2'2 Biconditional Statements

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2.2 Biconditional Statements
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Biconditional Statementa statement that
contains the phrase if and only if. Equivalent
to a conditional statement and its converse.
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Examples

• It is Saturday if and only if I am working at the
  restaurant
• An angle is acute if and only if it has a measure
  between 0 and 90.

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• Conditional Statement If it is Saturday, then I
  am working at the restaurant.
• Converse If I am working at the restaurant,
  then it is Saturday.

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In order for a biconditional statement to be
TRUE, both the conditional statement and its
converse must be true.
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Example 1.1Two lines intersect if and only if
their intersection is exactly one point.
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Conditional Statement

• If two lines intersect, then their intersection
  is exactly one point.

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ConverseIf their intersection is exactly one
point, then two lines intersect.
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Example 1.2Three lines are coplanar if and only
if they lie in the same plane.
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Conditional Statement If three lines are
coplanar, then they lie in the same plane.
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ConverseIf three lines lie in the same plane,
then they are coplanar.
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Example 2Give a counterexample which
demonstrates that the converse is false.
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1. If x2lt49, then xlt7

• Converse
• If xlt7, then x2lt49.
• Counterexample
• let x-8 then (-8) 264gt49

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2. If an angle measures 34, then the angle is
acute.Converse If an angle is acute, then it
measures 34.Counterexample mltA40, angle is
acute
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3. If Paul lives in Langhorne, then he lives in
PA.Converse If Paul lives in PA, then he lives
in Langhorne.CounterexamplePaul could live in
Newtown
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4. If two angles measure 33 and 57, then they
are complementary.Converse If two angles are
complementary, then they measure 33 and
57.Counterexampleangles 40 and 50
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1. If SR is perpendicular to QR, then their
intersection forms a right angle.Write Converse
and Biconditional Statement.
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Converse If SR and QRs intersection forms a
right angle, then they are perpendicular.
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Biconditional StatementSR is perpendicular to
QR if and only if their intersection forms a
right angle.
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2. If an animal is a leopard, then it has spots.
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Converse If an animal has spots, then it is a
leopard.Biconditional Statement none, converse
is false. Animal could be a Dalmatian.
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3. If PQ and PS both measure 8 cm, then PQ is
equal to PS.
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Converse If PQ is equal to PS, then PQ and PS
both measure 8 cm.Biconditional Statement
none, converse is false. Segments could measure
4 cm.