2.1 Conditional Statements
What you should learn
GOAL
1
Recognize and analyze a conditional
statement.
GOAL
2
Write postulates about points, lines, and
planes using conditional statements.
Why you should learn it
Point, line, and plane postulates help you analyze
real-life objects, such as a research buggy (pg. 71).
2.1 Conditional Statements
GOAL
1
RECOGNIZING CONDITIONAL
STATEMENTS
VOCABULARY
hypothesis
A conditional statement has two parts, a __________
conclusion If the statement is written in if-then
and a __________.
form, the “if” part contains the __________
hypothesis and the
“then” part contains the __________.
conclusion
If HYPOTHESIS, then CONCLUSION.
EXAMPLE 1
Extra Example 1
Rewrite in if-then form:
All mammals breathe oxygen.
Hypothesis:
Conclusion:
An animal is a mammal.
It breathes oxygen.
If-then form:
If an animal is a mammal, then it breathes oxygen.
TRUE OR FALSE?
For a conditional statement to be TRUE, the conclusion
must be true for all cases that fulfill the hypothesis. It will
be YOUR JOB to show this.
For a conditional statement to be FALSE, only one
counterexample is necessary.
EXAMPLE 2
Extra Example 2
Write a counterexample:
If a number is odd, then it is divisible by 3.
Sample answer:
7 is odd and 7 is not divisible by 3.
• Converse: The hypothesis and conclusion are switched.
EXAMPLE 3
NEGATIONS
Inverse: Both hypothesis and conclusion are negated.
Contrapositive: The hypothesis and conclusion of a
converse are negated.
CAUTION: When writing negations, do not be
more specific than the original statement.
The negation of “It is raining” is “It is not raining,”
NOT “It is sunny.” (It could be night, foggy, etc.).
EQUIVALENT STATEMENTS
• Two statements are either both true or both false.
• A conditional is equivalent to its contrapositive, and the
inverse and converse of any conditional statement are
equivalent.
EXAMPLE 4
Extra Example 4
Write the a) inverse, b) converse, and c) contrapositive of
the statement.
If the amount of available food increases,
the deer population increases.
Hint: Identify the hypothesis and conclusion first.
Inverse: If the amount of available food does not increase,
the deer population does not increase.
Converse: If the deer population increases, the amount of
available food increases.
Contrapositive: If the deer population does not increase, the
amount of available food does not increase.
Checkpoint
1. Rewrite in if-then form: All monkeys have tails.
If an animal is a monkey, then it has a tail.
2. Write a counterexample:
If a number is divisible by 2, then it is divisible by 4.
Sample: 14 is divisible by 2 but not divisible by 4.
3. Write the inverse, converse, and contrapositive of the
statement: If an animal is a fish, then it can swim.
Inverse: If an animal is not a fish, then it cannot swim.
Converse: If an animal can swim, then it is a fish.
Contrapositive: If an animal cannot swim, then it is not a fish.
2.1 Conditional Statements
GOAL
2
USING POINT, LINE, AND PLANE
POSTULATES
Study the postulates on page 73, and be sure you
understand what they mean. Example 5 will help you.
EXAMPLE 5
Extra Example 5
Give examples of Postulates 5-11.
P5: There is exactly one line (m) through A and B.
P6: m contains at least two points (A and B).
P7: m and n intersect at C.
P8: Q passes through A, B, and D.
P9: Q contains at least A, B, and D.
P
P10: A and B lie in Q. So m, which
contains A and B, also lies in Q.
P11: P and Q intersect in line n.
D
C
A
m
EXAMPLE 6
n
B
Q
Extra Example 6
Rewrite Postulate 6 in if-then form, then write its inverse,
converse, and contrapositive.
If-then form: If a figure is a line, then it contains at
least two points.
Inverse:
If a figure is not a line, then it does not
contain at least two points.
Converse:
If a figure contains at least two points,
then it is a line.
Contrapositive: If a figure does not contain at least two
points, then it is not a line.
EXAMPLE 7
Extra Example 7
Decide if the statement is true or false. If it is false, give a
counterexample.
Three points are always contained in a line.
FALSE
Sample answer:
C
B
A
Checkpoint
1. Write the inverse, converse, and contrapositive of
Postulate 8.
Inverse: If three noncollinear points are not distinct, then it is
not true that there is exactly one plane that passes
through them.
Converse: If exactly one plane passes through three
noncollinear points, then the three points are distinct.
Contrapositive: If it is not true that exactly one plane passes
through three noncollinear points, then the three
points are not distinct.
2. Decide whether the statement is true or false. If it is
false, give a counterexample.
A line can contain more than two points.
TRUE
Vocabulary Check
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Conditional Statement
If-then form
Hypothesis
Conclusion
Converse
Inverse
Contrapositive
Negation
Equivalent Statements