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Day 1
Monday, January 25
1. Opener
a) Give three relationships between <1 and <2.
1
2
b) What is the next term in the sequence: 1, 1, 2, 3, 5, 8…
Name each figure by its number of sides and classify it as convex or
concave:
c)
d)
f) How many black triangles will there be in the 6th shape of this
sequence?
g) What is the highest-paid mascot in professional sports? How much is
he/she/it paid?
2. Notes - Two Types of Reasoning
Deductive
Inductive
Start with lots of rules.
Establish a rule.
Make another rule.
Start with lots of observations.
6. Notes - Two Types of Reasoning
Deductive
Gravity makes things fall downwards.
Things that fall from a great height get hurt.
If I jump off this building, I
will fall downwards.
Inductive
Things I throw off the roof fall down.
I threw a ball off the roof and it fell down.
I threw a rock off the roof and it fell down.
I threw a cat off the roof and it fell down.
6. Notes - Two Types of Reasoning
Deductive
Inductive
All smart people are rich.
Rich people have lots of friends.
All smart people are rich.
All smart people have lots of
friends.
Mrs. Caldwell is smart and rich.
Bill Gates is smart and rich.
Conjecture
Make a Guess (But defend it)
Draw a picture and make a conjecture
about the following information:
<ABC is a right angle.
Conjecture
Make a Guess (But defend it)
Draw a picture and make a conjecture
about the following information:
AB = 7, BC = 3 and AC=10.
Counterexample
Statement
Math Teachers are Dorks
Counterexample
Mrs. Limburg
Movie Stars are Beautiful
Science Fiction is for Dorks
The opposite of a number is
always negative.
The opposite of -4
is 4.
Make a conjecture about the next number
based on the pattern.
2, 4, 12, 48, 240
Find a pattern:
2
4
12
48
240
×
×
×
×
2
3
4
5
The numbers are multiplied by 2, 3, 4, and 5.
Conjecture: The next number will be multiplied
by 6. So, it will be
or 1440.
Answer: 1440
Make a conjecture about the next number
based on the pattern.
Answer: The next number will
be
For points L, M, and N,
and
,
make a conjecture and draw a figure to illustrate
your conjecture.
Given: points L, M, and N;
Examine the measures of the segments.
Since
the points can be collinear with point N
between points L and M.
Answer:
Conjecture: L, M, and N are collinear.
ACE is a right triangle with
Make a
conjecture and draw a figure to illustrate
your conjecture.
Answer:
Conjecture: In ACE, C is a right angle and
is the hypotenuse.
Classwork:
Page 64 #11 – 24, 29 - 32
Tuesday, May 2,
2017
1. Opener
a) Every cheerleader at Washington High School is a
junior. Mark is a senior. Is Mark a cheerleader?
b) Edith, Ernie, and Eva have careers as an economist,
electrician, and engineer, but not necessarily in that order.
The economist does consulting work for Eva’s business.
Ernie hired the electrician to rewire his new kitchen. Edith
earns less than the engineer but more than Ernie. Match
names to occupations.
c) What percent of a panda’s diet is bamboo?
2-2 Logic
Statement
-vs-
Opinion
Apples are good.
California is a state.
Bill Clinton was a great president.
Thursday is the day after Sunday.
Walker Valley is the best school.
Determine the truth value of the statements.
Negation
(Don’t be so negative!)
Statement
Wednesday is chicken casserole day.
Spock is a Vulcan.
Negation
Wednesday is not chicken casserole day.
Spock is not a Vulcan.
Edward is a vampire.
Edward is not a vampire.
p
~p
It wouldn’t be inaccurate to
assume that I couldn’t exactly not
say that it is or isn’t almost
partially incorrect.
Truth Tables
p
~p
T
F
F
T
Compound Statements
Conjunctions (and)
p: Tennessee is a state.
q: Mrs. Limburg lives in Tennessee.
Write the statement
pq
Tennessee is a state and Mrs. Limburg lives
in Tennessee.
Truth Table
Conjunction
p
q
T
T
F
F
T
F
T
F
p^q
Compound Statements
Disjunction (or)
p: Tennessee is a state.
q: Mrs. Limburg lives in Tennessee.
Write the statement
pq
Tennessee is a state or Mrs. Limburg lives
in Tennessee.
Truth Table
Disjunctions
p
q
T
T
F
F
T
F
T
F
pq
Write statements:
p:
q:
r:
s:
Freshmen are young.
Robert is a freshman.
Mr. Coggin is our principal.
Football players are tough.
p ~ q
~rs
p  q ~ s
Create Truth Tables
p ^ ~q
p
q
T
T
F
F
T
F
T
F
~q
p ^ ~q
Tuesday, May 2, 2017
p: Homework is fun.
q: Math is my favorite.
r: The Jonas Brothers rock!
1.
p ~ r
2.
r p
3.
~ rq p
Construct a truth table for
~ pq
2-3 Conditional
Statements
If….., then…..
Hypothesis…, Conclusion…
Hypothesis
Conclusion
If it is Friday, there is a football game.
It is Friday
There is a football game.
We will have ice cream if you are good.
You are good
We will have ice cream.
Punk rock makes me sick.
It is punk rock
It makes me sick.
Conditional
Statements in Disguise
Mrs. Limburg is tall.
If she is Mrs. Limburg, then she is tall.
Perpendicular lines intersect.
If two lines intersect, they are perpendicular.
Logic
p: The person is a senior.
q: The person is mature.
If the person is a senior then the person is mature.
If p, then q
p
q
2. Logician’s Shorthand
Let
Let
Let
Let
P
P: The month is April.
Q: The sun is shining.
R: The moon is full.
S: The night is young.
Q
If the month is April then the sun is shining.
~P
The month is not April.
Therefore the night is young.
S
~R
S
If the moon isn’t full then the night is young.
Logic
Let
Let
Let
Let
p: I get a job.
q: I will earn money.
r: I will go to the movies.
s: I will spend my money.
Translate to English
1. If p then q.
If I get a job then I will earn money.
2. If q then r.
If I will earn money then I will go to the movies.
3. If p then r.
If I get a job then I will go to the movies.
4. If r then s.
If I will go to the movies then I will spend my money.
2. Logic
Let
Let
Let
Let
p: Today is Wednesday.
q: Tomorrow is Thursday.
r: Friday is coming.
s: Yesterday was Tuesday.
Translate to logic statements.
5. If today is Wednesday, then tomorrow is Thursday.
6. If tomorrow is Thursday, then Friday is coming.
7. If yesterday was Tuesday, then tomorrow is Thursday.
8. If yesterday was Tuesday, then Friday is coming.
p  q
q  r
s  q
s  r
Converse
Converse - A statement made by switching the hypothesis and
conclusion of a conditional statement.
“If a movie stars Bill Paxon, then Mrs. Caldwell hates it.”
hypothesis
conclusion
“If Mrs. Caldwell hates a movie, then it stars Bill Paxton.”
IS THIS CONVERSE TRUE?
False
Discuss With Someone Nearby
1. Is the converse of the Linear Pair Conjecture true?
“If two angles add up to 180°, then they are a linear pair.”
118°
62°
FALSE
2. Is the converse of the Vertical Angles Conjecture true?
“If two angles are congruent, then they are vertical angles.”
FALSE
300°
300°
Inverse
Inverse - A statement made by negating the
hypothesis and conclusion of a
conditional statement.
“If a movie stars Bill Paxton, then Mrs. Caldwell
hates it.”
pq
If a movie does not star Bill Paxton, then Mrs.
Caldwell does not hate it.
~ pFalse
~ q
Contrapositive
Contrapositive - A statement made by negating and
switching the hypothesis and
conclusion of a conditional statement.
“If a movie stars Bill Paxton, then Mrs. Caldwell hates it.”
If Mrs. Caldwell does not hate a movie, then it does
not star Bill Paxton.
True
If a conditional is true, then the contrapositive is
always true.
Your Turn
If my dog dies, then I am sad.
Write the converse,
inverse, and contrapositive
and label each.
Tuesday, May 2, 2017
If it is Monday, I am tired.
Write the converse, inverse, and contrapositive of the
above conditional statement.
Construct a truth table for: ~p ^ q
Into how many standard time zones is the world
divided?
Truth Tables
If you scrape the gum off of my desks, I will
give you $10.
Case 1: You scrape the gum. I give you $10.
Case 2: You scrape the gum. I do not give you $10.
Case 3: You do not scrap the gum. I give you $10.
Case 4: You do not scrape the gum. I do not give you $10.
p
q
T
T
F
F
T
F
T
F
pq
T
F
T
T
Alice in Wonderland
Classwork
Tuesday, May 2, 2017
Write the statement in if then form and then
write the converse, inverse, and
contrapositive.
Fairbanks is in Alaska.
Determine whether each related conditional is
true or false. If it is false, find a
counterexample.
What is the GWR for Most T-Shirts worn at
the same time? How many? How long did
it take to put them on?
Deductive Reasoning
Uses facts, rules, definitions, or properties to reach
logical conclusions.
Example: Doctors use this method to determine how
much medicine to take
Law of Detachment
If p
q is true and p is true, then q is
also true.
p
p
q
q
Conditional: If you are a freshman,
you are young.
Statement: Mark is a freshman.
Valid Conclusion: Mark is young.
Conditional: If you are a
freshman, then you are young.
Statement: Mark is young.
Invalid Conclusion: Mark is a
freshman.
Conditional:
All boys like cars.
Statement: Mark is a boy .
Valid Conclusion: Mark likes cars.
Conditional:
All boys like cars.
Statement: Markette is not a boy.
Invalid Conclusion: Markette does not
like cars.
Work Page 84 #’s 4-5
Law of Syllogism
If p q and q r are true, then
p r is also true.
p
q
p
q
r
r
Statement: If it rains, we will
stay inside.
Statement: If we stay inside,
we will play checkers.
Valid Conclusion: If it rains,
we will play checkers.
Statement: If WV wins, we
will have a party.
Statement: If WV wins, I will
cry with joy.
Invalid Conclusion: If we have
a party, I will cry with joy.
If my dog dies, I am sad.
If I am sad, my mom will
buy me a puppy.
If my mom buys me a
puppy, I will be happy.
If my dog dies, I will be
happy.
Work Page 84 #’s 6-7
Determine whether statement (3) follows
from statements (1) and (2) by the Law of
Detachment or the Law of Syllogism. If it
does, state which law was used. If it does
not, write invalid.
(1) If the sum of the squares of two sides of a
triangle is equal to the square of the third
side, then the triangle right triangle.
(2) For XYZ, (XY)2 + (YZ)2 = (ZX)2.
(3) XYZ is a right triangle.
p: the sum of the squares of the two sides of a
triangle is equal to the square of the third side
q: the triangle is a right triangle
By the Law of Detachment, if p
is true, then q is also true.
q is true and p
Answer: Statement (3) is a valid conclusion by
the Law of Detachment
Determine whether statement (3) follows
from statements (1) and (2) by the Law of
Detachment or the Law of Syllogism. If it
does, state which law was used. If it does
not, write invalid.
(1) If Ling wants to participate in the wrestling
competition, he will have to meet an extra three
times a week to practice.
(2) If Ling adds anything extra to his weekly
schedule, he cannot take karate lessons.
(3) If Ling wants to participate in the wrestling
competition, he cannot take karate lessons.
p: Ling wants to participate in the wrestling
competition
q: he will have to meet an extra three times a
week to practice
r: he cannot take karate lessons
By the Law of Syllogism, if
true. Then
is also true.
and
are
Answer: Statement (3) is a valid conclusion by
the
Law of Syllogism.
Determine whether statement (3) follows
from statements (1) and (2) by the Law of
Detachment of the Law of Syllogism. If it
does, state which law was used. If it does
not, write invalid.
(1) If a children’s movie is playing on Saturday,
Janine will take her little sister Jill to the
movie.
(2) Janine always buys Jill popcorn at the
movies.
(3) If a children’s movie is playing on Saturday,
Jill will get popcorn.
Answer: Law of Syllogism
b. (1) If a polygon is a triangle, then the sum of the
interior angles is 180.
(2) Polygon GHI is a triangle.
(3) The sum of the interior angles of polygon
GHI is 180.
Answer: Law of Detachment
Tuesday, May 2,
2017
Determine if a valid conclusion can be reached. If
so, determine which law you used.
If you are 18 years old, you are in
college.
You are in college.
Right angles are congruent.
<A and <B are right angles.
If you go to school, you are cool.
If you are cool, you get a diploma.
162
Japanese
died of
Karoshi in
2002. What
is it?
2.5 Postulates and
Paragraph Proofs.
• Postulate (axiom): Statement that
describes the fundamental relationship
between the basic terms in Geometry.
• Postulates are accepted as true.
• Basic ideas about points, lines, and
planes can be stated as postulates
Postulate 2.1: Through any two points there is
exactly one line.
Postulate 2.2: Through any three points not on the
same line, there is exactly one plane.
P
More Postulates
• Postulate 2.3: A line contains at least two
points.
• Postulate 2.4: A plane contains at least
three points not on the same line.
• Postulate 2.5: If two points lie in a plane,
then the entire line containing those points
lie in that plane.
• PostuIate 2.6: If two lines intersect, then
their intersection is exactly one point.
• Postulate 2.7: If two planes intersect, then
their intersection is a line.
Use postulates to determine if the
statement is sometimes, always, or never
true.
1) A line contains exactly one point.
2) Any two lines l and m intersect.
3) Planes R and S intersect at point P.
Do page 91 #6
Theorem
Official Vocabulary
•A statement that can be
shown or proven to be true.
•It can be used like a definition
or postulate to justify that
other statements are true.
Midpoint Theorem:
(Theorem 2.8)
If M is the midpoint of
AB, then AM = MB.
Proof
A proof is a logical argument in which
each statement you make is
supported by a statement that is
accepted as true.
One type of proof is called a
paragraph proof or informal proof.
Five Parts of a Proof
• State the theorem or conjecture to be
proven.
• List the given information.
• If possible, draw a diagram to illustrate the
given information.
• State what is to be proved.
• Develop a system of deductive reasoning.
In other words, to
prove something to be
true you must go step
by step and make a
logical progression.
Remember the definition of
Congruence: angles or segments
that have the same measure or
equal measures are said to be
congruent.
Remember, angles and segments
are congruent.
The measures of these things are
congruent.
In <ABC, BD is an angle bisector. Write a paragraph
proof to show that <ABD  <CBD.
A
D
B
C
By definition, an angle bisector divides an angle into
two congruent angles. Since BD is an angle
bisector, <ABC is divided into two congruent angles.
Thus, <ABD  <CBD.
Tuesday, May 2,
2017
Determine whether each statement is sometimes,
always, or never true.
There is exactly one Plane that contains points A, B,
and C.
Points E and F are contained in exactly one line.
Two lines intersect in two distinct points M and N.
Max is currently the oldest dog in the world. How
old is he in people years and dog years?
Chapter 2 Section 6
Algebraic Proofs
Properties of Equality
Reflexive
Symmetric
Transitive
Addition
Subtraction
Multiplication
Division
Substitution
Distributive
a=a
If a = b, then b = a.
If a = b and b = c, then a = c.
If a = b, then a + c = b + c.
If a = b, then a – c = b – c.
If a = b, then a c = b c.
If a = b, then a / c = b / c.
If a = b, then a may be
replaced with b.
a (b + c) = ab + ac
Work Page 97 #’s 4-7
Tell which property of equality justifies each
statement.
Let’s solve this equation and
justify each step by the properties
of equality
2(5 – 3a) – 4(a + 7) = 92
Work page 97 #8
**Two column proof (formal proof):
contains statements and reasons
organized in two columns.
**In a two-column proof, each step is
called a statement and the properties
that justify each step are called
reasons.
**In other words, left column is a stepby-step process that leads to a
solution, and the right column is the
reason for each step
A Few Notes
You must show each step!!!!!!
After using the Add, Sub, Multi, or Div
Properties of Equality, the next step is done
by Substitution.
Given: 6x + 2(x – 1) = 30
Prove: x = 4
Statements
Reasons
Given: 4x + 8 = x + 2
Prove: x = -2
Work Page 97 #’s 9
and 10
Geometric Proof
Many of the properties of equality used in Algebra
are also true in Geometry.
Reflexive: AB = AB
m<A = m<A
Symmetric: If AB = CD, then CD = AB.
If m<A = m<B, then m<B = m<A.
Transitive: If AB = CD and CD = EF, then AB = EF.
If m<1 = m<2 and m<2 = m<3, then m<1 = m<3.
State the property that justifies each statement.
If m<1 = m<2, then m<2 = m<1.
Symmetric
RS = RS
Reflexive
If AB = RY and RY = WS, then AB = RY.
Transitive
Look at page 96
Example 4
Let’s do page 98 #31
together
Tuesday, May 2,
2017
State the property that justifies each statement.
1. 2(LM + NO) = 2LM + 2NO
2. If m<R = m<S, then m<R + m<T = m<S + m<T
3. If 2PQ = LQ, then PQ = ½ LQ
4. m<Z = m<Z
5. If BC = CD and CD = EF, then BC = EF.
6. Where is the city of Batman?
Why Proofs?
2.7 Proving Segment
Relationships
A
B
C
Add a point, B, anywhere on AC.
What do you know about the
relationship between AB, BC, and
AC?
Segment Addition
Postulate
If B is between A and C, then AB + BC = AC.
Is the converse also true?
You can only add
Segment lengths and
not the actual
segments:
Change segment congruent statements to equal
statements first. (Definition of Congruency)
Prove the following.
Given: PR = QS
Prove: PQ = RS
Proof:
Statements
1. PR = QS
2. PR – QR = QS – QR
3. PQ = RS
Reasons
1. Given
2. Subtraction Prop.
3. Segment Addition
Postulate
Segment Congruence
Theorems
Reflexive
Property:
AB  AB
Symmetric
Property:
If
Transitive
Property:
AB  CD
then
CD  AB
Prove the following.
Given:
Prove:
Proof:
Statements
1. AC = AB, AB = BX
2. AC = BX
3. CY = XD
4. AC + CY = BX + XD
5. AY = BD
Reasons
1. Given
2. Transitive Property
3. Given
4. Addition Property
5. Segment Addition
Property
Prove the following.
Given:
Prove:
Proof:
Statements
1.
2.
3.
4.
5.
Reasons
1. Given
2. Definition of congruent segments
3. Given
4. Transitive Property
5. Transitive Property
Prove the following.
Given:
Prove:
Proof:
Statements
1.
2.
3.
4.
5.
Reasons
1. Given
2. Transitive Property
3. Given
4. Transitive Property
5. Symmetric Property
Tuesday, May 2, 2017
Justify each statement with a property of equality
or a property of congruence.
1. If AB  CD and CD  EF , then AB  EF
2. RS  RS
3. If H is between G and L, then GH + HL = GL.
State a conclusion that can be drawn from the
statements given using the property indicated.
4. W is between X and Z; Segment Addition Post
5. LM=NO and NO=PQ; Transitive Prop.
6. What do Eskimos use to prevent food from
freezing?
2.8 Reasoning and
Proof
Angle Addition Postulate: If R is in the interior of
<PQS, then m<PQR + m<RQS = m<PQS.
The converse is also true.
Theorems
Theorem 2.6: Angles supplementary to the
same angle or to congruent angles are
congruent.
A
C
B
m<A + m<B = 180
m<B + m<C = 180
m<A = m<C
Theorem
Theorem 2.7: Angles complementary to the
same angle or to congruent angles are
congruent.
A
m<A + m<B = 90
m<B + m<C = 90
B
m<A = m<C
C
Theorem
Theorem 2.8: If two angles
are vertical angles, then
they are congruent.
2
1
-orVert. <‘s are cong.
Theorems
Theorem 2.3: Supplement Theorem: If two
angles form a linear pair, then they are
supplementary.
Linear Pair
Supplementary
Add up to 180 degrees
Theorems
Theorem 2.4: Complement Theorem: If the
noncommon sides of two adjacent angles
form a right angle, then the angles are
complementary angles.
Right Angles
Theorem 2.9: Perpendicular lines intersect to form
four right angles.
Theorem 2.10: All right angles are congruent.
Theorem 2.11: Perpendicular lines form congruent
adjacent angles.
Theorem 2.12: If two angles are congruent and
supplementary, then each angle is a right angle.
Theorem 2.13: If two congruent angles form a
linear pair, then they are right angles.
If
and
form a linear pair and
find
Supplement Theorem
Subtraction Property
Answer: 14
If
and
.
find
are complementary angles and
Answer: 28
If 1 and 2 are vertical angles and m1
and m<2
find m1 and m2.
m1 m2
1 2
Definition of congruent
angles
Vertical Angles Theorem
Substitution
Add 2d to each side.
Add 32 to each side.
Divide each side by 3.
Answer: m1 = 37 and m2 = 37
If <A and <Z are vertical angles and the m<A =
3b – 23 and m<Z = 152 – 4b, find m<A and m<Z.
Answer: mA = 52; mZ = 52
Work Page 111 #’s 3-5
In the figure, and
form a linear pair, and
Prove that
and are congruent.
Given:
Prove:
form a linear pair.
Proof:
Statements
1.
2.
3.
4. m<3  m<4
Reasons
1. Given
2. Linear pairs are
supplementary.
3. Definition of
supplementary angles
4. <‘s supp same <‘s
are 
Work Page 111 #6
Homework Pages 112113 #’s 16-32