Notes
Sections 2.1, 2.2 and 5.4: Statements and Reasoning
Name:
Date:
CONDITIONAL STATEMENTS
is an if-then statement that contains two parts.
The part following the if is the
.
The part following the then is the
.
Example 1:
If you live in a country that borders the northern
United States, then you live in Canada.
Hypothesis:
Example 2:
If two lines are parallel, then the lines are coplanar.
Hypothesis:
Conclusion:
Conclusion:
Example 3: Write the statement as a conditional.
An acute angle measures less than 90 degrees.
Example 4: Write a counterexample to prove the
conditional false.
If x2 > 0, then x >0.
Hypothesis:
First part of the Conditional:
Conclusion:
Second part of the Conditional:
(The counterexample for the conditional is when the
is true but the
is false.)
Conditional Statement:
CONVERSE
The
of a conditional switches the hypothesis and the conclusion.
Example 1: Writing the converse of the conditional.
Conditional: If
, then
.
Converse: If
, then
.
Example 2: Writing the converse of the conditional.
Conditional: If
, then
.
Converse: If
, then
.
Example 3: Write the converse of the conditional
and determine the truth value of each.
Conditional: If
T or F
Converse: If
T or F
, then
, then
.
.
BICONDITIONALS
… When a conditional and its converse are true, you can combine them as a true
biconditionals. You can combine them by using the phrase
.
Example 1: Consider the conditional. Write its converse. If they are both true, combine the statements as a
biconditionals.
Conditional: If two angles
, then the angles
.
Converse: If two angles
Biconditional: Two angles
, then the angles
.
if and only if the angles
.
Example 2: Write the two statements that form the biconditionals.
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
.
, then you live in
Converse: If you live in
in
, then you live
.
NEGATIONS
The
of a statement has the opposite meaning of the original statement.
Example:
Original statement: Two angles are
Negation: Two angles are
.
.
INVERSE
The inverse of a conditional statement negates both the hypothesis and the conclusion.
Example:
Conditional: If a figure is
, then it is
Inverse: If a figure is
.
, then it is
.
CONTRAPOSITIVE
The
both.
of a conditional statement switches the hypothesis and the conclusion and negates
Example 1:
Conditional: If a figure is
, then it is
Contrapositive: If a figure is
.
, then it is
.
Example 2: Write the inverse and contrapositive of Maya Angelou’s statement.
“If you don’t stand for something, then you’ll fall for anything.”
Inverse:
Contrapositive:
IDENTIFYING CONTRADICTIONS
Example: Identify the two statements that contradict each other.
I. FG || KL
II. FG  KL
III. FG  KL
Two segments ________be parallel and congruent. So I and II
Two segments ________ be congruent and perpendicular. So II and III
Two segments ________ be parallel and perpendicular. So I and III
each other.
each other.
____________each other.