Deductive Reasoning

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Deductive Reasoning
GEOMETRY LESSON 2-3
A gardener knows that if it rains, the garden
will be watered. It is raining. What conclusion
can he make?
The first sentence contains a conditional
statement. The hypothesis is It rains. Because
the hypothesis is true, the gardener can conclude
that the garden will be watered.
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Deductive Reasoning
GEOMETRY LESSON 2-3
For the given statements, what can you conclude?
Given If ?A is acute, m?A lt 90. ?A is acute.
A conditional and its hypothesis are both given
as true. By the Law of Detachment, you can
conclude that the conclusion of the conditional
m?A lt 90 is also true.
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Deductive Reasoning
GEOMETRY LESSON 2-3
Does the following argument illustrate the Law
of Detachment?
Given If you make a field goal in basketball,
you score two points.Jenna scored two points in
basketball.
You conclude Jenna made a field goal. The two
given statements mean that a conditional and its
conclusion are both true. The Law of Detachment
applies only if a conditional and its hypothesis
are true. You can make no conclusion. You cannot
conclude that Jenna made a field goal.
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Deductive Reasoning
GEOMETRY LESSON 2-3
Use the Law of Syllogism to draw a conclusion
from the following true statements
If a quadrilateral is a square, then it contains
four right angles. If a quadrilateral contains
four right angles, then it is a rectangle.
The conclusion of the first conditional is the
hypothesis of the second conditional. This means
that you can apply the Law of Syllogism. The Law
of Syllogism If p ? q and q ? r are true, then p
? r is a true statement. So you can conclude If
a quadrilateral is a square, then it is a
rectangle.
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Deductive Reasoning
GEOMETRY LESSON 2-3
Use the Laws of Detachment and Syllogism to draw
a possible conclusion.
If the circus is in town, then there are tents at
the fairground. If there are tents at the
fairground, then Paul is working as a night
watchman. The circus is in town.
Because the conclusion of the first statement is
the hypothesis of the second statement, you can
apply the Law of Syllogism to write this new
conditional If the circus is in town, then Paul
is working as a night watchman. The third
statement means that the hypothesis of the new
conditional is true. You can use the Law of
Detachment to form the conclusion Paul is
working as a night watchman.
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Deductive Reasoning
GEOMETRY LESSON 2-3
Use the three statements below. A. If games are
canceled, then Maria reads a book. B. If it
snows, then games are canceled. C. It is
snowing. 1. Using only statements A and B, what
can you conclude? 2. Using only statements B
and C, what can you conclude? 3. Using
statements A, B, and C, what can you
conclude? 4. Suppose both statement B and
games are canceled are true. Can you conclude
that statement C is true? Explain.
If it snows, then Maria reads a book.
Games are canceled.
Maria is reading a book.
No sample you cannot apply the Law of
Detachment.