Download q - rreidymath

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 syllogism style
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
Syllogism tester
A valid syllogism follows this structure.
p
q
L1 __________________________.
q
r
L2 ____________________________.
p
r
Conclusion:._______________________
Is this a valid syllogism?
L1
If W is late, then G will eat the cake.
L2
If G eats the cake, then X will cry.
Conclusion: If w is late, then x will cry.
Is this a valid syllogism? yes
p
q
L1 If W is late, then G will eat the cake.
q
r
L2 If G eats the cake, then X will cry.
p
r
Conclusion: If w is late, then x will cry.
o
o
o
1. 45 – 45 - 90 right triangles have 2 congruent
sides/angles.
2. Triangles with 2 or 3 congruent sides/angles are
isosceles triangles.
Can you restate these statements in the form of “if
then” statements and put together a syllogism
argument?
Line 1:
p
Line 2:
q
Conclusion: p
q
r
r
1. If a right triangle has
angles of (45-45-90), then
it will be a triangle with 2
congruent sides/angles.
Syllogism Structure:
Line 1:
p
Line 2:
q
Conclusion: p
q
r
r
2. If a triangle has 2 or 3
congruent sides/angles,
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 infinitely long
syllogism as long as follow the structure:
Line 1:
p
q
Line 2:
q
r
Line 3:
r
s
Line 4:
s
t
Conclusion:
p
t
Using the Law of Syllogism.
If a valid conclusion is possible, write it.
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.
Use the Law of Syllogism. If a valid conclusion
is possible, write it.
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.
Using the Law of Syllogism.
If a valid conclusion is possible, write it.
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.
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 …
Homework : construct 2 valid argument/proofs of your
own to present to class.
Ex #1 : Is this valid?
(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.
Homework : construct 2 valid argument/proofs of your
own to present to class.
ex #2 Is this valid?
(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.