Geometry

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Geometry

• 2.1 If-Then Statements
• Converses

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• This is the first introduction to logical
  reasoning

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If-Then Statements

• Your friend says, If it rains after school,
  then I will give you a ride home.
• Your Dad says, If you get a B average, then
  you can get your drivers license.
• These are examples of if-then statements, which
  are also called conditionals.

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• In Geometry, a student might read,
• If B is between A and C, then
• AB BC AC
• (You know this as the Segment Addition Postulate)

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Vocabulary

• If- then statements conditional statements
  conditionals
• If it is sunny outside, then I will go out and
  play.
• Hypothesis it is sunny outside
• Conclusion I will go out and play
• Geometry If B is between A and C, then AB BC
  AC.
• Generic statement If p, then q.

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If-Then
To represent if-then statements symbolically, we
use the basic form below If p, then q.
p hypothesis q conclusion
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State whether each conditional is true or false

• If you live in San Francisco, then you live in
  California.
• True
• If points are coplanar, then they are collinear.
• False
• If AB BC, then B is the midpoint of AC.
• False

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Changing the conditional

• Converse of a conditional is formed by
  interchanging the hypothesis and the conclusion.
• If AB BC AC, then B is between A and C.
• If q, then p.
• Some true conditionals have false converses.
• Converse of first statement. (If it is sunny
  outside, then I will go out and play.)
• If ___________________, then __________.
• Is this true?

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Converse

• The converse of a conditional is formed by
  switching the hypothesis and the conclusion
• Statement If p, then q.
• Converse If q, then p.
• hypothesis conclusion

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• A statement and its converse say different
  things. A true statement may have a false
  converse. A false statement may have a true
  converse.
• Lets look at the converses of our previous
  examples.

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State whether each conditional is true or false

• Statement If you live in San Francisco, then you
  live in California.
• True
• Converse If you live in California, then you
  live in San Francisco.
• False
• Statement If points are coplanar, then they are
  collinear.
• False
• Converse If points are collinear, then they are
  coplanar.
• True
• If AB BC, then B is the midpoint of AC.
• False
• Converse If B is the midpoint of AC, then AB
  BC.
• True

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• A counterexample is an example where the
  hypothesis is true, but the conclusion is false.
• It takes only ONE counterexample to disprove a
  statement.
• Find counterexamples for the false conditionals
  that follow.

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Find a counterexample

• If you live in California, then you live in San
  Francisco.
• Danville, Alamo, Los Angeles, San Diego, etc.
• If points are coplanar, then they are collinear.
• Describe a plane with points on it that are not
  colinear
• If a line lies in a vertical plane, then the
  line is vertical.
• Describe a horizontal line crossing a vertical
  plane.
• If x 49, then x 7.
• How about -7 ?

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Other Forms of Conditional Statements

• General form Example
• If p, then q. If 6x 18, then x 3.
• p implies q. 6x 18 implies x 3.
• p only if q. 6x 18 only if x 3.
• q if p. x 3 if 6x 18.
• These all say the same thing.

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The Biconditional

• If a conditional and its converse are
• BOTH TRUE, then they can be combined
• into a single statement using the words
• if and only if. This is a biconditional.
• p if and only if q.
• Example Tomorrow is Saturday if and only if
  today is Friday.

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Counterexample

• An example that proves a statement to be false is
  called a counterexample.
• If I go out to play does that mean it is sunny?
• Ex. Identify the hypothesis, conclusion, and
  write the converse of the statement. Determine if
  it is true or false. If false give a
  counterexample. (5 parts)
• If I dive in the water, then I will get wet.
• If a c b c, then a b.

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Practice

• Tell whether the statement is true or false.
  Write the converse and determine if it is true or
  false. Can you provide a counterexample for each
  false statement?
• If today is Friday, then tomorrow is Saturday.
• If a number is divisible by 10, then it is
  divisible by 5.
• If x lt 0, then x² gt 0.

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