This document provides an overview of deductive reasoning, including the Law of Detachment and Law of Syllogism. It defines deductive reasoning as using facts, rules, and definitions to reach a logical conclusion. The Law of Detachment states that if p implies q is true and p is true, then q must be true. The Law of Syllogism states that if p implies q and q implies r are both true, then p implies r must be true. The document provides examples and problems applying these laws of deductive reasoning.

1. Section 2-4
Deductive Reasoning

2. Essential Questions
How do you use the Law of Detachment?
How do you use the Law of Syllogism?

3. Vocabulary
1. Deductive Reasoning:
2. Valid:
3. Law of Detachment:

4. Vocabulary
1. D e d u c t i v e R e a s o n i n g : Using facts, rules, definitions,
and properties to reach a logical conclusion from
given statements
2. Valid:
3. Law of Detachment:

5. Vocabulary
1. D e d u c t i v e R e a s o n i n g : Using facts, rules, definitions,
and properties to reach a logical conclusion from
given statements
2. V a l i d : A logically correct statement or argument;
follows the pattern of a Law
3. Law of Detachment:

6. Vocabulary
1. D e d u c t i v e R e a s o n i n g : Using facts, rules, definitions,
and properties to reach a logical conclusion from
given statements
2. V a l i d : A logically correct statement or argument;
follows the pattern of a Law
3. L a w o f D e t a c h m e n t : If p q is a true statement and p
is true, then q is true.

7. Vocabulary
1. D e d u c t i v e R e a s o n i n g : Using facts, rules, definitions,
and properties to reach a logical conclusion from
given statements
2. V a l i d : A logically correct statement or argument;
follows the pattern of a Law
3. L a w o f D e t a c h m e n t : If p q is a true statement and p
is true, then q is true.
Example: Given: If a computer has no battery, then it
won’t turn on.
Statement: Matt’s computer has no battery.
Conclusion: Matt’s computer won’t turn on.

8. Vocabulary
4. Law of Syllogism:

9. Vocabulary
4. L a w o f S y l l o g i s m : If p q and q r are true
statements, then p r is a true statement

10. Vocabulary
4. Law of Syllogism:
If p q and q r are true
statements, then p r is a true statement
Example: Given: If you buy a ticket, then you can get in
the arena. If you get in the arena, then you can watch
the game.
Conclusion: If you buy a ticket, then you can watch the
game.

11. Example 1
Determine whether each conclusion is based on
inductive or deductive reasoning. Explain your
choice.
a. In Matt Mitarnowski’s town, the month of March
had the most rain for the past 6 years. He thinks
March will have the most rain again this year.

12. Example 1
Determine whether each conclusion is based on
inductive or deductive reasoning. Explain your
choice.
a. In Matt Mitarnowski’s town, the month of March
had the most rain for the past 6 years. He thinks
March will have the most rain again this year.
This is inductive reasoning, as it is using past
observations to come to the conclusion.

13. Example 1
Determine whether each conclusion is based on
inductive or deductive reasoning. Explain your
choice.
b. Maggie Brann learned that if it is cloudy at night
it will not be as cold in the morning than if there are
no clouds at night. Maggie knows it will be cloudy
tonight, so she believes it will not be cold tomorrow.

14. Example 1
Determine whether each conclusion is based on
inductive or deductive reasoning. Explain your
choice.
b. Maggie Brann learned that if it is cloudy at night
it will not be as cold in the morning than if there are
no clouds at night. Maggie knows it will be cloudy
tonight, so she believes it will not be cold tomorrow.
This is deductive reasoning, as Maggie is using
learned information and facts to reach the
conclusion.

15. Example 2
Determine whether the conclusion is valid based on
the given information. If not, write invalid. Explain
your reasoning.
Given: If a figure is a right triangle, then it has two
acute angles.
Statement: A figure has two acute angles.
Conclusion: The figure is a right triangle.

16. Example 2
Determine whether the conclusion is valid based on
the given information. If not, write invalid. Explain
your reasoning.
Given: If a figure is a right triangle, then it has two
acute angles.
Statement: A figure has two acute angles.
Conclusion: The figure is a right triangle.
Invalid conclusion. A quadrilateral can have two
acute angles, but it is not a right triangle.

17. Example 3
Determine whether the conclusion is valid based on
the given information. If not, write invalid. Explain
your reasoning with a Venn diagram.
Given: If a figure is a square, it is also a rectangle.
Statement: The figure is a square.
Conclusion: The square is also a rectangle.

18. Example 3
Determine whether the conclusion is valid based on
the given information. If not, write invalid. Explain
your reasoning with a Venn diagram.
Given: If a figure is a square, it is also a rectangle.
Statement: The figure is a square.
Conclusion: The square is also a rectangle.
A square has four right angles
and opposite parallel sides, so
it is also a rectangle

19. Example 3
Determine whether the conclusion is valid based on
the given information. If not, write invalid. Explain
your reasoning with a Venn diagram.
Given: If a figure is a square, it is also a rectangle.
Statement: The figure is a square.
Conclusion: The square is also a rectangle.
Rectangles
Squares
A square has four right angles
and opposite parallel sides, so
it is also a rectangle

20. Example 4
Draw a valid conclusion from the given statements, if
possible. If no valid conclusion can be drawn, write
no conclusion and explain your reasoning.
Given: An angle bisector divides an angle into two
congruent angles. If two angles are congruent, then
their measures are equal.
Statement: BD bisects ∠ABC.

21. Example 4
Draw a valid conclusion from the given statements, if
possible. If no valid conclusion can be drawn, write
no conclusion and explain your reasoning.
Given: An angle bisector divides an angle into two
congruent angles. If two angles are congruent, then
their measures are equal.
Statement: BD bisects ∠ABC.
Conclusion: m∠ABD = m∠DBC

22. Problem Set

23. Problem Set
p. 119 #1-39 odd, 45
“Success does not consist in never making blunders,
but in never making the same one a second time.”
- Josh Billings