Section 2.2 Conditional Statements

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Section 2.2 Conditional Statements
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Goals

• Recognize and analyze a conditional statement
• Write postulates about points, lines, and planes
  using conditional statements

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Conditional Statement

• A conditional statement has two parts, a
  hypothesis and a conclusion.
• When conditional statements are written in
  if-then form, the part after the if is the
  hypothesis, and the part after the then is the
  conclusion.
• p ? q

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Examples

• If you are 13 years old, then you are a teenager.
• Hypothesis
• You are 13 years old
• Conclusion
• You are a teenager

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Rewrite in the if-then form

• All mammals breathe oxygen
• If an animal is a mammal, then it breathes
  oxygen.
• A number divisible by 9 is also divisible by 3
• If a number s divisible by 9, then it is
  divisible by 3.

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Writing a Counterexample

• Write a counterexample to show that the following
  conditional statement is false
• If x2 16, then x 4.
• As a counterexample, let x -4.
• The hypothesis is true, but the conclusion is
  false. Therefore the conditional statement is
  false.

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Converse

• The converse of a conditional is formed by
  switching the hypothesis and the conclusion.
• The converse of p ? q is q ? p

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Negation

• The negative of the statement
• Example Write the negative of the statement
• ?A is acute
• ?A is not acute
• p represents not p or the negation of p

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Inverse and Contrapositive

• Inverse
• Negate the hypothesis and the conclusion
• The inverse of p ? q, is p ? q
• Contrapositive
• Negate the hypothesis and the conclusion of the
  converse
• The contrapositive of p ? q, is q ? p.

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Example

• Write the (a) inverse, (b) converse, and (c)
  contrapositive of the statement.
• If two angles are vertical, then the angles are
  congruent.
• (a) Inverse If 2 angles are not vertical, then
  they are not congruent.
• (b) Converse If 2 angles are congruent, then
  they are vertical.
• (c) Contrapositive If 2 angles are not
  congruent, then they are not vertical.

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Equivalent Statements

• When 2 statements are both true or both false
• A conditional statement is equivalent to its
  contrapositive.
• The inverse and the converse of any conditional
  are equivalent.

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Vocabulary

• A biconditional statement is a statement
  containing the phrase if and only if. It is
  the equivalent of writing a conditional statement
  and its converse.
• A biconditional statement should only be written
  if both the conditional statement and its
  converse are true. If either one is false, then
  the biconditional would also be false.

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Example
Statement If it is Saturday, then George is
working at the restaurant.
Converse If George is working at the
restaurant, then it is Saturday.
Biconditional Statement It is Saturday if and
only if George is working at the restaurant.