Download Geometry 2-1 Inductive Reasoning and Conjecturing A. Definitions 1

Geometry 2-1 Inductive Reasoning and Conjecturing
A. Definitions
1. A conjecture is an ___________ guess.
2. Looking at several specific situations to arrive at a _____________ is called
inductive reasoning.
Ex 1: For points A, B, and C, AB = 10, BC = 8, and AC = 5. Make a conjecture and
draw a figure to illustrate your conjecture.
Given: Points A, B, and C
AB = 10, BC = 8, and AC =5
Conjecture:
3.
If a conjecture is made, and can be determined that it is false, it takes only one
false example to show that a conjecture is not true. The false example is called
a counterexample.
Ex 2: Find a counterexample based on the given information.
Given: points P, Q, and R are collinear
Conjecture: Q is between P and R.
Example 3: Based on the table showing unemployment rates for various counties in
Kansas, find a counterexample for the following statement.
The unemployment rate is highest in counties with the most people.
County
Civilian Labor Rate
Force
Shawnee
Jefferson
Jackson
Douglas
Osage
Wabaunsee
Pottawatomie
HW: Geometry 2-1 p. 64-66
11-37 odd, 43-44, 45-67 odd
Honors: 38-40
90,254
9,957
8,915
55,750
10,182
3,575
11,025
3.1%
3.0%
2.8%
3.2%
4.0%
3.0%
2.1%
Geometry Section 2-2 - Logic
A. Determine Truth Values
1. Definitions
a.) A statement is a sentence that is true or false, but not both.
b.) The truth or falsity of a statement is called its truth value.
c.) Statements are often represented by using the letters p and q.
p: Visalia is a city in California.
d.) The negation of a statement has the opposite meaning as well as the
opposite truth value.
Not p: Visalia is _______________________.
(this could also be written ~p.)
e.) Two or more statements can be joined to form a compound statement.
p: Sacramento is a city in California.
q: Sacramento is the capitol of California.
The two statements can be joined by the word and.
p and q: Sacramento is a city in California and Sacramento is the capital of
California.
f.) A conjunction is a compound statement formed by joining two or more
statements with the word and.
p ∧ q , read p and q.
Example 1: Use the following information to write a compound statement
for each conjunction. Then find its truth value.
p: one foot is 14 inches
q: September has 30 days.
r: A plane is defined by 3 noncollinear points.
a.) p and q
b.) r and p
c.) ~ q ∧ r
d.) ~ p ∧ r
g.) A disjunction is a compound statement formed by joining two or more
statements with the word or.
1. A disjunction is true if one or more of the statements are true.
2. p ∨ q is read p or q.
Example 3: Use the following information to write compound statements
for each disjunction. Then find its truth value.
p: AB is the proper notation for “line AB”
q: Centimeters are metric units.
r: 9 is a prime number.
a.) p or q
b.) q ∨ r
HW: Geometry 2-2 p. 72-74
18-29 all, 41-44, 54-55, 57-73 odd
Hon: 45-47
Section 2-3, If –Then Statements and Postulates
A. If-Then Statements
1. Statements can be changed into If-Then statements
a. Football players are jocks. _________________________________
b. Smart kids are geeks.
_________________________________
c. Boys are mean
_________________________________
d. Adjacent angles have a common side. _________________________
2. If - Then statements are used to clarify confusing statements
3. If –Then Statements are called conditional statements or conditionals.
a. The part following the “if” is called the hypothesis.
b. The part following the then is called the conclusion.
c. p → q represents the conditional statement “if p, then q”
Example 1: Identify the hypothesis and conclusion of the following statements:
a.) “If this had been an actual emergency, then the attention signal you just heard
would have been followed by official news, information or instruction.”
Hypothesis:
Conclusion:
b.) John will advance to the next level of play if he completes the maze on his
computer game.
Hypothesis:
Conclusion
Example 2: Write each statement in “if-then” form and identify the hypothesis and
conclusion.
a.) A five-sided polygon is a pentagon.
B. Alterations of If-Then Statements
1. You can change an if-then statement by switching the hypothesis and conclusion.
This new statement is called the converse.
Example 2: Write the converse of the true conditional “An angle that measures
120o is obtuse.
Step 1: Write the conditional in If-Then form.
Step 2. Write the converse of the true conditional.
Step 3. Determine if your new statement is true or false.
Step 4. Give a counterexample if it is not true.
2. The denial of a statement is called a negation.
a. the negation of the statement “ an angle is acute” is “ An angle is not
acute”
b. : p represents “not p” or the negation of p
3. Given a conditional statement, the negation of both the hypothesis and
conclusion is called the inverse.
a. The inverse of p → q is : p → : q
Example 3: Write the inverse of the true conditional “Vertical angles are
congruent.” Determine if the inverse is true or false. If false, give a
counterexample.
If-Then form:
Negate both the hypothesis and conclusion
Determine if they are true or false
4. Given a conditional statement, its contrapositive can be formed by negating
both the hypothesis and conclusion of the converse of the given statement.
a. The contrapositive of p → q is : q → : p
Example 4: Write the contrapositive of the true conditional “If two angles are
vertical, then they are congruent.”
Statement:
Converse:
Contrapositive:
True or false?
-The contrapositive of a true statement is always true and the contrapositive
of a false statement is always false.
HW Geometry 2-3 p. 78-80
16-27 all, 34-45 all, 50-51, 53-67 odd
Hon: 46-47
Geometry 2-4 Deductive Reasoning
A. Definitions
1. Deductive reasoning uses facts, rules, definitions, or properties to reach logical
conclusions.
2. (Inductive reasoning uses examples to make a conjecture)
3. A form of deductive reasoning that is used to make conclusions from conditional
statements is called the Law of Detachment.
4. The law of detachment says if
p ◊ q is a true conditional,
and there is a specific example where p is true,
then q must be true.
Example 1: Determine if the following conclusions are true from the Law of
Detachment.
a.) If you are at least 16 years old, you can get a drivers license.
b.) Peter is 16 years old.
c.) Peter can get a drivers license.
a.) If you like pizza, you love Chicago style deep dish.
b.) Jen loves Chicago style deep dish.
c.) Jen likes pizza.
5. Another form of deductive reasoning is the law of syllogism, which states that:
If p ◊ q is true,
and q ◊ r is true,
then p ◊ r must be true.
Example 2: Determine if the following conclusions are true based on the law of
syllogism.
a.) If you are a careful bicycle rider, then you wear a helmet.
b.) If you wear a helmet, then you look kind-of silly.
c.) If you are a careful bicycle rider, then you look kind-of silly.
a.) If you run every day, you will get healthier.
b.) If you run every day, you will get faster.
c.) If you get healthier, you will get faster.
Example 3: Determine whether statement 3.) follows from statements 1.) and 2.)
by the law of syllogism or the law of detachment. If it does, state which law was
used. If it does not, write invalid.
1.) If Sue wants to participate in the wrestling competition, she will have to meet
an extra 3 times a week to practice.
2.) If Sue adds anything to her weekly schedule, she cannot take karate lessons.
3.) If Sue wants to participate in the wrestling competition, she cannot take karate
lessons.
1.) If the girls want to win the volleyball game, they will have to play better.
2.) Shania is playing better.
3.) The girls will win the volleyball game.
1.) If you want to get an A in Geometry, you will have to study hard.
2.) Faith is studying hard.
3.) Faith will get an A in Geometry.
1.) If two angles are vertical, then they are congruent.
2.) ∠1 and ∠2 are vertical angles.
3.) ∠1 and ∠2 are congruent
HW: Geometry 2-4 p. 85-87
12-31 all, 34, 35, 37-57 odd
Hon: 32
Geometry 2-5 Postulates and Paragraph Proofs
A. Points lines and planes
1. In geometry, a postulate is a statement that describes a fundamental relationship
between the basic terms of geometry.
2. Postulates are accepted as true.
a.) Postulate 2-1 – Through any two points, there is exactly 1 line.
b.) Postulate 2-2 – Through any three points not on the same line, there is exactly
one plane.
Example 1: Some snow crystals are shaped like regular hexagons. How many
lines must be drawn to interconnect all vertices of a hexagonal snow crystal.
c.) Postulate 2-3 – A line contains at least two points
d.) Postulate 2-4 – A plane contains at least three points not on the same line.
e.) Postulate 2-5 – If two points lie in a plane, then the entire line containing those
two points lies in that plane.
f.) Postulate 2-6 – If two lines intersect, then their intersection is exactly one
point.
g.) Postulate 2-7 – If two planes intersect, then their intersection is a line.
Example 2: Determine if each statement is always, sometimes, or never true.
suu
r
suu
r
a. If plane T contains EF , and EF contains point G, then plane T
contains point G.
suu
r
b. For XY , if X lies in plane Q and Y lies in plane R , then plane Q
intersects plane R.
suuu
r
c. GH contains three noncollinear points.
B. Paragraph proofs
1. Once a statement or conjecture has been proven to be true, it is called a theorem.
2. A proof is a logical argument in which each statement you make is supported by a
statement that is accepted as true.
a. One type of proof is called a paragraph proof, where you write a paragraph to
explain why a conjecture for a given situation is true.
3. Proof writing -There are five parts to writing a good proof.
a. State the theorem to be proved.
b. List the given information.
c. If possible, draw a diagram to illustrate the given information.
d. State what is to be proved.
e. Develop a system of deductive reasoning.
suur
suur
Example 3: Given AC intersecting CD , write a paragraph proof to show ACD is
a plane.
suur
suur
AC and CD must intersect _________________ because if 2 lines intersect, then
suur
______________________________________________. Point A is on AC , and
_________________________________. Therefore, points ____________ are
not __________________. Therefore, ACD __________________________
because it contains 3 points that are not __________________________.
4. Theorem 2-1 - Midpoint Theorem - If M is is the midpoint of AB , then
AM ≅ MB .
HW: Geometry 2-5 p. 92-93
12-31 all, 33-48 all
Geometry 2-6 Algebraic Proofs
A. Algebraic Proofs
1. Properties of Equality
Properties of Equality for Real Numbers
Reflexive Property
For every number a, a = a.
Symmetric Property
For all numbers a and b, if a = b, then b = a.
Transitive Property
For all numbers a, b, and c, if a = b and b = c, then
a = c.
Addition and Subtraction
Properties
Multiplication and Division
Properties
For all numbers a, b, and c,
if a = b, then a + c = b + c and a - c = b - c.
For all numbers a, b, and c, if a = b, then
a gc = bgc , and if c ≠ 0, then
a b
=
c c
Substitution Property
For all numbers a and b, if a = b, then a may be
replaced by b in any equation or expression.
Distributive Property
For all numbers a, b, and c, a(b+c) = ab + ac.
Example 1: Give a reason for each statement.
Statements
a.) If AB + BC = CE + BC, then AB = CE
Reasons
a.)
b.) m∠ABC = m∠ABC
b.)
c.) If XY = PQ and XY = RS, then PQ = RS
c.)
d.) If 1/3x = 5, then x = 15
d.)
e.) If 2x = 9, then x = 9/2.
e.)
2. These types of proofs are called two column proofs.
Example 2: Justify each step in solving
Given:
3x + 5
=7
2
3x + 5
= 7.
2
Prove: x = 3
Statements
Reasons
1.
1.
2.
3.
4.
5.
3x + 5
=7
2
 3x + 5 
2
 = 2(7)
 2 
3x + 5 = 14
3x = 9
x =3
2.
3.
4.
5.
B. Geometric Proof
1. We can apply these properties to geometric expressions.
Property
Segments
Angles
Reflexive
PQ = PQ
mπ 1 = _____
Symmetric
If AB = CD, then __________
If mπ A = mπ B, then mπ B = mπ A
Transitive
If GH = JK and JK = LM, then
GH = LM
If mπ 1 = mπ 2 and _________,
then mπ 1 = mπ 3.
Example 3: Justify the steps for the proof.
Given: ∠ABD and ∠DBC are complimentary angles.
Prove: ∠ABC is a right angle.
A
Statements
1.) ∠ABD and ∠DBC are
complimentary.
2.) m∠ABD + m∠DBC = 90
3.) m∠ABD + m∠DBC = m∠ABC
4.) m∠ABC = 90
5.) ∠ABC is a right angle
B
HW: Geometry 2-6 p. 97-100
14-25 all, 26, 37-38, 39-53 odd
Hon: 27, 30-32
Reasons
1.)
2.)
3.)
4.)
5.)
D
C
Geometry 2-7 Proving Segment Relationships
A. Segment Addition
1. Postulate 2-8 - Ruler Postulate - the points on any line or line segment can be
paired with real numbers so that, given any two points A and B on a line, A
corresponds to zero, and B corresponds to a positive real number.
2. Postulate 2-9 - Segment Addition Postulate - If B is between A and C, then AB
+ BC = AC.
A
B
C
Example 1:
Given: Points A, B, C, and D
are collinear.
Prove: AB = AD − BD
Proof:
Statements
1. Points A, B, C, and D
are collinear.
2. AD = AB + BD
3. AD – BD = AB
4. AB = AD – BD
A
B
C
D
Reasons
1.
2.
3.
4.
3. Theorem 2-2 - Congruence of segments is reflexive, symmetric, and transitive.
a.) Reflexive Property - AB ≅ AB
b.) Symmetric Property - If AB ≅ CD , then CD ≅ AB .
c.) Transitive Property - If AB ≅ CD , and CD ≅ EF , then AB ≅ EF .
Example 2:
Given: AC ≅ DF, AB ≅ DE
A
B
C
Prove: BC ≅ EF
D
Proof:
Statements
1. AC ≅ DF, AB ≅ DE
2. AC = DF, AB = DE
3. AC – AB = DF – DE
4. AC = AB + BC
DF = DE + EF
5. AC – AB = BC
DF – DE = EF
6. DF – DE = BC
7. BC = EF
8. BC ≅ EF
HW: Geometry 2-7 p. 104-106
12-18 all, 21, 29-30, 31-45 all
Hon: 19, 22
Reasons
1.
2.
3.
4.
5.
6.
7.
8.
E
F
Geometry 2-8 - Proving Angle Relationships
A. Supplementary and Complementary Angles
1. Postulate 2-11 - Angle Addition Postulate - If
R is in the interior of ∠PQS , then m∠PQR +
m∠RQS = m∠PQS . The converse is also true: If
m∠PQR + m∠RQS = m∠PQS , then R is in the
interior of ∠PQS .
Example 1: If m∠PQR = 2 x + 12 ,
m∠RQS = 3 x − 2 , and m∠PQS = 50,
find the value of x and m∠PQS .
P
Q
R
S
P
Q
R
S
2. Theorem 2-3 – Supplemental Theorem - If two angles form a linear pair, then
they are supplementary.
3. Theorem 2-4 - Complement Theorem - If the noncommon sides of two
adjacent angles form a right angle, then the angles are complimentary angles.
Example 2: If ∠1 and ∠2 form a linear
pair, and m ∠1 = 4 x − 5 and
m ∠2 = 14 x + 5 , find x, and m ∠1 .
2
1
4. Theorem 2-5 - Congruence of angles is reflexive, symmetric, and transitive
a. Reflexive Property - ∠1 ≅ ∠1
b. Symmetric Property - If ∠1 ≅ ∠2 , then ∠2 ≅ ∠1 .
c. Transitive Property - If ∠1 ≅ ∠2 , and ∠2 ≅ ∠3 , then ∠1 ≅ ∠3 .
5. Theorem 2-6 - Angles supplementary to the
same angle or to congruent angles are congruent.
a. If m∠1 + m∠2 = 180 , and
m∠2 + m∠3 = 180 , then ∠1 ≅ ∠3
2
1
3
Example 3: Proof of Theorem 2-6
Given: ∠1 and ∠2 are supplementary
∠2 and ∠3 are supplementary
Prove:
Proof:
Statements
1. ∠1 and ∠2 are supplementary
∠2 and ∠3 are supplementary
m
2. ∠1 + m∠2 = 180
m∠2 + m∠3 = 180
3.
4. m∠1 = m∠3
5. ∠1 ≅ ∠3
y
3
2
1
=
6. Theorem 2-7 - Angles complementary
To the same angle or to congruent
angles are congruent.
a. If m∠1 + m∠2 = 90o , and
m∠2 + m∠3 = 90o , then ∠1 ≅ ∠3
Reasons
1. Given
2.
3. Substitution
4.
5.
1
2
3
7. Theorem 2-8 - If two angles are vertical angles, then they are congruent.
Example 4: If ∠1 and ∠2 are vertical angles, and m∠1 = d − 32 , and
m∠2 = 175 − 2d , find m∠1 and m∠2 .
8. Theorem 2-9 - Perpendicular lines intersect to form four _______________.
9. Theorem 2-10 - All right angles are congruent.
10. Theorem 2-11 - Perpendicular lines form congruent adjacent angles.
11. Theorem 2-12 - If 2 angles are congruent and supplementary, then each
angles is a right angle.
12. If two congruent angles form a linear pair, then they are _____________.
HW: Geometry 2-8 p. 112-114
16-32 all, 39, 41, 44-45, 46-55 all
Hon: 38