a Start by identifying the hypothesis and the conclusion of the given statement.
B
b Think of the relative positions of two lines in the space.
A
aTable:
Type of Statement
Statement
Converse statement
If two lines do not intersect, then the lines are parallel.
Inverse statement
If two lines are not parallel, then the lines intersect.
Contrapositive statement
If two lines intersect, then the lines are not parallel.
B
bTable:
Type of Statement
Statement
True or False?
Counterexample
Converse statement
If two lines do not intersect, then the lines are parallel.
False
Skew lines
Inverse statement
If two lines are not parallel, then the lines intersect.
False
Skew lines
Contrapositive statement
If two lines intersect, then the lines are not parallel.
True
-
Practice makes perfect
a We are given a conditional statement, p→ q, in an if-then form.
If two lines are parallel,
then the lines do not intersect.
Let's consider each of the statements one at a time using p and q.
p =& two lines are parallel
q =& the lines do not intersect
Converse
The converse of a conditional statement, q→ p, exchanges the hypothesis and the conclusion of the conditional statement.
If two lines do not intersect,
then the lines are parallel.
Inverse
The inverse of a conditional statement, ~ p→ ~ q, requires us to negate the hypothesis and the conclusion of the conditional statement.
If two lines are not parallel,
then the lines intersect.
Contrapositive
The contrapositive of a conditional statement, ~ q→ ~ p, starts out with the converse of the conditional statement. Then we have to negate the hypothesis and the conclusion.
If two lines intersect,
then the lines are not parallel.
b Let's determine the truth value of each statement.
Converse
The converse of the given conditional statement is false because the lines can be skew lines.
Inverse
The inverse is a false statement because there are lines that do not intersect and are not parallel. Again, we can give the example of skew lines.
Contrapositive
By the definition of parallel lines, we know that parallel lines do not intersect. Therefore, if they intersect, they cannot be parallel. This is a true statement.
