Download UNIT 2 - Peru Central School

Geometry UNIT 2
Reasoning and Proof
Day 1
Inductive Reasoning and Conjecture
Page 93-95: 15-25 odds, 31-34, 36, 38, 47a
Day 2
Logic
Page 101-102: 17, 19, 21, 31, 33
Day 3
Conditional/Biconditional/Inverse/Converse/Contrapositive
Page 109-: 19-31 odd, 35-45 odd, 47, 48
Day 4
Review for QUIZ
QUIZ
Day 5
Postulates and Paragraph Proofs
Page 129: 16, 17, 20, 24, 28, 29, 30*(typo in book)
Day 6
Algebraic Proof
Page 137-139: 9-11, 13-17, 25, 26
Day 7
Proving Segment Relationships
Page 146-147: 9 – 12, 15 (Do most in class)
Day 8
Proving Angle Relationships
Worksheet
Day 9
Review
Worksheet and Page 161-162: #35, 37, 38
Day 11
TEST
VOCABULARY
Inductive Reasoning
Statement
Negation
Truth Table
Conclusion
Contrapositive
Law of Detachment
Two-Column (Formal) Proof
Conjecture
Truth Value
Conjunction
Conditional
Converse
Logically Equivalent
Postulate
Counterexample
Compound Statement
Disjunction
Hypothesis
Inverse
Deductive Reasoning
Axiom
Subtraction Property
Reflexive Property
Substitution Property
Multiplication Property
Symmetric Property
Distributive Property
Properties of Real Numbers
Addition Property
Division Property
Transitive Property
Postulates/Axioms (accepted without proof)
Segment Addition
Angle Addition
Theorems (can be proven)
Supplement Theorem
Right Angle Thereoms
Complement Theorem
Vertical Angles Theorem
Inductive Reasoning and Conjecture
Inductive Reasoning:
_____________________________________________________________________________
Conjecture:
Step 1: Look for a pattern
Step 2: Make a conjecture
Ex 1: Write a conjecture that describes the pattern in the sequence. Then use your
conjecture to fine the next item in the sequence.
a.
Movie show times: 8:30 a.m., 9:45 a.m., 11:00a.m., 12:15 p.m.,…
b.
10, 4, -2, -8, …
Ex 2: Make a conjecture about each value or geometric relationship. List or draw some
examples to support your conjecture.
a.
The sum of two odd numbers
b.
Segments joining opposite vertices of a rectangle
Counterexample:
Ex 4: Find a counterexample to show that the conjecture is false.
a.
If n is a real number, then n2 > n.
b.
Jf JK = KL, the K is the midpoint of JL.
c.
If n is a real number, then -n is a negative.
Sentences, Statements, and Truth Values, Connectives and Negations
Mathematical Sentence: A sentence that states a fact or contains a complete idea.
Truth Value: True or False.
Statement: A mathematical sentence that is either true or false.
Open Sentence: A mathematical sentence that contains a variable.
Closed Sentence (Statement): A sentence that can be judged as either true or false.
Compound Statement: Two or more statements joined by ‘and’ or ‘or’.
Ex 1:
Ex 2:
Ex 3:
Ex 4:
Ex 5:
(Open)
(Open)
(Open)
(Closed)
(Closed)
Liver tastes really good.
3x + 2 = 17
He is my friend.
3(3) + 2 = 17
3(5) + 2 = 17
Truth value?
Variable?
Variable?
Truth Value?
Truth Value?
Symbol ~
Negation:
p: There are seven days in a week.
Symbol Λ
Conjunction:
p
T
T
F
F
q
T
F
T
F
pΛq
p: A right angle measures 90o.
pΛq :
Truth Value:
~p V q:
Truth Value:
~p: ____________________________
Symbol V
Disjunction:
p
T
T
F
F
q
T
F
T
F
pVq
q: Supplementary angles have a sum of 180o.
Conditional/Biconditional/Inverse/Converse/Contrapositive
Conditional:
Symbol → (Can also stand for “implies” )
Hypothesis:
Conclusion:
p: You pass the test.
q: I buy you a pizza.
p→ q: _______________________________________________________________________
p → ~q: _____________________________________________________________________
p
q
p→q
Related Conditionals:
a.
Inverse:
b.
Converse:
c.
Contrapositive
d.
Logically Equivalent Statements:
_____________________________________________________________________
Biconditional:
Symbol ↔
p: There is no school.
q: It is Saturday.
p↔ q: _______________________________________________________________________
p ↔ ~q: _____________________________________________________________________
p
q
p↔q
Reasoning and Proof QUIZ REVIEW
Use the information below to: a.
b.
c.
Let m represent:
Let s represent:
Let b represent:
Let a represent:
1.
Name
Write a compound statement in symbolic form.
Tell whether the compound sentence is true or false.
For # 3 – 4, write the inverse, converse, and
contrapositive in symbolic form.
A segment is bisected at its midpoint. (True)
Congruent segments are equal in length. (True)
An angle bisector forms two congruent angles. (True)
Congruent angles are equal in measure. (True)
A segment is bisected at its midpoint or congruent segments are equal in length.
Symbolic Form:
Truth Value:
2.
If an angle bisector does not form two congruent angles then congruent angles are
equal in measure.
Symbolic Form:
Inverse:
Truth Value:
3.
Truth Value:
Converse:
Truth Value:
Contrapositive:
Truth Value:
If you study, then you pass the test. (Hyp. is true, conc. is true).
Hypothesis:_________________________________________________________________
Conclusion:_________________________________________________________________
Inverse:____________________________________________________________________
Truth Value:
Converse:___________________________________________________________________
Truth Value:
Contrapositive:______________________________________________________________
Truth Value:
Postulates and Paragraph Proofs
Words
Picture
Through any two points, there is exactly one <------------------------------------------------------->
line.
A
B
C
D
Through any three noncollinear points there
is exactly one plane.
A line contains at least two points.
A plane contains at least 3 noncollinear
points.
C
A
m
B
<------------------------------------------------------->
A
B
C
D
C
n
B
A
If two point lie in a plane, then the entire
line containing those points lies in that
plane.
A
B
k
k
If two lines intersect, then their intersection
is exactly one point.
m
n
X
If two planes intersect, then their
intersection is a line.
Proof:
Theorem:
Midpoint Theorem:
If M is the midpoint of AB, then AM = MB
ALGEBRAIC PROOFS
Property of Real Numbers
Example
Addition Property of Equality
Subtraction Property of Equality
Multiplication Property of Equality
Division Property of Equality
Reflexive Property of Equality
Symmetric Property of Equality
Transitive Property of Equality
Substitution Property of Equality
Distributive Property of Equality
Example:
Solve -5(x + 4) = 70 with justification at each step.
-5(x + 4) = 70
-5x + (-5●4) = 70
-5x – 20 = 70
-5x – 20 = 70
+20 +20
-5x = 90
-5x = 90
-5 -5
x = -18
__________________________
__________________________
__________________________
__________________________
__________________________
__________________________
__________________________
8. If -4(x – 3) + 5x = 24, then x = 12.
Geometry Unit 2 Day 7
Proving Segment Relationships
Property
Example
Reflexive Property of Congruence
Symmetric Property of Congruence
Transitive Property of Congruence
Substitution Property
Segment Addition Postulate
Segment Subtraction Postulate
A whole is equal to the sum of all its parts.
1.
Line segment AE and DB intersect at C. (DRAW PICTURE)
Given:
C is the midpoint of AE.
C is the midpoint of BD.
AE = BD
Prove:
AC = CD
Statement
2.
Reason
In ΔJNL, point M is on JN and point K is on JL. (DRAW PICTURE)
Given:
LK = NM, KJ = MJ
Prove:
Statement
LJ = NJ
Reason
Geometry Unit 2 Day 8
Proving Angle Relationships
Complementary Angles
Complement Theorem
Congruent Complements Theorem
Supplementary Angles
Supplements Theorem
Congruent Supplements Theorem
Vertical Angles
Vertical Angles Theorem
Angle Addition Postulate
Right Angles Theorem: Perpendicular lines
intersect to form right angles.
Right Angles Theorem: All right angles are
congruent.
Right Angles Theorem: Perpendicular lines
form congruent adjacent angles.
Right Angles Theorem: If two angles are
congruent and supplementary, then each
angle is a right angle.
Right Angles Theorem: If two congruent
angles form a linear pair, then they are right
angles.
Reasoning and Proof Test Review
1.
What is the negation of the statement “The Sun is shining”?
[1] It is cloudy.
[2] It is daytime.
2.
[3] It is not raining.
[4] The Sun is not shining
If S = "It is snowing." and C = "It is cold.", which statement is ~C V S?
[1] It is not cold, and it is snowing.
[2] If it is not cold, then it is snowing.
[3] It is not snowing or it is cold.
[4] It is not cold or it is snowing.
3.
What is the converse of the statement: "If ABCD is a rectangle, then the diagonals bisect
each other."
[1] If the diagonals of ABCD bisect each other, then ABCD is a rectangle
[2] If ABCD is not a rectangle, then the diagonals do not bisect each other.
[3] ABCD is a rectangle and the diagonals bisect each other.
[4] If the diagonals of ABCD do not bisect each other, then ABCD is not a rectangle.
4.
Which statement is logically equivalent to “If Andrea gets a job, she buys a new car”?
[1] Andrea gets a job and she buys a new car.
[2] If Andrea does not buy a new car, she does not get a job.
[3] If Andrea does not get a job, she does not buy a new car.
[4] If Andrea buys a new car, she gets a job.
5.
The sentence "If __________, then 6 + 3 = 9." is TRUE. Which of the following
statements could be used to fill the blank and maintain the truth value of the sentence?
[1] 10/5 = 2
[2] 12/3 = 6
[3] both choice [1] and choice [2] could be used
[4] neither choice [1] nor choice [2] could be used
6.
7.
8.
9.
10.
11.
“If two sides of a triangle are congruent, then the angles opposite those sides are
congruent.”
Inverse:
Converse:
Contrapositive:
Which one is logically equivalent to the original statement?
Write a Two-Column Proof for each:
12.
Given:
Prove:
JK = AB, KL = BC
JL = AC
A
B
C
J
K
L
13.
14.
Given:
5x – 3 = 2x + 1
6
Prove:
x = - 9 /7
Given:
B is the midpoint of AC.
C is the midpoint of BD.
Prove:
AB = CD
A
B
C
D
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