2.3. Deductive Reasoning http://www.ck12.org
Example 4:Here are two true statements.Be careful!
If^6 A and^6 B are a linear pair, then m^6 A+m^6 B= 180 ◦.
m^61 = 90 ◦and m^62 = 90 ◦.
What conclusion can you draw from these two statements?
Solution:Here there is NO conclusion. These statements are in the form:
p→q
qWecannotconclude that^6 1 and^6 2 are a linear pair. We are told thatm^61 = 90 ◦andm^62 = 90 ◦and while
90 ◦+ 90 ◦= 180 ◦, this does not mean they are a linear pair. Here are two counterexamples.
In both of these counterexamples,^6 1 and^6 2 are right angles. In the first, they are vertical angles and in the second,
they are two angles in a rectangle.
This is called theConverse Errorbecause the second statement is the conclusion of the first, like the converse of a
statement.
Law of Contrapositive
Example 5:The following two statements are true.
If a student is in Geometry, then he or she has passed Algebra I.
Daniel has not passed Algebra I.
What can you conclude from these two statements?
Solution:These statements are in the form:
p→q
∼qNotqis the beginning of the contrapositive(∼q→∼p), therefore the logical conclusion isnot p: Daniel is not in
Geometry.
This example is called the Law of Contrapositive.
Law of Contrapositive:Suppose thatp→qis a true statement and given∼q. Then, you can conclude∼p.
Recall that the logical equivalent to a conditional statement is its contrapositive. Therefore, the Law of Contrapositive
is a logical argument.