Sec' 54 Inverses, Contrapositives,

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Sec. 5-4Inverses, Contrapositives, Indirect
Reasoning
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Conditional Statements
Let p and q be statements.

• Name Symbolic Form
• Conditional p ? q
• Converse q ? p
• Inverse p ? q
• Contrapositive q ? p

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Conditional If a figure is a square, then it is
a rectangle. Notation (p ? q) True or
False Converse IF a figure is a rectangle then
it is a square. Notation (q ? p) True or False
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Writing the negation of a statement.

• ?ABC is obtuse.
• Write the negation of the statement. (p)
• Add not to the statement.
• ?ABC is not obtuse.
• A figure is not a square.
• (p)
• A figure is a square.

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Writing the inverse of a conditional statement.

• Negate both the hypothesis and the conclusion.
• (p ? q)
• Conditional If a figure is a triangle, then
  it has exactly 180 in it.
• Inverse If a figure is not a triangle, then
  it does not have a 180 in it.
• True or False

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Writing the Contrapositive of a conditional
statement.

• Switch the hypothesis and conclusion negate
  both of them.
• (q ? p)
• Conditional If a figure is a triangle, then
  it has exactly 180 in it.
• Contrapositve If a figure does not have a
  180 in it, then it is not a triangle.
• True or False

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• If the conditional is true, then the
  contrapositive is also true.
• If the conditional is false, then the
  contrapositive is also false.
• They are known as equivalent statements.

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Example 1

• Conditional If an ? is a straight ?, then
  its measure is 180. (T or F)
• Converse
• If an ? has a measure of 180, then it is a
  straight ?. (T or F)
• Inverse
• If an ? is not a straight ?, then its measure is
  not 180. (T or F)
• Contrapositive
• If an ? does not have a measure of 180, then it
  is not a straight ?. (T or F)

p ? q
q ? p
p ? q
q ? p
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Using Indirect Reasoning

• Indirect Reasoning All possibilities are
  considered then all but one are proved false.
  The remaining possibility must be true.
• Indirect Proof A proof involving indirect
  reasoning.
• A statement and its negation often are the only
  possibilities.

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Writing an Indirect Proof

• State the negation of what you want to prove.
• Show that this assumption leads to a
  contradiction.
• Conclude that the assumption must be false and
  that what you want to prove must be true.

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Example 2 Write the first step of an indirect
proof.

• Prove A ? cannot have more than one right ?.
• Step 1 A ? can have more than one right ?.
• Can this be true???

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Example 3

• Identify the 2 statements that contradict each
  other.
• ?ABC is Acute.
• ?ABC is scalene.
• ?ABC is equiangular.

These two contradict each other.
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p ? q q ? p p ? q q ? p