Statements

Report
Chapter 2
Student Notes
Friday, 2/3/12
Dress for Success for Extra Credit
2.1
Inductive Reasoning
and Conjecture
Conjecture Make a conjecture from the given statement.
Given: The toast is burnt.
Conjecture: ___________________________
Given: It is winter.
Conjecture: ___________________________
Given: Angle A is a right angle.
Conjecture: ___________________________
Counterexample Write a counterexample for each conjecture.
Conjecture: The sky is blue.
Counterexample: ________________________________
Conjecture: Angle 1 and Angle 2 are congruent.
Counterexample: ________________________________
Conjecture: l and m are parallel
Counterexample: ________________________________
Determine if each conjecture is true or false. Give a
counterexample for any false conjecture.
1. Given: A, B, C are collinear.
Conjecture: A, B, C are on the same line.
2. Given: 1 is a right angle.
Conjecture: m1 = 90.
3. Given: AB = BC.
Conjecture: B is the midpoint of AC.
T/F
T/F
T/F
Determine if each conjecture is true or false. Give a
counterexample for any false conjecture.
1. Given: The dog is brown.
Conjecture: It is a chocolate lab.
T/F
2. Given: 3 and 4 form a linear pair.
Conjecture: 3  4
T/F
3. Given: 1 and 2 are complementary
Conjecture: m1 = 45, m2 = 45.
m1 = 48, m2 = 42
T/F
2.2
Logic
Statement  Truth Value –
 Negation  Compound Statement –
Conjunction  Symbol for And:
Disjunction  Symbol for Or:
Circle the statement that is true.
p: Angle A is a right angle.
r: Angle A is an obtuse angle.
r: Angle A is an acute angle.
p
q
r
2.
p
q
r
>
>
1.
>
>
Truth Table  Examples of Truth Tables.
p
~p
T
F
F
T
Disjunction
p
q
T
T
T
F
F
T
F
F
p
q
p
q
T
T
T
F
F
T
F
F
p
q
>
Negation
Conjunction
2.3
Conditional Statements
Conditional Statement Statement: A right angle has a measure of 90 degrees.
If-then:
Statement: A car has four wheels.
If-then:
Statement: A triangle has 3 sides.
If-then:
Parts of a Conditional Statement
 Hypothesis
 Conclusion
If it is a car, then it has four wheels.
Converse Conditional: If it is a car then it has 4 wheels.
Converse:
Conditional: If it is a pig, then it can fly.
Converse:
Conditional: If it is a right angle, then it measure 90.
Converse:
Inverse Conditional: If it is a car then it has 4 wheels.
Inverse:
Conditional: If it is a pig, then it can fly.
Inverse:
Conditional: If it is a right angle, then it measure 90.
Inverse:
Contrapositive Conditional: If it is a car, then it has 4 wheels.
Contrapositive:
Conditional: If it is a pig, then it can fly.
Contrapositive:
Conditional: If it is a right angle, then it measure 90.
Contrapositive:
Identify the converse, inverse and contrapositive of each
conditional statement. Determine if each statement is true
or false.
T / F If you go to WMHS, then you are a hornet.
T / F Converse: __________________________
__________________________
T / F Inverse: __________________________
__________________________
T / F Contrapositive: ______________________
______________________
Identify the converse, inverse and contrapositive of each
conditional statement. Determine if each statement is true
or false.
T / F If it is a right angle, then it measures 90.
T / F Converse: __________________________
__________________________
T / F Inverse: __________________________
__________________________
T / F Contrapositive: ______________________
______________________
2.4
Deductive Reasoning
Deductive Reasoning -
Law of Detachment
1.
2.
3.
Law of Syllogism
1.
2.
3.
Examples of the Laws of Detachment and Syllogism.
 Detachment
1. If it is a triangle,
then it has 3 sides.
 Syllogism
1. If it is a Jeep,
then it has 4 wheel drive.
Determine whether the 3rd statement is valid based
on the given information. If not, write invalid.
1. If it is a dog, then it has 4 legs.
2. Rover is a dog.
3. Rover has 4 legs.
Is it valid?
Does it follow one of our
Laws?
Determine whether the 3rd statement is valid based
on the given information. If not, write invalid.
1. If you are 18 or older, then you are an adult.
2. If you are an adult, then you can vote.
3. If you are 18 or older, then you can vote.
Is it valid?
Does it follow one of our Laws?
Use the Law of Detachment or the Law of Syllogism to determine if
a valid conclusion can be reached. If it can, state it and the law
used. If not, write no conclusion.
1. If it is a car, then it has 4 wheels.
2. A Ferrari is a car.
3.
__________________________
1. If you go to the store, then you will go to the post office.
2. If you go to the post office, then you will buy stamps.
3.
_________________________________________
Use the Law of Detachment or the Law of Syllogism to determine if
a valid conclusion can be reached. If it can, state it and the law
used. If not, write no conclusion.
1. If you are in college, then you are at least 18.
2. Pete is in college.
3.
______________________________
1. Right angles are congruent.
2. Angle 1 and Angle 2 are congruent.
3. _______________________________
2.5
Postulates
Postulate – Statement that is
accepted without proof.
Postulate 2.1 -
A
B
Postulate 2.2 -
A
P
B
C
Plane P
Plane ABC
Postulate 2.3
_________________________
_____________________________________
Postulate 2.4
_________________________
_____________________________________
_____________________________________
Postulate 2.5
________________________
____________________________________
____________________________________
____________________________________
Postulate 2.6
_________________________
_____________________________________
Postulate 2.7
_________________________
_____________________________________
P
R
Midpoint Theorem -
A
M
B
Determine if each statement is always, sometimes
or never true.
1.
A, B, and C are collinear.
2.
A, B, and C, are coplanar.
3.
RST is a right angle.
4.
Two planes intersect to form a line.
5.
If AB = BC, the B is the midpoint of AC.
6.
Vertical angles are adjacent.
7.
If B is the midpoint of AC, then AB = BC.
Determine the number of segments that can be
drawn connecting each pair of points.
1.
2.
2.6
Algebraic Proof
Properties
 Reflexive:
 Symmetric
 Transitive
 Substitution
Properties
 Distribution
 Addition /
Subtraction
 Multiplication /
Division
Identify each property that justifies each statement.
1.
If 7 = x, then x = 7.
2.
If x + 5 = 7, then x = 2
3.
If x = 7 and 7 = y, then x = y.
4.
If m1 + m2 = 180
and m2 = m3,
then m1 + m3 = 180.
Identify each property that justifies each statement.
1.
2x + 1 = 2x + 1
2.
If x – 6 = 7, then x = 13
3.
If 2(x + 3) = 7, then 2x + 6 = 7.
4.
If 2x = 16, then x = 8.
Given: 2x – 5 = 13
Prove: x = 9
Statements
Reasons
1. _____________
1.
___________
2. _____________
2.
___________
3. _____________
3.
___________
Given: 8 – n = 4 – 2 n
3
Prove: n = 12
Statements
Reasons
1. ___________________
1. ______________
2. ___________________
2. ______________
3. ___________________
3. ______________
4. ___________________
4. ______________
5. ___________________
5. ______________
Given: 2x + 1 = 7
3
Statements
Prove: x = 10
Reasons
1.
2x + 1 = 7
3
1. Given
2.
_________________
2. _________________
3.
_________________
3. _________________
4.
_________________
4. _________________
2.7
Proving Segment Relationships
Ruler Postulate
 The points on any line or line segment can be
_______________________________________________________
______________________________________
Betweenness of Points
 A point can only be between two _________
_________________________________
C
A
B
Segment Addition Postulate
If B is between A and C,
C
B
A
Segment Addition Postulate Statements
 Make a statement using the previous postulate about
each figure.
Y
M
X
J
R
K
L
XY =
JK =
T
S
RS =
Theorem 2.2
Reflexive
Symmetric
Transitive
Given: AB  XY, AC  XZ
Prove: BC  YZ
A
B
Z
Y
Reasons
Statements
1.
AB  XY, AC  XZ
2. ______________
3. ______________
1. ______________
2. ______________
3. ______________
4.
5.
6.
7.
4.
5.
6.
7.
______________
______________
______________
______________
C
______________
______________
______________
______________
X
O
Given: MO  PO, MN  PR
Prove: NO  RO
R
N
Statements
M
Reasons
1.
MO  PO, MN  PR
2. ________________
3. ________________
1. Given
2. ______________
3. ______________
4.
5.
6.
7.
4.
5.
6.
7.
________________
________________
________________
________________
______________
_____________
______________
______________
P
2.8
Proving Angle Relationships
Angle Addition Postulate
If R is in the interior of PQS, then
P
OR
Q
1
R
2
mPQS =
S
Make an angle addition postulate statement
about each figure.
P
mMNO =
O
M
N
K
J
M
L
mJKL =
Supplement Theorem
m1 + m2 = ____o
Complement Theorem
3
4
1
2
If the noncommon sides of two
adjacent angles form a right
angle, then the angles are
complementary.
m3 + m4 = ____o
Theorem 2.5
 Reflexive
 Symmetric
 Transitive
Theorem 2.6
2
3
1
If m1 + m2 = 180o,
Theorem 2.7
3
2
1
If m1 + m2 = 90o,
Vertical Angles Theorem
1
4
2
3
 Abbreviation:
Theorem 2.9
Theorem 2.10
Theorem 2.13
Given: MNO  RST, MNQ  RSP
Prove: QNO  PST
Q
O
M
Statements
N
S
R
T
P
Reasons
1. MNQ  RSP, MNQ  RSP
1. Given
2. _________________________________
2. _______________
3. _________________________________
3. _______________
4. _________________________________
4. _______________
5. _________________________________
5. _______________
6. _________________________________
6. _______________
7. _________________________________
7. _______________
Find the measure of each numbered angle.
2. 3 and 4 are
complementary,
m4 = 48.
1. m1 = 72
2
1
3
5
4
m2 =
m3 =
m5 =
Find the measure of each numbered angle.
3. m6 = x – 5,
m7 = 2x – 4
6
7
m6 =
m7 =
Find the measure of each numbered angle.
4. m8 = 52.
9
8
m9 =
Find the measure of each numbered angle.
5. 10 and 11 are
complementary.
13  11, m12= 38.
m13 =
12
13
m11 =
10
11
m10 =

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