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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?
Day 1 Monday, January 25
2. Notes - Two Types of Reasoning
Deductive Inductive
Start with lots of rules.
Make another rule.
Establish a rule.
Start with lots of observations.
6. Notes - Two Types of Reasoning
Deductive Inductive
Gravity makes things fall downwards.Things that fall from a great height get
hurt.
If I jump off this building, I will fall downwards.
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 have lots of friends.
All smart people are rich.
Mrs. Caldwell is smart and rich. Bill Gates is smart and
rich.
ConjectureMake a Guess (But defend it)
Draw a picture and make a conjecture about the following information:
<ABC is a right angle.
ConjectureMake a Guess (But defend it)
Draw a picture and make a conjecture about the following information:
AB = 7, BC = 3 and AC=10.
CounterexampleStatement CounterexampleMath Teachers are Dorks
Movie Stars are Beautiful
Science Fiction is for Dorks
The opposite of a number is always negative.
The opposite of -4 is 4.
Mrs. Limburg
Make a conjecture about the next number based on the pattern.2, 4, 12, 48, 240
Answer: 1440
Find a pattern:2 4 12 48 240
×2The numbers are multiplied by 2, 3, 4,
and 5.Conjecture: The next number will be multiplied by 6. So, it will be or 1440.
×3
×4
×5
Make a conjecture about the next number based on the pattern.
Answer: The next number will be
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.
For points L, M, and N, and ,
make a conjecture and draw a figure to illustrate your conjecture.
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
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?
Wednesday, April 19, 2023
2-2 LogicStatement -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 Negation
Wednesday is chicken casserole day. Wednesday is not chicken casserole day.
Spock is a Vulcan. 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
FF
T
Compound Statements
Conjunctions (and)p: Tennessee is a state.q: Mrs. Limburg lives in Tennessee.
Write the statement
qp
Tennessee is a state and Mrs. Limburg lives in Tennessee.
Truth TableConjunction
p q p^qT T
T F
F T
F F
Compound Statements
Disjunction (or)p: Tennessee is a state.q: Mrs. Limburg lives in Tennessee.
Write the statement
qp
Tennessee is a state or Mrs. Limburg lives in Tennessee.
Truth TableDisjunctions
p qT T
T F
F T
F F
qp
Write statements:p: Freshmen are young.q: Robert is a freshman.r: Mr. Coggin is our principal.s: Football players are tough.
sqp
sr
qp
~
~
~
Create Truth Tablesp ^ ~q
p q ~q p ^ ~qT T
T F
F T
F F
Wednesday, April 19, 2023
p: Homework is fun.q: Math is my favorite.r: The Jonas Brothers rock!
1.
2.
3.
Construct a truth table for
rp ~pr pqr ~
qp ~
2-3 Conditional Statements
If….., then…..Hypothesis…, Conclusion…
HypothesisConclusion
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
goodWe will have ice
cream.Punk rock makes me sick.
It is punk rock
It makes me sick.
Conditional Statements in
DisguiseMrs. 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 P: The month is April.Let Q: The sun is shining.Let R: The moon is full.Let S: The night is young.
P Q If the month is April then the sun is shining.
~P The month is not April.
Therefore the night is young. S
If the moon isn’t full then the night is young.~R S
Logic
Let p: I get a job.Let q: I will earn money.Let r: I will go to the movies.Let s: I will spend my money.
Translate to English
1. If p then q.
2. If q then r.
3. If p then r.
4. If r then s.
If I get a job then I will earn money.
If I will earn money then I will go to the movies.
If I get a job then I will go to the movies.
If I will go to the movies then I will spend my money.
2. Logic
Let p: Today is Wednesday.Let q: Tomorrow is Thursday.Let r: Friday is coming.Let 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. rs
qs
rq
qp
ConverseConverse - 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 Nearby1. Is the converse of the Linear Pair Conjecture true?
“If two angles add up to 180°, then they are a linear pair.”
FALSE
2. Is the converse of the Vertical Angles Conjecture true?
“If two angles are congruent, then they are vertical angles.”
118° 62°
300° 300°
FALSE
Inverse
If a movie does not star Bill Paxton, then Mrs. Caldwell does not hate it.
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.”qp
qp ~~ False
Contrapositive
If Mrs. Caldwell does not hate a movie, then it does not star Bill Paxton.
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.”
TrueIf a conditional is true, then the contrapositive is always true.
Your TurnIf my dog dies, then I am sad.
Write the converse, inverse, and
contrapositive and label each.
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?
Wednesday, April 19, 2023
Truth TablesIf 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 qp
TT
TT
TTT
F FF
FF
Alice in Wonderland
Classwork
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?
Wednesday, April 19, 2023
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 qpq
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 qq rp 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.
(3) XYZ is a right triangle.
(2) For XYZ, (XY)2 + (YZ)2 = (ZX)2.
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 q is true and p is true, then q is also true.
Answer: Statement (3) is a valid conclusion by the Law of Detachment
(1) If Ling wants to participate in the wrestling competition, he will have to meet an extra three times a week to practice.
(3) If Ling wants to participate in the wrestling competition, he cannot take karate lessons.
(2) If Ling adds anything extra to his weekly schedule, he cannot take karate lessons.
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.
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 and are true. Then is also true.
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.
(3) If a children’s movie is playing on Saturday, Jill will get popcorn.
(2) Janine always buys Jill popcorn at the movies.
Answer: Law of Syllogism
b. (1) If a polygon is a triangle, then the sum of the interior angles is 180.
(3) The sum of the interior angles of polygon GHI is 180.
(2) Polygon GHI is a triangle.
Answer: Law of Detachment
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.
Wednesday, April 19, 2023
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
TheoremOfficial 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.
ProofA proof is a logical argument in whicheach statement you make is supported by a statement that isaccepted 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
BC
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.
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?
Wednesday, April 19, 2023
Chapter 2 Section 6Algebraic Proofs
Properties of Equality
Reflexive a = aSymmetric If a = b, then b = a.Transitive If a = b and b = c, then a = c.Addition If a = b, then a + c = b +
c.Subtraction If a = b, then a – c = b – c.Multiplication If a = b, then a c = b c.Division If a = b, then a / c = b / c.Substitution If a = b, then a may be
replaced with b.Distributive 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 step-by-step process that leads to a solution, and the right column is the reason for each step
A Few NotesYou 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) = 30Prove: x = 4Statements Reasons
Given: 4x + 8 = x + 2Prove: x = -2
Work Page 97 #’s 9 and 10
Geometric ProofMany 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 = RSReflexive
If AB = RY and RY = WS, then AB = RY.Transitive
Look at page 96 Example 4
Let’s do page 98 #31 together
State the property that justifies each statement.
1.2(LM + NO) = 2LM + 2NO2.If m<R = m<S, then m<R + m<T = m<S +
m<T3.If 2PQ = LQ, then PQ = ½ LQ4.m<Z = m<Z5.If BC = CD and CD = EF, then BC = EF.
6.Where is the city of Batman?
Wednesday, April 19, 2023
Why Proofs?
2.7 Proving Segment Relationships
Add a point, B, anywhere on AC.
A CB
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)
Given: PR = QSProve the following.
Prove: PQ = RS
1. GivenPR = QS1.
2. Subtraction Prop.
PR – QR = QS – QR2.
3. Segment Addition Postulate
3. PQ = RS
Proof:Statements Reasons
Segment Congruence Theorems
Reflexive Property:
Symmetric Property:
Transitive Property:
ABAB
CDAB ABCD If then
Prove the following.
Prove:
Given:
Proof:Statements Reasons
1. Given2. Transitive
Property
3. Given4. Addition Property
AC = AB, AB = BXAC = BXCY = XDAC + CY = BX + XD
5. Segment Addition Property
AY = BD
1. 2. 3. 4. 5.
Prove the following.
Prove:
Given:
1. Given1. 2. Definition of congruent
segments2.
3. Given3. 4. Transitive
Property4.
5. Transitive Property5.
Proof:Statements Reasons
Prove the following.
Prove:
Given:
Proof:Statements Reasons
1. Given2. Transitive Property3. Given4. Transitive Property5. Symmetric
Property
1. 2. 3. 4. 5.
Wednesday, April 19, 2023
Justify each statement with a property of equality or a property of congruence.1.If and , then2. 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 Post5.LM=NO and NO=PQ; Transitive Prop. 6.What do Eskimos use to prevent food from freezing?
CDAB EFCD EFAB RSRS
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 B C
m<A + m<B = 180m<B + m<C = 180 m<A = m<C
Theorem
A CB
Theorem 2.7: Angles complementary to the same angle or to congruent angles are congruent.
m<A + m<B = 90m<B + m<C = 90 m<A = m<C
Theorem
12
Theorem 2.8: If two angles are vertical angles, then they are congruent.
-or-
Vert. <‘s are cong.
Theorems
Theorem 2.3: Supplement Theorem: If two angles form a linear pair, then they are supplementary.
Linear PairSupplementary
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 AnglesTheorem 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.
Supplement Theorem
Subtraction Property
Answer: 14
form a linear pair and find If and
Answer: 28
are complementary angles and .
andIffind
Substitution
Add 2d to each side.
Add 32 to each side.
Divide each side by 3.
If 1 and 2 are vertical angles and m1 and m<2 find m1 and m2.
Vertical Angles Theorem1 2
Definition of congruent angles
m1 m2
Answer: m1 = 37 and m2 = 37
Answer: mA = 52; mZ = 52
If <A and <Z are vertical angles and the m<A = 3b – 23 and m<Z = 152 – 4b, find m<A and m<Z.
Work Page 111 #’s 3-5
Given: form a linear pair.
Prove:
In the figure, form a linear pair, and Prove that are congruent.
andand
1. Given1.
3. Definition of supplementary angles
3.
Proof:Statements Reasons
2. Linear pairs are supplementary.
2.
4. m<3 m<4 4. <‘s supp same <‘s are
Work Page 111 #6
Homework Pages 112-113 #’s 16-32
