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OTCQ
True or False
1. An Isosceles triangle has 2 congruent
sides and no congruent angles.
2. An Equiangular triangle has three angles
◦
that measure 70 .
3. A right triangle may be shaped with
◦
◦
◦
angles of 1 , 89 and 90 .
OTCQ 092409
True or False
1. An Isosceles triangle has 2 congruent sides and
no congruent angles. False.
An isosceles
has 2 or 3 congruent sides and
2 or 3 congruent angles.
2. An Equiangular triangle has three angles that
◦
◦
measure 70 . False. 3 angles of 70 would add
3.
◦
up to 210 . In every triangle, the 3 angles add up
◦
to exactly 180 . No more no less.
◦
A right triangle may be shaped with angles of 1 ,
◦
◦
◦
◦
◦
◦
89 and 90 . True 1 + 89 + 90 = 180
Aims 2-1 and 2-2
How do we define the rules of
logic? (part 2)
Homework : construct 2 valid argument/proofs of
your own to present to class.
NY GG 24 & GG 25
Objectives:
1. SWBAT think/reason with logic.
2. SWBAT define terms in logic and
apply logic rules in their arguments.
Summary of part 1.
Inductive Reasoning: from observed examples to
conjectures.
Counterexamples: one example that disproves an
inductive conjecture.
Deductive reasoning: from assumed truths to
conjectures.
Conditional statements:
Basic: If (premise), then (conclusion): p
Converse:
q
Inverse:
~p
Contrapositive:
~q
q
p
~q
~p
Consider Inductive Example:
Observation: The sun was out all week.
Conclusion: It is always sunny.
Can you disprove with a
counterexample?
Constructing Logical Arguments.
Law of Syllogism: A valid form of deductive
reasoning is the Law of Syllogism.
It allows you to draw conclusions from two
conditional statements when the conclusion of
one is the hypothesis of the other.
Law of Syllogism
If p  q and q  r are true statements, then
p  r is a true statement.
Law of Syllogism
If p  q and q  r are true statements, then
p  r is a true statement.
Example:
If I go to Miami, then I will be in Florida.
p
q
If I am in Florida, then I will be south of NY.
q
r
If I go to Miami, then I will be south of NY.
p
r
Consider Deductive Example:
Assumptions/Givens:
1. A right triangle may have 2 congruent
sides/angles (45-45-90 right triangle).
2. An isosceles triangle has 2 or 3
congruent sides/angles.
Can you restate these in the form of a
syllogism?
Steps to create a valid 2
hop proof argument
with the law of
Syllogism.
1. Order and translate all
given/assumptions into
p
q, q r and
p
r form.
2. Write proof as follows:
Line 1:
p
q
Line 2:
q
r
Conclusion: p
r
Step 1.
Try to order from general to
specific. Comes with practice.
Try to keep translations short:
1. If a right triangle has angles of
(45-45-90), then it will have 2
congruent sides.
2. If a triangle has 2 congruent
sides, then it is an isosceles
triangle.
Conclusion:
If a right triangle has angles of
(45-45-90), then it is an
isosceles triangle.
Law of Syllogism
If p  q and q  r are true statements, then
p  r is a true statement.
In theory you can have an infinite hop proof
as long as you do not skip a hop:
Line 1:
p
q
hop1
Line 2:
q
r
hop2
Line 3:
r
s
hop3
Line 4:
s
t
hop4
Conclusion:
p
t
One way to check your proof is to draw the
red lines. If you cannot not connect to the
bottom line before your conclusion, your
proof is incomplete
Line 1:
p
q
hop1
Line 2:
q
r
hop2
Line 3:
r
s
hop3
Line 4:
s
t
hop4
Conclusion:
p
t
Using the Law of Syllogism. If a valid
conclusion is possible, write it.
1. If two angles are complementary, then the sum
of their measures is 90.
If the sum of the measures of two angles is 90,
then both of the angles are acute.
2. If the heat wave continues, then air conditioning
will be used more frequently.
If air conditioning is used more frequently, then
energy costs will be higher.
Use the Law of Syllogism. If a valid conclusion
is possible, write it.
1. If two angles are complementary, then the sum
of their measures is 90.
If the sum of the measures of two angles is 90,
then both of the angles are acute.
If two angles are complimentary, then both of the
angles are acute.
2. If the heat wave continues, then air conditioning
will be used more frequently.
If air conditioning is used more frequently, then
energy costs will be higher.
If the heat wave continues, then energy costs …
Explain which is valid.
Homework : construct 2 valid argument/proofs of your
own to present to class.
Ex #1
(1) If it is Tuesday, then Marla tutors
chemistry.
(2) If Marla tutors chemistry, then she
arrives home at 4 P.M.
(3) If Marla arrives at home at 4 P.M., then
it is Tuesday.
Explain which is valid.
Homework : construct 2 valid argument/proofs of your
own to present to class.
ex #2
(1) If a marine animal is a starfish, then
it lives in the intertidal ocean zone.
(2) If an animal is in the intertidal zone,
then it lives in the least stable ocean
zone.
(3) If a marine animal is a starfish, then
it lives in the least stable ocean zone.