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Transcript
Biconditionals and Definitions
Chapter 2 Section 2
Biconditional
 When both the conditional and its converse are true, you can
combine them into a biconditional.
 Biconditional: the statement that you get by connecting the
conditional and its converse using the word and
 You can shorten a biconditional further by joining the two
parts of the conditional (the hypothesis and conclusion) with
the phrase if and only if
Examples
1.
2.
3.
Consider each statement and determine if it is true or false
Write the converse, determine if the converse is true or false
Combine the statements into a biconditional

Conditional: If two angles have the same measure, then the angles are
congruent.
True or false?


two angles are congruent
Converse: If __________________________________,
then
________________________________________.
they have the same measure





True
True or false?
True
Biconditional: If two angles have the same measure, then the angles are
congruent AND if two angles are congruent, then they have the same
measure.
Two angles have the same measure if and only if the angles are congruent.
Examples:
 If three points are collinear, then they lie on the same line.
 True or false?
 True
three points lie on the same line
 Converse: If ___________________________________,
they are collinear
then ________________________________________.
 True or false?

True
 Biconditional: If three points are collinear, then they lie on the
same line and if three points lie on the same line, then they are
collinear.
 Three points are collinear if and only if they lie on the same line.
Examples:
 If x=5, then x+15=20.
 True or false?
 True
x+15=20
 Converse: If ___________________________________,
x=5
then ________________________________________.
 True or false?

True
 Biconditional: If x=5, then x+15=20 and if x+15=20, then
x=5.
 x=5 if and only if x+15=20.
Writing a biconditional as two conditionals that are
inverses of each other
 Example:
 Biconditional: A number is divisible by 3 if and only if the sum
of its digits is divisible by 3.
a number is divisible by 3
 Conditional: If _______________________________,
the sum of its digits is divisible by 3
then _______________________________________.
the sum of a numbers digits is divisible by 3
 Converse: If _________________________________,
The number is divisible by 3
then _______________________________________.
Writing a biconditional as two conditionals that are
inverses of each other
 Example:
 Biconditional: A number is prime if and only if it has two
distinct factors, 1 and itself.
a number is prime
 Conditional: If _______________________________,
it has two distinct factors, 1 and itself
then _______________________________________.
a number has two distinct factors, 1 and itself
 Converse: If _________________________________,
it is prime
then _______________________________________.
Recognizing Good Definitions
 Geometry starts with undefined terms such as point, line,
and plane, whose meaning you understand intuitively. You
then use those terms to define other terms, such as collinear
points.
 A good definition has several important components:
uses clearly defined terms which are either common
knowledge or already defined.
2. is precise. Avoids words such as large, sort of, and almost.
3. is reversible. You can write a good definition as a true
biconditional.
1.
Examples:
Write the definitions as conditionals.
Show that they are reversible by writing the converse.
Determine that both are true.
Write as a biconditional.
1.
2.
3.
4.
Definition: Perpendicular lines are lines that meet to form right
angles.




lines are perpendicular
Conditional: If ____________________________________,
they meet to form right angles
then ____________________________________________.
lines meet to form right angles
Converse: If ____________________________________,
they are perpendicular
then ____________________________________________.
Lines are perpendicular
Biconditional: _________________if
and only
they meet to form right angles
if_______________________________________________
_____________________
Example…
Definition: A right angle is an angle whose measure is 90°.




an angle is a right angle
Conditional: If ____________________________________,
its measure is 90°
then ____________________________________________.
the measure of an angle is 90°
Converse: If ____________________________________,
it is a right angle
then ____________________________________________.
An angle is a right angle
Biconditional: _________________if
and only
its measure is 90°.
if_______________________________________________
_____________________
Good definitions???
 One way to show that a statement is not a good definition is
to find a counterexample.
 Examples:
 An airplane is a vehicle that flies.
 A triangle has sharp corners.
 A square is a figure with four right angles.
Practice!!
p. 90, Selected exercises #1-23
WS 2-2