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Conditional Statements, Biconditionals, and Deductive Reasoning
Note Taking Guide
Part 2: Conditional Statements
Conditional:
Hypothesis:
Conclusion:
What is the hypothesis and conclusion of the conditional?
If an animal is a robin, then the animal is a bird
H:
C:
If an angle measures 130, then the angle is obtuse
H:
C:
Write the following statement as a conditional
Vertical angles share a vertex
Step 1: Identify the Hypothesis and Conclusion
H:
C:
Step 2: Write the Conditional
Try Your Own: How do you write “Dolphins are mammals” as a
conditional?
Truth Value:
•The truth value of a conditional is either true or false.
•To show a conditional is true, show that every time the hypothesis is
true, the conclusion is also true
•To show a conditional is false find one counter example for which the
hypothesis is true and the conclusion is false
Is the conditional true or false? If it is false find a counter example.
1.) If a Women is Hungarian, then she is European
2.) If a number is divisible by 3, then it is odd
3.) If a month has 28 days, then it is February
4.) If two angles form a linear pair, then they are supplementary
***SEE CHARTS***
Negation
•The negation of a statement p is the opposite of that statement, the
symbol is ~p and is read “not p”
–Example:
•The negations of the statement “the sky is blue” is “the sky
is not blue”
•You use the negation to write statements related to a condition
Write the Converse, Inverse, and Contrapositive statements for this
conditional.
“If a vegetable is a carrot, then it contains beta carotene”
Part 3: Biconditionals
What is a Biconditional?
Writing a Biconditional
To write a biconditional first determine what is the converse of the
following true conditional. If the converse is true then write a
biconditional statement
–Conditional: If the sum of the measure of two angles is 180, then the
two angles are supplementary
–Converse:
- Biconditional:
You Try:
What is the converse of the following conditional; if the converse is true
write a biconditional statement
Conditional: If two angles have equal measures, then the angles are
congruent
Converse:
Biconditional:
What are the two statements that form a biconditional
–A ray is an angle bisector if and only if it divides and angle into two
congruent angles
Step 1: Find p and q
•P –
•Q –
Step 2: Write the conditional and Converse
–Conditional:
–Converse:
What are the two conditionals that form this biconditional?
–Two numbers are reciprocals if and only if their product is one.
•Conditional:
•Converse:
Part 4: Deductive Reasoning
Deductive Reasoning:
Law Of Detachment:
Using the Law of Detachment
What can you conclude from the given true statements?
–If a student gets an A on a final exam, then the student will pass the
course. Felicia got an A on her history Final
–If a ray divided an angle into two congruent angles, then the ray is an
angle bisector. Ray RS divides <ARB so that <ARS ≅ <SRB
–If two angles are adjacent, then they share a common vertex. <1 and <2
share a common vertex.
–If there is lightning, then it is not safe to be out in the open. Marla sees
lightning from the soccer field.
–If a figure is a square, then its sides have equal lengths, figure ABCD
has sides of equal length.
Law of Syllogism
Using the Law of Syllogism
What can you conclude from the given true statements?
•If a figure is a square, then the figure is a rectangle. If a figure is a
rectangle than the figure has four sides
•If you do gymnastics, then you are flexible. If you do ballet then you are
flexible.
•If a whole number ends in 0, then it is divisible by 10. If a whole
number is divisible by 10, than it is divisible by 5.
•If Ray AB and Ray AD are opposite rays, then the two rays form a
straight angle. If two rays are opposite rays, then the two rays form a
straight angle.