1. 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

2. 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

3. 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.
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4. Vocabulary Biconditional Statement:
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”
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Chapter 2: Reasoning in Geometry 4

5. 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.
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Chapter 2: Reasoning in Geometry 5

6. Example #1 What conclusion can you draw based on the two conditional statements below?
Law of Syllogism
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry

7. Example #2
Solve the equation for x. Give a reason for each step in the process.
Original Equation
Combine Like Terms
Distribute
Combine Like Terms
Subtract 12
Add 5x
Divide by 15
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Chapter 2: Reasoning in Geometry 7

8. Example #3 Acute Acute Acute
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. D A C B D C A B A B D C
Acute
Acute
Acute
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Chapter 2: Reasoning in Geometry 8

9. If an obtuse angle is bisected,
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,
then the two newly formed congruent angles are _________.
acute
Statements
Reasons
1. Given
2. Definition of Angle Bisector
3. Definition of Acute Angle
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Chapter 2: Reasoning in Geometry 9