Biconditionals and Definitions

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Biconditionals and Definitions
GEOMETRY LESSON 2-2
Identify the hypothesis and the conclusion of
each conditional statement. 1. If x gt 10, then x
gt 5. 2. If you live in Milwaukee, then you live
in Wisconsin. Write each statement as a
conditional. 3. Squares have four sides. 4. All
butterflies have wings. Write the converse of
each statement. 5. If the sun shines, then we go
on a picnic. 6. If two lines are skew, then they
do not intersect. 7. If x 3, then x3 27.
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Biconditionals and Definitions
GEOMETRY LESSON 2-2
1. The hypothesis follows if and the conclusion
follows then so the hypothesis is x gt 10 and
the conclusion is x gt 5. 2. The hypothesis
follows if and the conclusion follows then so
the hypothesis is you live in Milwaukee and
the conclusion is you live in Wisconsin. 3. Rewr
ite the statement as an if-then statement If a
figure is a square, then it has four
sides. 4. Rewrite the statement as an if-then
statement If something is a butterfly, then it
has wings. 5. Switch the hypothesis and
conclusion If we go on a picnic, then the sun
shines. 6. Switch the hypothesis and conclusion
If two lines do not intersect, then they are
skew. 7. Switch the hypothesis and conclusion If
x3 27, then x 3.
Solutions
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Ch 2 Reasoning and Proof

• A proof is a convincing logical argument that
  uses deductive reasoning.

• Types of statements used in a logical argument
• Conditional statements
• Bi conditional statements
• Definitions
• Properties
• Postulates
• Theorems

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2.1 Biconditionals and Definitions

• A biconditional
• Occurs when a conditional AND its converse are
  both true.
• It is identified by the if and only if
  statement.
• It is Tuesday if and only if yesterday was
  Monday.
• It is a triangle if and only if it has three
  sides.

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Consider this true conditional statement. Write
its converse. If the converse is also true,
combine the statements as a biconditional.
Conditional If x 5, then x 15 20. To write
the converse, exchange the hypothesis and
conclusion. Converse If x 15 20, then x
5. When you subtract 15 from each side to solve
the equation, you get x 5. Because both the
conditional and its converse are true, you can
combine them in a true biconditional using the
phrase if and only if. Biconditional original
hypothesis if and only if conclusion
Answer x 5 if and only if x 15
20.
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Write the two statements that form this
biconditional.
Biconditional Lines are skew if and only if they
are noncoplanar.
conditional hypothesis if and only if conclusion.
A biconditional is written as two conditionals
that are converses of each other. Conditional If
lines are skew, then they are noncoplanar. Convers
e If lines are noncoplanar, then they are skew.
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• A definition accurately describes the character
  of something.
• A good definition should follow a few simple
  rules
• Uses clear terms that are commonly understood or
  are already defined.
• Is precise. It avoids words such as big, small,
  few, many, etc.
• Is reversible. It is a biconditional statement.

Definition A quadrilateral is a figure with four
enclosed sides. Perpendicular lines intersect to
form a 90º angle.
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Show that this definition of triangle is
reversible. Then write it as a true biconditional.
Definition A triangle is a polygon with exactly
three sides.
The original conditional is true.Conditional
concerning polygons If it is a triangle, then it
has exactly three sides. The converse is also
true.Converse concerning polygons If it has
exactly three sides, then it is a
triangle. Because both statements are true, they
can be combined to form a biconditional.
A polygon is a triangle
if and only if it has exactly three sides.
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Is the following statement a good definition?
Explain.
An apple is a fruit that contains seeds.
The statement is true as a description of an
apple. Now exchange An apple and a fruit that
contains seeds, and the reverse reads A fruit
that contains seeds is an apple. There are many
other fruits containing seeds that are not
apples, such as lemons and peaches. These are
counterexamples, so the reverse of the statement
is false. The original statement is not a good
definition because the statement is not
reversible.
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What is a good definition of apple?

• Good Definitions
• Uses clear terms that are commonly understood or
  are already defined.
• Is precise. It avoids words such as big, small,
  few, many, etc.
• Is reversible. It is a biconditional statement.

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Re Cap

• A biconditional
• Occurs when a conditional AND its converse are
  both true.
• It is identified by the if and only if
  statement
• Good Definitions
• Uses clear terms that are commonly understood or
  are already defined.
• Is precise. It avoids words such as big, small,
  few, many, etc.
• Is reversible. It is a biconditional statement.

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Additional Practice
GEOMETRY LESSON 2-2
1. Write the converse of the statement. If it
rains, then the car gets wet. 2. Write the
statement above and its converse as a
biconditional. 3. Write the two conditional
statements that make up the biconditional.
Lines are skew if and only if they are
noncoplanar. Is each statement a good
definition? If not, find a counterexample. 4. Th
e midpoint of a line segment is the point that
divides the segment into two congruent segments.
5. A line segment is a part of a line.
If the car gets wet, then it rains.
It rains if and only if the car gets wet.
If lines are skew, then they are noncoplanar if
lines are noncoplanar, then they are skew.
yes
No the statement is not reversible a ray.
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