Geometry

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Geometry

• Chapter 2 Logic and Reasoning

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2.2 If-Then Statements and Postulates

• Objectives
• Write a statement in if-then form Identify
  hypothesis and conclusion
• Write the inverse, converse and contrapositive
  and test their validity

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Vocabulary

• If-then statements
• Conditional Statements
• hypothesis
• conclusion
• converse
• negation
• inverse
• contrapositive
• Postulate

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Vocabulary

• If-then statements
• if ______ then _____.
• Conditional statements
• if ______ then _____.

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Vocabulary

• Hypothesis (given)
• If ____ then ______.

• Conclusion (conjecture or prove)
• If ____ then ______.

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Topic 1 IF-Then Statement

• If-Then Statements A statement in If-Then form
• Often called Conditional statement
• If part is called the hypothesis
• The if part is often given the variable p
• The if part is always the GIVEN in reasoning
• Then part is called the conclusion
• The then part is given the variable q
• The then part is always the CONJECTURE or PROVE
  in reasoning
• In Words If p then q
• In Symbols p ? q
• Re-writing a statement in if-then form. Ask
  yourself two questions
• What are we talking about? (this is the If)
• What about it? (this is the then)

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Examples

• Identify the hypothesis and the conclusion
• If it is raining then it is cloudy.

• Write the statement in if- then form
• Adjacent angles have a common side
• What are we talking about?
• Adjacent angles
• What about it?
• They have a common side
• If angles are adjacent then they have a common
  side

hypothesis
conclusion
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Vocabulary

• Negation To say something is NOT
• Example
• It is raining.
• Negation It is not raining
• Symbol

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Vocabulary

• Postulate Rules that cannot be proven for
  sureno counterexample has been found

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Re-writing in If-Then Statement

• Re-writing a statement in if-then form. Ask
  yourself two questions
• What are we talking about? (this is the If)
• What about it? (this is the then)
• Example Vertical Angles are congruent
• What are we talking about?
• What about it?

Vertical Angles
They are congruent
they are congruent
Angles are vertical
If ____________________ then __________________
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Example

• Write the inverse, converse and contrapositive of
  the statement vertical angles are congruent.
  Then determine if the statement is true
• Inverse
• Converse
• Contrapositive

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You try

• Write the inverse, converse and contrapositive of
  the statement a line contains at least two
  points. Then determine if the statement is
  true
• Inverse
• Converse
• Contrapositive

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Example

• Write the inverse of the conditional Vertical
  angles are congruent. Determine if the inverse
  is true or false. If false, give a
  counter-example.
• Statement
• If (what are we talking about) then (what about
  it)
• If angles are vertical then they are congruent
• Inverse (p?q)
• If angles are not vertical then they are not
  congruent
• FALSE

• Write the contrapostive of the statement a line
  contains at least two points. Determine if the
  contrapositive is true or false. If false, give
  a counter-example.
• Statement
• If (what are we talking about) then (what about
  it)
• If it is a line then it contains at least two
  points
• Contrapositive (q?p)
• If it does not contain at least two points then
  it is not a line
• FALSE

A
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Vocabulary

• Postulate Rules that cannot be proven for
  sureno counterexample has been found
• Through any two points there is exactly one line

B
A

• Through any three points not on the same line
  there is exactly one plane

• A line contains at least two points
• A plane contains at least three points not on the
  same line
• If two points lie in a plane, then the entire
  line containing those two points lie in the plane

• If two planes intersect, then their intersection
  is a line

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Homework

• Page 81 28, 36