and q be - SCHOOLinSITES
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Answers to the HW
p. 75 #10-20 even, 25-28 all, 55 & 56
10. If an object weighs one pound, then it weights
16 ounces.
12. If a fish is a blue trunkfish, then it lives in the
waters of a coral reef.
14. True
16. True
18. If 1 is obtuse, then 1 measures 123.
20. If I go to the mall, then it is not raining.
25. One 26. two 27. line 28. a line
55. D
56. D
Answers to the HW
p. 82 #14 - 26 even, 54 & 55
14. True 16. true 18. true
20. (conditional) If two angles are congruent, then they
have the same measure; (converse) If two angles
have the same measure, then they are congruent.
22. (conditional) If two lines are perpendicular, then they
intersect to form right angles; (converse) If two lines
intersect to form right angles, then the two lines are
perpendicular.
24. A 100 angles is obtuse, but doesn’t measure 94.
26. Terry could live in Orlando, FL, not in Tampa, FL.
54. D
55. B
You may use your notes
1.
Identify the conclusion. If the weather is warm, then we
should go swimming.
2.
Write the contrapositive. If you like purple, then you will
like this shirt.
3.
Rewrite the biconditional statement as a conditional
statement and its converse. Two segments are
congruent if and only if they have the same measure.
4.
Write the inverse. If you like hockey, then you go to the
hockey game.
5.
Give a counterexample that demonstrates that the
converse of the statement is false. If a vehicle is a car,
then it has wheels.
2.3 Deductive Reasoning
Objective: Apply logic to true statements to
make valid conclusions
Let p be “Today is Tuesday”
and q be “There is school.
1. What is p q ?
If today is Tuesday, then there is school.
2. What is ~p ~q ?
If today is not Tuesday, then there is no school.
3. What is q p ?
If there is school, then today is Tuesday.
4. What is ~q ~p ?
If there is no school, then today is not Tuesday.
Deductive Reasoning
Deductive Reasoning uses facts,
definitions, and accepted properties
to write logical arguments (proofs).
How is this different from
Inductive Reasoning?
Law of Detachment
If a conditional statement is true
and the hypothesis is true, then the
conclusion is automatically true.
pq
p is true
q is true
EX 1: Use the Law of Detachment to determine a
conclusion.
If a triangle is equilateral, then the
measure of each angle is 60.
Triangle ABC is an equilateral triangle.
The measure of each angle is 60.
EX 2: Use the Law of Detachment to determine a
conclusion.
If two lines are parallel, then the lines
do not intersect.
k is parallel to m.
k and m do not intersect.
EX 3: Use the Law of Detachment to determine a
conclusion.
If Robby is taller than Ben, then
Robby is at least 6 feet tall.
Ben is older than Robby.
NO VALID CONCLUSION!
Law of Syllogism
Similar to the transitive property in algebra
Has 3 statements:
1st and 2nd are true, 2nd and 3rd are
true, so 1st and 3rd must also be
true.
p q is true
q r is true
p r is true
EX 4: Use the Law of Syllogism to determine a
conclusion.
If Donnie asks Pam, then she will say yes.
If she says yes, then they will get married.
If Donnie asks Pam, then they will
get married.
EX:5 Use the Law of Syllogism to determine a
conclusion.
If Tim gets stung by a bee, then he will
get very ill.
If he gets very ill, then he will go to the
hospital.
If Tim gets stung by a bee, then
he will go to the hospital.
EX 6: Use the Law of Syllogism to determine a
conclusion.
If Jay doesn’t work hard, then he will
not play.
If he doesn’t play, then he will quit the
team.
If Jay doesn’t work hard, then
he will quit the team.
EX 7: Use the Law of Syllogism to determine a
conclusion.
If Susan screams, then the dog will run
away.
If the dog licks Susan, then Susan will
scream.
NO VALID CONCLUSION!!
Use the true statements to determine
whether the conclusion is true or false.
•If it looks like rain, then I will bring my
umbrella to school with me.
•If there are clouds in the sky and the sky is
dark, then it looks like rain.
•If I bring my umbrella to school with me, then
I will hang it in the classroom closet.
•This morning, there are clouds in the sky and
the sky is dark.
Conclusion: My umbrella is hanging in the
classroom closet.
In order…
•This morning, there are clouds in the sky and
the sky is dark.
•If there are clouds in the sky and the sky is
dark, then it looks like rain.
•If it looks like rain, then I will bring my
umbrella to school with me.
•If I bring my umbrella to school with me, then I
will hang it in the classroom closet.
Conclusion: My umbrella is hanging in the
classroom closet.
2.4 Reasoning with Algebraic
Properties
Objective: Use Properties of Algebra to
explain multi-step equations
Algebraic Properties of Equality
p. 96 and 98
KNOW THEM!
Let a, b, and c be real numbers
Addition Property
If a = b, then a + c = b + c
add the same thing to both sides of an eqn. & they will still be equal
Subtraction Property If a = b, then a - c = b - c
subtract the same thing from both sides of an eqn. & they will still be
equal
Multiplication Property If a = b, then ac = bc
multiply the same thing to both sides of an eqn & they will still be equal
Division Property
If a = b and c 0, then
ac= bc
divide the same thing from both sides of an eqn & they will still be equal
Let a, b, and c be real numbers
Reflexive Property For any real number a, a = a
A number, length, or measure will always equal itself
Symmetric Property
If a = b, then b = a
If two things are equal then order does not matter
Transitive Property If a = b and b = c then a = c
This is true by the Law of Syllogism
If a = b, then a can be substituted
for b in any equation or expression
Think about substituting 3 in for x in an equation when x = 3
Substitution Property
Write a reason for each step.
10 2 x 3 x 2 4 Given
10 2x 3x 6 4 Distributive Property
10 2x 3x 2 Addition Property
10 5x 2 Subtraction Property
5x 12
12
x
5
Subtraction Property
Division Property
Use the property to complete the statement.
1. Addition property of equality: If AB = 5,
15
then 10 + AB = ____.
2. Multiplication property of equality: If
1/2 (mC) = 15 .
mC = 30, then ____
3. Reflexive property of equality: AF = ___.
AF
4. Symmetric property of equality: If
mMJC = mDCF
mDCF = mMJC, then ________________.
5. Transitive property of equality: If YZ = DB
YZ = JK
and DB = JK, then _________.
6. Substitution property of equality: If
5(3)
MN = 3, then 5(MN) = _____.
Homework:
p. 91-93 (13 problems) #8-32 even
p.99–101 (9 problems) #4-8, 10-14 even,
32 a-e