Deductive Reasoning
SOL: G.1 d, e, f
Sec: 2.4
E.Q: What is the difference between the Law of
syllogism and the law of Detachment?
Deductive Reasoning
Definition – Use facts, definitions, accepted
proportions and the laws of logic to form
a logical argument.
There are there laws of logic:
1. The law of Detachment
2. The law of Syllogism
3. The law of Contrapositive
Law of Detachment
Definition – If the Hypothesis of a true conditional
statement is true, then the conclusion is also true.
So, if p → q is true and p is true, then q is true.
Symbolic Representation: p → q
p
q
Ex:
 Angles that are supplementary have measures
with a sum of 180°.
 < A and < B are Supplementary
< A and < B measures are a sum of 180°

Law of Syllogism
Definition –
If hypothesis p, then conclusion q.
If hypothesis q , then conclusion r.
(if both above statements are true)
If hypothesis p, then conclusion r.
(Then the above is also true)
Symbolic Representation: p → q
q→r
pr
Ex:
 The sun is a star
 Stars are in constant motion.
The sun is in constant motion.

Law of Contrapositive
Definition –So, if p → q is true and q is not
true, then p is not true.
Symbolic Representation: p → q
~q
~p
Ex:
(1) If you do your home work, then you will
pass this class.
(2) Shane did not pass the class.
(3) Shane did not do his homework
Determine if statement (3) follows from statements (1)
and (2) by the Law of Detachment, the Law of Syllogism
or the Law of Contrapositive. If it does, state which law
was used. If it does not, write invalid.
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In – line skates live dangerously.
If you live dangerously, then you like to dance.
If you are an in – line skater, then you like to dance.
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If you drive safely, the life you save may be your own.
The life Casey saves may not be her own.
She does not drive safely.
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If a figure is a rectangle, then its opposite sides are congruent.
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AB  DC and AD  BC.
ABCD is a rectangle.
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Determine if a valid conclusion can be reached from the
two true statements using the Law of Detachment, the
Law of Syllogism or the Law of Contrapositive. If a valid
conclusion is possible, state it and the law that is used. If
a valid conclusion does not follow, write no conclusion.
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If two angles are vertical, then they do not form a linear pair.
If two angles are vertical, then they are congruent.
If you eat to live, then you live to eat.
Paula does not live to eat.
Cars are useful.
Useful cars are practical.
Assignments
Classwork: Worksheet
Homework: Pg 84-85 8-9, 24-29