This document provides information on deductive reasoning and the laws of detachment and syllogism. It defines deductive reasoning as using facts and rules to reach a logical conclusion. The law of detachment states that if p implies q and p is true, then q must be true. The law of syllogism states that if p implies q and q implies r, then p implies r. The document includes examples applying these concepts and determining whether conclusions are valid deductive reasoning or not.

1. Section 2-3
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. Deductive Reasoning: Using facts, rules, definitions,
and properties to reach a logical conclusion from
given statements
2. Valid:
3. Law of Detachment:

5. Vocabulary
1. Deductive Reasoning: Using facts, rules, definitions,
and properties to reach a logical conclusion from
given statements
2. Valid: A logically correct statement or argument;
follows the pattern of a Law
3. Law of Detachment:

6. Vocabulary
1. Deductive Reasoning: Using facts, rules, definitions,
and properties to reach a logical conclusion from
given statements
2. Valid: A logically correct statement or argument;
follows the pattern of a Law
3. Law of Detachment: If p q is a true statement and p
is true, then q is true.

7. Vocabulary
1. Deductive Reasoning: Using facts, rules, definitions,
and properties to reach a logical conclusion from
given statements
2. Valid: A logically correct statement or argument;
follows the pattern of a Law
3. Law of Detachment: If p q is a true statement and p
is true, then q is true.
Example: Given: If an iPhone has no charge, then it
won’t turn on.
Statement: Matt’s iPhone has no charge.
Conclusion: Matt’s iPhone won’t turn on.

8. Vocabulary
4. Law of Syllogism:

9. Vocabulary
4. Law of Syllogism:If p q and q r are true
statements, then p r is a true statement

10. Vocabulary
4. Law of Syllogism:
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.
If p q and q r are true
statements, then p r is a true statement

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. Fails the Laws of Detachment
and Syllogism. 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.
Valid. Passes the Law of
Detachment. 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
Square
s
Valid. Passes the Law of
Detachment. 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