Download Geometry Chapter 2: Geometric Reasoning

Geometry Chapter 2: Geometric Reasoning
Lesson 1: Using Inductive Reasoning to Make Conjectures
Learning Targets
Success Criteria
LT-1: Use inductive reasoning to identify patterns
and make conjectures.
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Inductive Reasoning:
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Conjecture:
Use inductive reasoning to identify a
pattern.
Make a conjecture.
Find a counterexample
Advantages:
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can draw conclusions from limited information
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helps us to organize our thinking
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human beings know how to use inductive reasoning naturally
Disadvantages:
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sometimes we draw the wrong conclusion from the information
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our conclusions are only as strong as the evidence that we choose to consideration
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in mathematics, just one example which contradicts our conclusion tells us that our conclusion is
wrong
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cannot be used to “prove” something mathematically
Ex: #1: Identify a Pattern. Find the next item in each pattern.
A. January, March, May, ...
B. 7, 14, 21, 28, ...
C.
D.
Ex: #2: Make a Conjecture.
A. The sum of two positive numbers is
_____________________.
B. The number of lines formed by four points, no
three of which are collinear, is _____________.
C. The quotient of one positive number and one
negative number is ____________________.
D. If the side length of a square is doubled, the
perimeter of the square is _________________.
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Ex: #3: Make a Conjecture. Biology Application
A biologist observed blue-whale spouts of 25ft, 29ft, 27ft, and 24ft. Another biologist recorded
humpback-whale spouts of 8ft, 7ft, 8ft, and 9ft. Make a conjecture based on the data.
To show that a conjecture is TRUE,, you must prove it for ALL possibilities.
To show that a conjecture is FALSE you only have to show just ONE counterexample.
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Counterexample:
Ex #4: Find a counterexample. Show that each conjecture is FALSE by finding a COUNTEREXAMPLE.
A. For every integer n, n³ is positive.
B. Two complementary angles are not congruent.
C. Based on the data in the table,
the monthly high temperature
in Abilene is never below 90° F
for two months in a row.
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Chapter 2 Lesson 2: Conditional Statements
Learning Targets
Success Criteria
LT-2: Identify, write, and analyze the truth value of
conditional statements.
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Identify the hypothesis and conclusion in a
conditional statement.
Write a conditional statement.
Analyze the truth value of a conditional
statement.
Conditional Statement
Definition
Symbols
Venn Diagram
Conditional Statement:
Hypothesis:
Conclusion:
Ex: #1 Identify the parts of a conditional statement. Underline the hypothesis once and the conclusion
twice.
A. If today is Thanksgiving Day, then today is Thursday.
B. A number is a rational number if it is an integer.
C. If an angle measures 40º, then it is an acute angle.
D. A number is divisible by 3 if it is divisible by 6.
Ex: #2 Write a conditional statement from each of the following.
A. An obtuse triangle has exactly one obtuse angle. B. Congruent segments have equal measures.
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Truth value:
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When is a conditional false?
Ex: #3 Determine if each conditional is true. If false, give a counterexample.
A. If this month is August, then next month is
B. If two angles are acute, then they are congruent.
September.
C. If an even number greater than 2 is prime, then
5 + 4 = 8.
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D. If a number is odd, then it is divisible by 3.
Negation:
Chapter 2 Lesson 3: Using Deductive Reasoning to Verify Conjectures
Learning Targets
Success Criteria
LT-3: Apply the Law of Detachment and Law of
Syllogism in logical reasoning.
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Determine whether a conclusion is the result
of deductive or inductive reasoning.
Verify conjectures by using the Law of
Detachment
Verify conjectures by using the Law of
Syllogism.
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Deductive Reasoning:
Advantages of Deductive Reasoning:
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proves that a statement is always true
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follows a systematic pattern of rules
Disadvantages of Deductive Reasoning:
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can be difficult to use
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a skill that can take a lot of practice
Ex: #1 Is each conclusion a result of inductive or deductive reasoning? Explain your answer.
A. There is a myth that you can balance an egg on B. There is a myth that the Great Wall of China is
its end only on the spring equinox. A person was
the only man-made object visible from the Moon.
able to balance an egg on July 8, September 21, and The Great Wall is barely visible in photographs
December 19. Therefore this myth is false.
taken from 180 miles above the Earth. The moon is
about 237,000 miles from the Earth. Therefore the
myth cannot be true.
C. There is a myth that an eelskin walllet will
demagnetize credit cards because the skin of the
electric eels used to make the wallet holds an
electric charge. However, eelskin products are not
made from electric eels. Therefore, the myth
cannot be true.
D. There is a myth that toilets and sinks drain in
opposite directions in the Southern and Northern
Hemispheres. However, if you were to observe
sinks draining in the two hemishperes, you would
see that this myth is false.
Law of Detachment
If p ➝ q is a true statement and p is true,
then q is true.
In your own words:
Law of Syllogism
If p ➝ q and q ➝ r are both true statements, In your own words:
then p ➝ r is a true statement.
Ex: #2 Determine if each conjecture is valid by the Law of Detachment.
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A. Given: In the World Series, if a team wins four games, then the team wins the series.
The Red Sox won four games in the 2004 World Series.
Conjecture: The Red Sox won the 2004 World Series.
B. Given: If the side lengths of a triangle are 5cm, 12cm, and 13cm, then the area of the triangle is
30cm².
The area of ΔPQR is 30cm².
Conjecture: The side lengths of ΔPQR are 5cm, 12, cm, and 13cm.
Ex: #3 Determine if each conjecture is valid by the Law of Syllogism.
Given: If a figure is a kite, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon.
Conjecture: If a figure is a kite, then it is a polygon.
Given: If a number is divisible by 2, then it is even. If a number is even, then it is an integer.
Conjecture: If a number is an integer, then it is divisible by 2.
Ex: #4 Apply the Laws of Deductive Reasoning. Draw a conclusion from the given information.
A. Given: If 2y = 4, then z = -1.
B. Given: If the sum of the measures of two
angles is 180º, then the angles are supplementary.
If x + 3 = 12, then 2y = 4.
If two angles are supplementary, they are not angles
of a triangle.
x + 3 = 12.
m∠A = 135º, and m∠B = 45º.
Chapter 2 Lesson 4: Biconditional Statements and Definitions
Learning Targets
Success Criteria
LT-4: Write and analyze biconditional statements.
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Identify the conditionals within a
biconditional statement.
Write a biconditional statement.
Analyze the truth value of a biconditional
statement.
Write definitions as biconditional
statements.
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Biconditional: A statement that can be written in the form “p if and only if q.”
This means “if p, then q” and “if q, then p.”
p
q and q
p means
p
q
All mathematical definitions can be written as biconditional statements.
An angle is obtuse if and only if its measure is greater than 90º and less than 180º.
A figure is circle if and only if it is the set of all points that are the same distance from a given
point.
When is a biconditional true?
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Definition:
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Polygon:
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Triangle:
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Quadrilateral:
Chapter 2 Lesson 5: Algebraic Proofs
Learning Target: (LT-5) Use properties of equality and congruence to write algebraic proofs.
Learning Targets
Success Criteria
LT-5: Use properties of equality and congruence to
write algebraic proofs.
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Proof:
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Solve an algebraic equation and justify the
steps used.
Solve an equation in geometry and justify
the steps used.
Identify Properties of Equality and
Congruence.
Properties of Equality
For any real number a, b, and c:
Example:
Addition Property
If a = b, then a + c = b + c.
(Subtraction Property)
If a = b, then a – c = b – c.
Multiplication Property
If a = b, then ac = bc.
(Division Property)
If a = b, and c ≠ 0, then
a
c
=
b
.
c
Reflexive Property
a=a
Symmetric Property
If a = b, then b = a.
Transitive Property
If a = b, and b = c, then a = c.
Substitution Property
If a = b, then b can be used to substitute for a
in any expression.
Distributive Property
a(b + c) = ab + ac
The solution to an algebraic equation is a type of proof. The steps must appear in the correct order, and
you must be able to justify each step.
Ex: #1 Solve the algebraic equation and write a justification for each step.
1. 4m – 8 = -12
Given
2.
3.
4.
5
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Ex: #2 Solve the algebraic equation and write a justification for each step.
1. n + 8 = -6
Given
5
2.
3.
4.
5.
Ex: #3 Write a justification for each step.
A.
4 x
-
B.
A
4
3x + 5
6x - 16
B
N
2 x
M
3 x
-
9
C
m ∠A B C = 8 x
O
NO = NM + MO
m∠ABC = m ∠ABD + m∠DBC
4x – 4 = 2x + (3x – 9)
8x = (3x + 5) + (6x – 16)
4x – 4 = 5x – 9
8x = 9x – 11
-4 = x – 9
-x = - 11
5=x
x = 11
Properties of Congruence
Reflexive Property of Congruence:
Symmetric Property of Congruence:
Transitive Property of Congruence:
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D
Each property of congruence has a corresponding property of equality.
Ex: #4 Identify the property that justifies each statement.
A. If EF = GH and GH = JK, then EF = JK
B. AB = 14 then 14 = AB
C. m∠EJM = m∠MJE
AB ≅ CD and CD ≅ EF , then
AB ≅ EF
E.
D. If 2( x – 5 ) = 7, then 2x – 10 = 7
F. ∠QRS ≅ ∠QRS
Chapter 2 Lesson 6: Geometric Proof
Learning Target: (LT-6) Prove geometric theorems by using deductive reasoning.
Learning Targets
Success Criteria
LT-6: Prove geometric theroems by using
deductive reasoning.
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Theorem:
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Two-Column Proof:
Write justifications for steps in a geometric
proof.
Write a two-column proof.
Theorems
Linear Pair Theorem: If two angles form a linear pair, then they are supplementary.
Right Angle Congruence Theorem: All right angles are congruent.
Vertical Angles Theorem: Vertical angles are congruent.
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Ex: #1 Fill in the blanks to complete the two-column proof.
Given: ∠A and ∠B are supplementary and m∠A = 45º.
Prove: m∠B = 135
Statements
1. ∠A and ∠B are supplementary and m∠A = 45º
1. Given
2. m∠A + m∠B = 180
2. ?
3.
45 + m∠B = 180
3. ?
4.
m∠B = 135
4. ?
Ex: #2 Fill in the blanks to complete the two-column proof.
Given: ∠2 ≅ ∠3
Prove: ∠1 and ∠3 are supplementary
Statements
Reasons
Reasons
1. ∠2 ≅ ∠3
1. Given
2. m∠2 = m∠3
2. ?
3. ?
3. Linear Pair Theorem
4. m∠1 + m∠2 = 180º
4. Definition of supplementary ∠'s
5. m∠1 + m∠3 = 180º
5. ?
6. ?
6. Definition of supplementary ∠'s
Ex: #3 Fill in the blanks to complete the two-column proof.
Given:
Prove:
XY
XY ≅
XY
Statements
Reasons
1. ?
1. Given
2. XY = XY
2. ?
3. ?
3. Def. of congruent segments.
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Ex: #4 Write a two-column proof.
Given: ∠1 and ∠2 are supplementary and ∠2 and ∠3 are supplementary
Prove: ∠1 ≅ ∠3
Statements
Reasons
E
Ex: #5 Write a two-column proof.
FD bisects ∠EFC and "
FC bisects ∠DFB
Given: "
Prove: ∠EFD ≅ ∠CFB
Statements
1.
"
FD bisects ∠EFC
"
FC bisects ∠DFB
D
F
C
B
Reasons
1
2.
2
3.
3
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Chapter 2 Extension: Introduction to Symbolic Logic
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Compound Statement:
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Truth Table:
Compound Statements
Term
Words
Symbols
Example
Conjunction
Disjunction
When is a conjunction true?
When is a disjunction true?
Examples: Use p, q, and r to write a compound statement, then find its truth value.
p: the month after April is May
q: the next prime number after 13 is 17
r: half of 19 is 9
p∧q
q∨r
Examples: Construct a truth table for the following compound statements.
u
v
p
q
r
∼u
∼v
∼u∨∼v
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p∧ q
(p∧ q) ∨ r