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Biconditionals &
Deductive Reasoning
Geometry
Biconditionals
When a conditional and its converse are true, you can combine them as
a true biconditional. You do this by connecting the two parts of the
conditional with the words “if and only if”
If two angles have the same measure, then the angles are congruent.
If two angles are congruent, then the angles have the same measure.
Since both of these are true, the biconditional statement would read:
Two angles are congruent if and only if the angles have the same measure.
Biconditionals
Write the converse of this statement. If both
statements are true, then rewrite it as a
biconditional.
In the United States, if it is July 4th,
then it is Independence Day.
Separating a Biconditional into Parts
• You can also rewrite a biconditional as a
conditional and its converse.
A number is divisible by 3 if and only if the sum of
its digits is divisible by 3.
• This would be separated into the two statements:
If a number is divisible by 3, then the sum of its
digits is divisible by 3.
If the sum of a number’s digits is divisible by 3, then
it is divisible by 3 .
Separating a Biconditional into Parts
• Write the 2 statements that form the
biconditional:
• A line bisects a segment if and only if the line
intersects the segment only at its midpoint.
Definitions
A good definition has several components:
 It uses clearly understood terms.
 It is precise and specific. (avoid opinion and use of words such as
large, sort of, and almost).
 It is reversible. This means that a good definition can be written as a
true biconditional.
Definition of Perpendicular Lines: Perpendicular Lines are two lines that intersect to
form right angles.
Conditional: If two lines are perpendicular, then they intersect to form right angles.
Converse: If two lines intersect to form right angles, then they are perpendicular.
Since the two are both true, the definition can be written as a
true biconditional:
Two lines are perpendicular if and only if they intersect to form
right angles.
Definitions
A dog is a good pet.
Parallel lines do not intersect.
An angle bisector is a ray that divides an angle
into two congruent angles.
Chapter 2.4 Deductive
Reasoning
Vocabulary
• Deductive Reasoning (logical reasoning) = the process of reasoning
logically from given statements to a conclusion. You conclude
from facts.
Example
• A physician diagnosing a patient’s illness uses deductive reasoning.
They gather the facts before concluding what illness it is.
Law of Detachment
• Law of Detachment = If a conditional is true and its hypothesis is
true, then its conclusion is true
• In symbolic form:
• If p q is a true statement and p is true, then q is true
Law of Detachment Example
• Use Law of Detachment to draw a conclusion
If a student gets an A on a final exam, then the student will pass the
course.
Felicia gets an A on the music theory final exam
Conclusion: She will pass Music Theory class
Law of Detachment: Try it
• Use Law of Detachment to draw a conclusion
If it’s snowing, then the temperature is below 0 degrees
It is snowing outside
Conclusion: It’s below 0 degrees
Law of Detachment Example: Careful!
• Use the Law of Detachment to draw a conclusion
If a road is icy, then driving conditions are hazardous
Driving conditions are hazardous
Conclusion: You can’t conclude anything!!!
Quick Summary
• You can only conclude if you are given the hypothesis!!!
• If you are given the conclusion you CAN NOT prove the hypothesis
Law of Detachment
• What conclusion can you draw from the following two statements?
• If a person doesn’t get enough sleep, they will be tired.
• Evan doesn’t get enough sleep.
A.Evan will get more sleep.
B.Evan will be tired
C.Evan will not be tired.
D.You cannot make a conclusion.
Law of Syllogism
If pq and qr are true statements, then pr is a true statement
Example:
If the electric power is cut, then the refrigerator does not work.
If the refrigerator does not work, then the food is spoiled.
So if the electric power is cut, then the food is spoiled.
Law of Syllogism example:
• Use law of syllogism to draw a conclusion
• If you are studying botany, then you are studying biology.
• If you are studying biology, then you are studying a science
• Conclusion: If you are studying botany, then you are studying a
science
Law of Syllogism: Try it
Use the law of Syllogism and draw a conclusion
If you play video games, then you don’t do your
homework
If you don’t do your homework, then your grade will
drop
Conclusion: If you play video games, then your grade will
drop
Law of Syllogism
• What conclusion can you draw from the following two statements?
• If you have a job, then you have an income.
• If you have an income, then you must pay taxes.
A.If you have a job then you must pay taxes
B.If you don’t have a job, then you don’t pay taxes.
C.If you pay taxes, then you have a job.
D.If you have a job, then you don’t pay taxes.