2.1, 2.2 and 5.4: Statements and
Reasoning
(2.1) Conditional Statements
Conditional is an if-then statement that contains two parts.
The part following the if is the Hypothesis.
The part following then is the Conclusion.
Example 1:
If you live in a country that borders the northern United States, then
you live in Canada.
Hypothesis: You live in a country that borders the
northern United States.
Conclusion: You live in Canada.
(2.1) Conditional Statements
Conditional is an if-then statement that contains two parts.
The part following the if is the Hypothesis.
The part following then is the Conclusion.
Example 2:
If two lines are parallel, then the lines are coplanar.
Hypothesis: Two lines are parallel.
Conclusion: The lines are coplanar.
(2.1) Conditional Statements
Conditional is an if-then statement that contains two parts.
The part following the if is the Hypothesis.
The part following then is the Conclusion.
Example 3: Write the statement as a conditional.
An acute angle measures less than 90 degrees.
Hypothesis: An angle is acute
First part of the Conditional: If an angle is acute
Conclusion: It measures less than 90 degrees
Second part of the Conditional: then it measures less than
90 degrees
Conditional Statement:
If an angle is acute, then it measures less than 90
degrees.
(2.1) Conditional Statements
Conditional is an if-then statement that contains two parts.
The part following the if is the Hypothesis.
The part following then is the Conclusion.
Example 4: Write a counterexample to prove the conditional false.
If x2 > 0, then x > 0.
The counterexample for the conditional is when the hypothesis is
true but the conclusion is false.
(2.1) Converse
The converse of a conditional switches the hypothesis and
the conclusion.
Example 1: Writing the converse of the conditional.
Conditional: If two lines intersect to form right angles, then
they are perpendicular.
Converse: If two lines are perpendicular, then they
intersect to form right angles.
(2.1) Converse
Converse of a conditional switches the hypothesis and
the conclusion.
Example 2: Writing the converse of the conditional.
Conditional: If x = 9, then x + 3 = 12
Converse: If x + 3 = 12 , then x = 9
(2.1) Converse
Converse of a conditional switches the hypothesis and
the conclusion.
Example 3: Write the converse of the conditional and determine the
truth value of each.
Conditional: If a = 5, then a2 = 25
Converse: If a2 = 25, then a = 5
*Sometimes the converse is false!!!
(2.2) Biconditionals
Biconditionals When a conditional and its converse are true, you
can combine them as a true biconditional. You can combine them by
using the phrase if and only if.
Example 1: Consider the conditional. Write its converse. If they are
both true, combine the statements as a biconditional.
Conditional: If two angles have the same measure, then the angles
are congruent.
Converse: If two angles are congruent, then the angles have
the same measure.
Biconditional: Two angles have the same measure if and only if
the angles are congruent.
(2.2) Biconditionals
Biconditionals When a conditional and its converse are true, you can
combine them as a true biconditional. You can combine them by
using the phrase if and only if.
Example 2: Write the two statements that form the biconditional.
Biconditional: You live in Washington, D.C., if and only if you live in
the capital of the United States.
Conditional: If you live in Washington, D.C., then you live in
the capital of the United States.
Converse: If you live in the capital of the United States,
then you live in Washington, D.C.
(5.4) Negation
The negation of a statement has the opposite
meaning of the original statement.
Example: Negate the following statement
Original statement: Two angles are congruent.
Negation: Two angles are not congruent.
(5.4) Inverse
The inverse of a conditional statement negates both
the hypothesis and the conclusion.
Example: Find the inverse of the following conditional
statement
Conditional: If a figure is a square, then it is a
rectangle.
Inverse: If a figure is not a square, then it is not a
rectangle.
(5.4) Contrapositive
The contrapositive of a conditional statement switches the
hypothesis and the conclusion and negates both.
Example: Find the contrapositive of the following conditional.
Conditional: If a figure is a square, then it is a rectangle.
Contrapositive: If a figure is not a rectangle, then it is not
a square.
(5.4) Contrapositive
The contrapositive of a conditional statement switches the
the hypothesis and the conclusion and negates both.
Example: Write the (a) the inverse and (b)
contrapositive of Maya Angelou’s statement.
“If you don’t stand for
something, then you’ll
fall for anything.”
(5.4) Identifying Contradictions
Example: Identify the two statements that contradict each
other.
I. FG || KL
II. FG  KL
III. FG  KL
Two segments can be parallel and congruent. So I and II do not
contradict each other.
Two segments can be congruent and perpendicular. So II and III do
not contradict each other.
Two segments cannot be parallel and perpendicular. So I and III
contradict each other.
2.1, 2.2 and 5.4: Statements and
Reasoning
HOMEWORK:
Page 83 #3, 10-18 even, 24-32 even;
Page 90 #2, 6, 10-16 even, Challenge: 27-30 even
Page 283 #2-6 even, 7-9 all, 18-19 all
TERMS:
Biconditional, Conclusion, Conditional,
Converse, Hypothesis,
*Truth Value: true or false