Deductive Reasoning

1
Section 2.3

• Deductive Reasoning

2
Using Symbolic Notation

• Conditional Statements
• If - then form
• statement with two parts
• Hypothesis
• Conclusion

3
Using Symbolic Notation

• Conditional Statements can be written
  symbolically
• p represents hypothesis
• q represents conclusion
• ? is read as implies

4
Using Symbolic Notation
If two angles have the same measurement, then
the angles are congruent
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Using Symbolic Notation
hypothesis
If two angles have the same measurement, then
the angles are congruent
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Using Symbolic Notation
hypothesis
If two angles have the same measurement, then
the angles are congruent
conclusion
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Using Symbolic Notation
p
If two angles have the same measurement, then
the angles are congruent
q
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Using Symbolic Notation

• Conditional statement written symbolically
• If p, then q
• OR
• p ? q

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Using Symbolic Notation

• Biconditional Statements
• Contains if and only if

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Using Symbolic Notation

• Biconditional Statement Example
• Two angles have the same measurement if and only
  if the angles are congruent.

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Using Symbolic Notation

• Biconditional Statement Symbolically
• If p, then q and if q, then p
• OR
• p ? q

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Using Symbolic Notation

• Biconditional Statement Symbolically
• p if and only if q
• OR
• p iff q

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Using Symbolic Notation

• Converse
• Switch the hypothesis and the conclusion
• Switch p and q

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Using Symbolic Notation
p
If
two angles have the same measurement
the angles are congruent
then
q
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Using Symbolic Notation
q
If
two angles are congruent
the angles have the same measurement
then
p
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Using Symbolic Notation

• Converse Written Symbolically
• If q, then p
• OR
• q ? p

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Using Symbolic Notation

• Inverse
• Negate hypothesis and conclusion
• means negation
• Read as not

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Using Symbolic Notation

• Inverse Example

If two
angles have the same measurement, then the angles
are congruent.
Conditional Statement
Inverse

• If two angles do not have the same
  measurement, then the angles are not congruent.

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Using Symbolic Notation

• Inverse Written Symbolically
• If p, then q
• OR
• p ? q

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Using Symbolic Notation

• Contrapositive
• Negate hypothesis and conclusion of the converse
• Negate p and q, switch

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Using Symbolic Notation

• Contrapositive Example
• Conditional Statement
• If two angles have the same measurement, then
  the angles are congruent.
• Contrapositive
• If two angles are not congruent, then the angles
  do not have the same measurement.

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Using Symbolic Notation

• Contrapositive Written Symbolically
• If q, then p
• OR
• q ? p

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Using Symbolic Notation

• In Review
• Conditional statement
• Biconditional Statement
• Inverse
• Converse
• Contrapositive

p ? q
p ? q
p ? q
q ? p
q ? p
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Understanding Symbolic Notation

• Conditional Statement
• p ? q

Inverse
q
p
?


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Understanding Symbolic Notation

• Conditional Statement
• p ? q

Converse
p
q
?
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Understanding Symbolic Notation

• Converse
• q ? p

Contrapositive
p
q
?


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Practice Using Symbolic Notation

• If two angles form a linear pair, then they are
  supplementary.
• Determine p and q
• p two angles form a linear pair
• q they are supplementary
• p ? q

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Practice Using Symbolic Notation

• If two angles form a linear pair, then they are
  supplementary.
• Determine inverse
• If two angles do not form a linear pair, then
  they are not supplementary
• p ? q

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Practice Using Symbolic Notation

• If two angles form a linear pair, then they are
  supplementary.
• Determine Contrapositive
• If two angles are not supplementary, then they do
  not form a linear pair.
• q ? p

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Practice Using Symbolic Notation

• p BD bisects ?ABC
• q ?ABD is congruent to ?DBC
• Determine p ? q
• BD bisects ?ABC if and only if ?ABD is congruent
  to ?DBC

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Practice Using Symbolic Notation

• p BD bisects ?ABC
• q ?ABD is congruent to ?DBC
• q ? p
• If ?ABD is not congruent to ?DBC, then BD does
  not bisects ?ABC

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Using Symbolic Notation

• Homework
• Page 91, 8 - 20