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Lesson Goals
Rewrite a definition as a biconditional statement.
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Biconditional Statement
The combination of a conditional statement
and its converse.
The phrase “if and only if” is used to
indicate this combination.
A biconditional is only true when both the
conditional and converse are true :
All geometric definitions are biconditional statements. .
Complementary Angles:
Two angles with measures that have a sum of 90o.
Conditional: If two angles are complementary,
then their measures have a sum of 90o.
Converse:
If the measures of two angles have a sum
of 90o, then the angles are complementary.
Biconditional: Two angles are complementary if and
only if their measures have a sum of 90o
Perpendicular lines: Two lines that intersect to
form a right angle.
m
m
Conditional:
Converse:
If two lines are perpendicular,
then they intersect to form a right angle.
If two lines intersect to form a right angle,
then they are perpendicular.
Biconditional: Two lines are perpendicular if and only if
they intersect to form a right angle.
Rewrite the definition as a conditional and its converse
A ray bisects an angle if and only if it divides the
angle into two congruent angles.
Conditional: If a ray bisects an angle, then it divides the
Angle into two congruent angles.
Converse: If ray divides the angle into two congruent
angles, then the ray bisects the angle.
Rewrite the postulate as a conditional and its converse.
Two lines intersect if and only if their
intersection is exactly one point.
Conditional:
If two lines intersect, then they
have exactly one point in common.
Converse:
If two lines have exactly one point in
common, then the lines intersect.
Decide whether each statement about the diagram is
true and explain using definitions.
U
Points R, S, and T are collinear.
R
S
Definition collinear: points that lie on the same line
R, S, and T are on the same line.
Therefore, it is true that R, S, and T are collinear.
T
Decide whether each statement about the diagram is
true and explain using definitions.
SU ^ RT
U
R
S
Definition perpendicular:
Two lines that intersect to form a right angle.
RSU is not a right angle
Therefore,itis false that SU ^ RT
T
Give a counterexample that demonstrates that the
converse of the statement is false.
If an angle measures 94 , then it is obtuse.
Converse: If an angle is obtuse, then it measures 94 .
Counterexample:
Let the angle measure 91 .
Determine whether the statement can be combined with
its converse to form a true biconditional.
If x 2  49, then x  7
True
Converse: If x  7, then x 2  49
False
Let x  8  8  7  , but  8   64
2
It cannot be a true biconditional since the converse is
false.
Determine whether the statement can be combined with
its converse to form a true biconditional.
If x 2  4, then x  2 or x  2
Converse:
True
If x  2 or x  2, then x  4
2
x  4 if and only if x  2 or x  2.
2
True
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