Chapter 2 Notes

Report
Chapter 2
Reasoning and Proof
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Objectives
1
2
• Recognize conditional statements
• To write converses of conditional
statements
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Key Concepts
A conditional statement is _______________________.
Every conditional statement has two parts.
•The part following the “If” is the ____________.
•The part following the “then” is the __________.
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Identify the hypothesis and the conclusion:
If two lines are parallel, then the lines are coplanar.
Hypothesis:
Conclusion:
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Write the statement as a conditional:
An acute angle measures less than 90º.
The subject of the sentence is “An acute angle.”
The hypothesis is “An angle is acute.” The first part of the
conditional is “If an angle is acute.”
The verb and object of the sentence are “measures less than
90°.”
The conclusion is “It measures less than 90°.” The second
part of the conditional is “then it measures less than 90°.”
“If an angle is acute, then it measures less than 90°.”
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Key Concepts
A _________________ is a case in which the hypothesis is
true and the conclusion is false.
To show that a conditional is false, you need to find
only one counterexample.
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Find a counterexample to show that this
conditional is false: If x2 ≥ 0, then x ≥ 0.
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Use the Venn diagram below. What does it
mean to be inside the large circle but outside the
small circle?
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Key Concepts
In the converse of a conditional statement the hypothesis
and conclusion are switched.
Conditional: If p, then q
pq
Converse:
q p
Chapter 2: Reasoning and Proof
If q, then p
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Lesson 2 – 1
Conditional Statements
The Mad Hatter states: “You might just as well
say that ‘I see what I eat’ is the same thing as ‘I
eat what I see’!” Provide a counterexample to
show that one of the Mad Hatter’s statements is
false.
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Write the converse of the conditional:
If x = 9, then x + 3 = 12.
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Write the converse of the conditional, and
determine the truth value of each:
If a2 = 25, a = 5.
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Lesson Quiz
Use the following conditional for Exercises 1–3.
If a circle’s radius is 2 m, then its diameter is 4 m.
1. Identify the hypothesis and conclusion.
2. Write the converse.
If a circle’s diameter is 4 m, then its radius is 2 m.
3. Determine the truth value of the conditional and its converse.
Both are true.
Show that each conditional is false by finding a counterexample.
4. If lines do not intersect, then they are parallel.
skew lines
5. All numbers containing the digit 0 are divisible by 10.
Sample: 105
Chapter 2: Reasoning and Proof
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Lesson 2 – 1
Conditional Statements
Homework
Pages 83-85
1, 2 -14 even, 28 -32 even, 40, 48-52 even
Chapter 2: Reasoning and Proof
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Lesson 2 – 2
Biconditionals and Definitions
Objectives
1
2
•To write biconditionals
•To recognize good definitions
Chapter 2: Reasoning and Proof
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Lesson 2 – 2
Biconditionals and Definitions
Key Concepts
If a conditional and its converse are both true, the
statement is said to be ________________.
Biconditional statements are often stated in the form “…if
and only if …”
IFF – short for if and only if
 - symbol for if and only if
An angle is a right angle if and only if it measures 90°.
Chapter 2: Reasoning and Proof
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Lesson 2 – 2
Biconditionals and Definitions
Consider this true conditional statement. Write
its converse. If the converse is also true,
combine the statements as a biconditional.
Chapter 2: Reasoning and Proof
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Lesson 2 – 2
Biconditionals and Definitions
Write the two statements that form this
biconditional.
Biconditional: Lines are skew if and only if they are
noncoplanar.
Conditional:
Converse:
Chapter 2: Reasoning and Proof
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Lesson 2 – 2
Biconditionals and Definitions
Key Concepts
The Reversibility Test
The reverse (converse) of a definition must be true.
If the reverse of a statement is false, then the
statement is not a good definition.
A good definition is reversible. That means that you
can write a good definition as a true biconditional.
Chapter 2: Reasoning and Proof
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Lesson 2 – 2
Biconditionals and Definitions
Show that this definition of triangle is reversible.
Then write it as a true biconditional.
Definition: A triangle is a polygon with exactly three sides.
Chapter 2: Reasoning and Proof
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Lesson 2 – 2
Biconditionals and Definitions
Is the following statement a good definition? Explain.
An apple is a fruit that contains seeds.
Chapter 2: Reasoning and Proof
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Lesson 2 – 2
Biconditionals and Definitions
Lesson Quiz
1. Write the converse of the statement.
If it rains, then the car gets wet.
2. Write the statement above and its converse as a biconditional.
3. Write the two conditional statements that make up the
biconditional.
An angle is a straight angle if and only if it measures 180°.
Is each statement a good definition? If not, find a counterexample.
4. The midpoint of a line segment is the point that divides the
segment into two congruent segments.
5. A line segment is a part of a line.
Chapter 2: Reasoning and Proof
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Lesson 2 – 2
Biconditionals and Definitions
Homework
Page 90
2-26 even
Chapter 2: Reasoning and Proof
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Lesson 2 – 3
Deductive Reasoning
Objectives
1
2
•To use the Law of Detachment
•To use the Law of Syllogism
Chapter 2: Reasoning and Proof
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Lesson 2 – 3
Deductive Reasoning
Key Concepts
Deductive Reasoning (or logical reasoning) is
• If the given statements are true, deductive
reasoning produces a true conclusion.
Chapter 2: Reasoning and Proof
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Lesson 2 – 3
Deductive Reasoning
Key Concepts
Law of Detachment
If a conditional is true and its hypothesis is true, then its
conclusion is true.
In symbolic form:
If p  q is a true statement and p is true, then q is true.
Chapter 2: Reasoning and Proof
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Lesson 2 – 3
Deductive Reasoning
A gardener knows that if it rains, the garden
will be watered. It is raining. What conclusion
can he make?
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Lesson 2 – 3
Deductive Reasoning
For the given statements, what can you
conclude?
Given: If an angle acute, then its measure is less
than 90°.
A is acute.
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Lesson 2 – 3
Deductive Reasoning
Does the following argument illustrate the
Law of Detachment?
Given: If you make a field goal in basketball, you score two
points.
Jenna scored two points in basketball.
Chapter 2: Reasoning and Proof
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Lesson 2 – 3
Deductive Reasoning
Key Concepts
Law of Syllogism
If p  q and q  r are true statements, then p  r is
a true statement.
Chapter 2: Reasoning and Proof
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Lesson 2 – 3
Deductive Reasoning
Use the Law of Syllogism to draw a conclusion from
the following true statements:
If a quadrilateral is a square, then it contains four right angles.
If a quadrilateral contains four right angles, then it is a rectangle.
Chapter 2: Reasoning and Proof
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Lesson 2 – 3
Deductive Reasoning
Use the Laws of Detachment and Syllogism to
draw a possible conclusion.
If the circus is in town, then there are tents at the fairground.
If there are tents at the fairground, then Paul is working as a
night watchman. The circus is in town.
Chapter 2: Reasoning and Proof
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Lesson 2 – 3
Deductive Reasoning
Lesson Quiz
Use the three statements below.
A. If games are canceled, then Maria reads a book.
B. If it snows, then games are canceled.
C. It is snowing.
1. Using only statements A and B, what can you conclude?
2. Using only statements B and C, what can you conclude?
3. Using statements A, B, and C, what can you conclude?
4. Suppose both statement B and “games are canceled” are true. Can
you conclude that statement C is true? Explain.
Chapter 2: Reasoning and Proof
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Lesson 2 – 3
Deductive Reasoning
Homework
Pages 96 – 97
1 – 21
Chapter 2: Reasoning and Proof
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Lesson 2 – 4
Reasoning in Algebra
Objectives
1
•To connect reasoning in
algebra and geometry
Chapter 2: Reasoning and Proof
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Lesson 2 – 4
Reasoning in Algebra
Key Concepts
Properties of Equality
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 a · c = b · c.
Division Property
a b
If a = b and c ≠ 0, then 
c c
Chapter 2: Reasoning and Proof
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Lesson 2 – 4
Reasoning in Algebra
Key Concepts
Properties of Equality continued
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 replace a in any expression.
Chapter 2: Reasoning and Proof
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Lesson 2 – 4
Reasoning in Algebra
Key Concepts
The Distributive Property
a(b + c) = ab + ac
Chapter 2: Reasoning and Proof
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Lesson 2 – 4
Reasoning in Algebra
Justify each step used to solve
5x – 12 = 32 + x for x.
Given: 5x – 12 = 32 + x
1. 5x = 44 + x
2. 4x = 44
3. x = 11
Chapter 2: Reasoning and Proof
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Lesson 2 – 4
Reasoning in Algebra
Suppose that points A, B, and C are collinear
with point B between points A and C.
Solve for x if AC = 21, BC = 15 – x, and
AB = 4 + 2x. Justify each step.
AB + BC = AC
(4 + 2x) + (15 – x) = 21
19 + x = 21
x= 2
Chapter 2: Reasoning and Proof
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Lesson 2 – 4
Reasoning in Algebra
Key Concepts
Properties of Congruence
Reflexive Property
AB ≅ AB
∠A ≅ ∠A
Symmetric Property
If AB ≅ CD, then CD ≅ AB
If ∠A ≅ ∠B, then ∠B ≅ ∠A.
Transitive Property
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.
Chapter 2: Reasoning and Proof
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Lesson 2 – 4
Reasoning in Algebra
Name the property that justifies each
statement.
a. If x = y and y + 4 = 3x, then x + 4 = 3x.
b. If x + 4 = 3x, then 4 = 2x.
Chapter 2: Reasoning and Proof
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Lesson 2 – 4
Reasoning in Algebra
(continued)
c. If ∠P ≅ ∠Q, ∠Q ≅ ∠R, and ∠R ≅ ∠S, then ∠P ≅ ∠S
Chapter 2: Reasoning and Proof
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Lesson 2 – 4
Reasoning in Algebra
Lesson Quiz
Name the justification for each statement.
1. ab = ab
2. If mABC + 40 = 85, then mABC = 45.
3. If k = m and k + w = 12, then m + w = 12.
4. If B is a point in the interior of AOC, then mAOB + mBOC =
mAOC.
5. Fill in the missing information.
Given: AC = 36
a. AB + BC = AC
i. ?
b. 3x + 2x + 1 = 36
ii. ?
c. ?
iii. ?
d. 5x = 35
iv. ?
e. x = ?
v. ?
Chapter 2: Reasoning and Proof
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Lesson 2 – 4
Reasoning in Algebra
Homework
Pages 105 – 107
2 – 22 even, 28, 31
Chapter 2: Reasoning and Proof
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Lesson 2 – 5
Proving Angles Congruent
Objectives
1
•To prove and apply
theorems about angles
Chapter 2: Reasoning and Proof
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Lesson 2 – 5
Proving Angles Congruent
Key Concepts
A __________ is a convincing argument that
uses deductive reasoning.
A statement that you prove true is a ____________.
A paragraph proof is written as sentences in a
paragraph.
• “Given”: lists what you know from the
hypothesis of the theorem
• “Prove”: the conclusion of the theorem
• Diagram: records the given information visually
Chapter 2: Reasoning and Proof
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Lesson 2 – 5
Proving Angles Congruent
Key Concepts
Theorem: Vertical angles are congruent.
Given: 1 and 2 are vertical angles
Prove: 1  2
Proof: By the Angle Addition Postulate,
m1 + m 3 = 180 and m 2 + m 3 = 180.
By substitution, m 1 + m 3 = m 2 + m 3.
Subtract m 3 from each side. You get
m 1 = m 2, or  1   2.
Chapter 2: Reasoning and Proof
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Lesson 2 – 5
Proving Angles Congruent
Find the value of x.
Chapter 2: Reasoning and Proof
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Lesson 2 – 5
Proving Angles Congruent
Key Concepts
Theorem: If two angles are supplements of the same
angle, then the two angles are congruent.
Given: 1 and 2 are supplementary
3 and 2 are supplementary
Prove: 1  3
Proof: By the definition of supplementary angles,
m1 + m 2 = 180 and m 3 + m 2 = 180.
By substitution, m 1 + m 2 = m 3 + m 2.
Subtract m 2 from each side. You get m 1 = m 3,
or  1   3.
Chapter 2: Reasoning and Proof
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Lesson 2 – 5
Proving Angles Congruent
Key Concepts
Theorem: If two angles are supplements of congruent
angles, then the two angles are congruent.
Given: 1 and 2 are supplementary
3 and 4 are supplementary
2  4
Prove: 1  3
Proof: By the definition of supplementary angles,
m1 + m 2 = 180 and m 3 + m 4 = 180.
By substitution, m 1 + m 2 = m 3 + m 4.
Since 2  4, by the definition of congruence m 2 = m 4.
By substitution m 1 + m 4 = m 3 + m 4.
Subtract m 4 from each side. You get m 1 = m 3, or
 1   3.
Chapter 2: Reasoning and Proof
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Lesson 2 – 5
Proving Angles Congruent
Key Concepts
Theorem: If two angles are complements of the same
angle (or of congruent angles), then the two
angles are congruent.
Theorem: All right angles are congruent.
Theorem: If two angles are congruent and
supplementary, then each is a right angle.
Chapter 2: Reasoning and Proof
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Lesson 2 – 5
Proving Angles Congruent
Write a paragraph proof using the given,
what you are to prove, and the diagram.
Given: WX = YZ
Prove: WY = XZ
Chapter 2: Reasoning and Proof
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●
●
●
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Lesson 2 – 5
Proving Angles Congruent
Lesson Quiz
Use the diagram and mABS = 3x + 6 and
mRBC = 5x – 20 for Exercises 1–4.
1. Find x.
2. Find mABS.
3. Find mSBC.
4. Without using the Vertical Angle Theorem, what
theorem can you use to prove that ABR  SBC?
Chapter 2: Reasoning and Proof
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Lesson 2 – 5
Proving Angles Congruent
Homework
Pages 112 – 114
1 – 7, 8 – 18 even,
21, 23 – 28
Chapter 2: Reasoning and Proof
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