Download Deductive Reasoning

2.2 Deductive Reasoning
Objective:
• I CAN use inductive and deductive
reasoning to make and defend
conjectures.
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry
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Vocabulary
Conditional Statement:
hypothesis & conclusion
If-Then Form:
If hypothesis, then conclusion.
Ex: All math classes use numbers.
If the class is math,
then it uses numbers.
Negation: opposite (not)
Ex: Math isn’t the best class ever.
Math is the best class ever.
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry
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Vocabulary
Converse:
If conclusion, then hypothesis.
Ex: If an angle is 90, then it’s right.
If an angle is right, then it’s 90.
Inverse:
If NOT hypothesis, then NOT conclusion.
Ex: If an angle is 90, then it’s right.
If an angle is not 90, then it’s not right.
Contrapositive:
If NOT conclusion, then NOT hypothesis.
Ex: If an angle is 90, then it’s right.
If an angle is not right, then it’s not 90.
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry
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Vocabulary
Perpendicular Lines:
If 2 lines intersect to form a right
angle, then they are perpendicular lines.
If 2 lines are perpendicular, then they
intersect to form a right angle.
2 lines are perpendicular if and only if
they intersect to form a right angle.
Biconditional Statement:
statement and its converse are true
“if and only if”
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry
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Vocabulary
Deductive Reasoning:
proof using definitions,
postulates, theorems
Law of Detachment:
If the hypothesis is true,
then the conclusion is true.
Law of Syllogism:
If p, then q.
If q, then r.
If p, then r.
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry
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Example #1
What conclusion can you
draw based on the two
conditional statements
below?
If a  4, then 4a  16.
If 3a  12, then a  4.
Law of Syllogism
If 3a  12, then a  4.
If a  4, then 4a  16.
If 3a  12, then 4a  16.
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry
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Example #2
Solve the equation for x. Give a reason for each step in the process.

 

3 2x  1  2 2x  1  7  42  5x

  
52x  1 7  42  5x
3 2x  1  2 2x  1  7  42  5x
10x  5  7  42  5x
10x  12  42  5x
10x  30  5x
15x  30
x2
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry
Original Equation
Combine Like Terms
Distribute
Combine Like Terms
Subtract 12
Add 5x
Divide by 15
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Example #3
In each diagram ray AC bisects obtuse angle BAD.
Classify each angle as acute, right, or obtuse.
Then write and prove a conjecture about the angles formed.
A
C
B
C
B
D
B
C
A
D
mBAD  120
120
mCAD 
2
mCAD  60
Acute
mBAD  92
92
mCAD 
2
mCAD  46
Acute
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry
A
D
mBAD  158
158
mCAD 
2
mCAD  79
Acute
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Example #3
In each diagram ray AC bisects obtuse angle BAD.
Classify each angle as acute, right, or obtuse.
Then write and prove a conjecture about the angles formed.
Conjecture:
If an obtuse angle is bisected,
acute
then the two newly formed congruent angles are _________.
Statements
Reasons
1. Given
1. mBAD  180


1
1
2.
mBAD  180  90 2. Definition of Angle Bisector
2
2
1
3. Definition of Acute Angle
3.
mBAD is acute.
2
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry
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