation 1: The ball is not red.
Ex.5: Statement 2: The cat is not black.
Negation 2: The cat is black.
• Verifying Statements:
• Conditional statements can be true or false.
• To show that a conditional statement is true, you
must prove that the conclusion is true every time
the hypothesis is true.
• To show that a conditional statement is false, you
need to give only one counterexample.
• Related Conditionals:
o
If mA  99 , then  A is obtuse.
• To write the converse of a conditional statement,
exchange the hypothesis and conclusion.
Converse: If angle A is obtuse, then the measure of
angle A is 99o.
• To write the inverse of a conditional statement,
negate both the hypothesis and the conclusion.
Inverse: If the measure of angle A is not 99o, then
angle A is not obtuse.
• To write the contrapositive, first write the converse
and then negate both the hypothesis and the
conclusion.
Contrapositive: If angle A is not obtuse, then the
measure of angle A is not 99o.
Ex.6: Write the conditional statement in if-then form,
the converse, the inverse, and the contrapositive of
the statement “Soccer players are athletes.” Decide
whether each statement is true or false.
Conditional: If a person is a soccer player, then they
are an athlete.
Converse: If a person is an athlete, then they are a
soccer player.
Inverse: If a person is not a soccer player, then they
are not an athlete.
Contrapositive: If a person is not an athlete, then they
are not a soccer player.
Ex.7: Write the if-then form, the converse, the inverse,
and the contrapositive of the conditional statement
“Guitar players are musicians.” Decide whether
each statement is true or false.
Conditional: If a person is a guitar player, then they
are a musician.
Converse: If a person is a musician, then they are a
guitar player.
Inverse: If a person is not a guitar player, then they
are not a musician.
Contrapositive: If a person is not a musician, then
they are not a guitar player.
Ex.8: Write the converse, the inverse, and the
contrapositive of the conditional statement, “If a
polygon is equilateral, then the polygon is regular.”
Converse: If a polygon is regular, then the polygon is
equilateral.
Inverse: If a polygon is not equilateral, then the
polygon is not regular.
Contrapositive: If a polygon is not regular, then the
polygon is not equilateral.
• Equivalent Statements:
• A conditional statement and its contrapositive are
either both true or both false.
• Similarly, the converse and inverse of a conditional
statement are either both true or both false.
• When two statements are both true or both false,
they are called equivalent statements.
• Definitions:
• You can write a definition as a conditional
statement in if-then form or as its converse.
• Both the conditional statement and its converse are
true.
• Perpendicular Lines:
• If two lines intersect to form a right angle, then they
are perpendicular lines.
• The definition can also be written using the
converse:
If two lines are perpendicular lines, then they
intersect to form a right angle.
• You can write “line l is perpendicular to line m” as
__________________.
• Biconditional Statements:
• When a conditional statement and its converse are
both true, you can write them as a single
biconditional statement.
• A biconditional statement is a statement that
contains the phrase “if and only if.”
– Any valid definition can be written as a
biconditional statement.
Ex.9: Write the definition of perpendicular lines as a
biconditional.
Definition: If two lines are perpendicular lines, then
they intersect to form a right angle.
Converse: If two lines intersect to form a right angle,
then they are perpendicular lines.
Biconditional: Two lines are perpendicular lines if and
only if they intersect to form a right angle.
Ex.10: Write the definition of supplementary angles as
a biconditional.
Definition: If two angles are supplementary angles,
then their measure is 180o.
Converse: If the measure of two angles is 180o, then
they are supplementary angles.
Biconditional: Two angles are supplementary angles if
and only if their measure is 180o.
Ex.11: Use the diagram shown. Decide whether each
statement is true. Explain your answer using the
definitions you have learned.
a. JMF and FMG are supplementary.
b. Point M is the midpoint of FH .
c. JMF and HMG are vertical angles.
d. FH  JG