1. LESSON 2–4
Deductive Reasoning

2. Five-Minute Check (over Lesson 2–3) TEKS Then/Now New Vocabulary
Example 1: Real-World Example: Inductive and Deductive Reasoning
Key Concept: Law of Detachment
Example 2: Law of Detachment
Example 3: Judge Conclusions Using Venn Diagrams
Key Concept: Law of Syllogism
Example 4: Law of Syllogism
Example 5: Apply Laws of Deductive Reasoning
Lesson Menu

3. Identify the hypothesis and conclusion. If 6x – 5 = 19, then x = 4.
A. Hypothesis: 6x – 5 = 19 Conclusion: x = 4
B. Hypothesis: 6x – 5 = 19 Conclusion: x  4
C. Hypothesis: x = 4 Conclusion: 6x – 5 = 19
D. Hypothesis: 6x – 5  19 Conclusion: x = 4
5-Minute Check 1

4. Identify the hypothesis and conclusion
Identify the hypothesis and conclusion. A polygon is a hexagon if it has six sides.
A. Hypothesis: The polygon is not a hexagon. Conclusion: A polygon has six sides.
B. Hypothesis: The polygon is a hexagon. Conclusion: A polygon has six sides.
C. Hypothesis: A polygon has six sides. Conclusion: The polygon is a hexagon.
D. Hypothesis: A polygon does not have six sides. Conclusion: The polygon is a hexagon.
5-Minute Check 2

5. A. If you exercise, then you will be healthier.
Which choice shows the statement in if-then form? Exercise makes you healthier.
A. If you exercise, then you will be healthier.
B. If you are healthy, then you exercise.
C. If you don’t exercise, you won’t be healthier.
D. If you don’t exercise, then you won’t be healthier.
5-Minute Check 3

6. A. If a figure has 4 sides, then it is a square.
Which choice shows the statement in if-then form? Squares have 4 sides.
A. If a figure has 4 sides, then it is a square.
B. If a figure is a square, then it has 4 sides.
C. A figure is not a square if it does not have 4 sides.
D. If a figure is not a square, then it does not have 4 sides.
5-Minute Check 4

7. A. the negation of the statement B. the inverse of the statement
Given the statement, “If you live in Miami, then you live in Florida,” which of the following is true?
A. the negation of the statement
B. the inverse of the statement
C. the converse of the statement
D. the contrapositive of the statement
5-Minute Check 5

8. Mathematical Processes G.1(F), G.1(G)
Targeted TEKS
Preparation for G.4(B) Identify and determine the validity of
the converse, inverse, and contrapositive of a conditional statement and recognize the connection between biconditional statement and a true conditional statement with a true
converse.
Mathematical Processes
G.1(F), G.1(G)
TEKS

9. You used inductive reasoning to analyze patterns and make conjectures.
Use the Law of Detachment.
Use the Law of Syllogism.
Then/Now

10. deductive reasoning valid Law of Detachment Law of Syllogism
Vocabulary

11. Inductive and Deductive Reasoning
A. WEATHER Determine whether the conclusion is based on inductive or deductive reasoning.
In Miguel’s town, the month of April has had the most rain for the past 5 years. He thinks that April will have the most rain this year.
Answer: Miguel’s conclusion is based on a pattern of observation, so he is using inductive reasoning.
Example 1

12. Inductive and Deductive Reasoning
B. WEATHER Determine whether the conclusion is based on inductive or deductive reasoning.
Sandra 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. Sandra knows it will be cloudy tonight, so she believes it will not be cold tomorrow morning.
Answer: Sandra is using facts that she has learned about clouds and temperature, so she is using deductive reasoning.
Example 1

13. A. Determine whether the conclusion is based on inductive or deductive reasoning. Macy’s mother orders pizza for dinner every Thursday. Today is Thursday. Macy concludes that she will have pizza for dinner tonight.
A. inductive
B. deductive
Example 1

14. B. Determine whether the conclusion is based on inductive or deductive reasoning. The library charges $0.25 per day for overdue books. Kyle returns a book that is 3 days overdue. Kyle concludes that he will be charged a $0.75 fine.
A. inductive
B. deductive
Example 1

15. Concept

16. q: The figure is a parallelogram.
Law of Detachment
Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. Given: If a figure is a square, then it is a parallelogram The figure is a parallelogram. Conclusion: The figure is a square.
Step 1 Identify the hypothesis p and the conclusion q of the true conditional.
p: A figure is a square.
q: The figure is a parallelogram.
Example 2

17. Step 2 Analyze the conclusion.
Law of Detachment
Step 2 Analyze the conclusion.
The given statement the figure is a parallelogram satisfies the conclusion q of the true conditional. However, knowing that a conditional statement and its conclusion are true does not make the hypothesis true. The figure could be a rectangle.
Answer: The conclusion is invalid.
Example 2

18. The following is a true conditional
The following is a true conditional. Determine whether the conclusion is valid based on the given information. Given: If a polygon is a convex quadrilateral, then the sum of the interior angles is ABCD is a convex quadrilateral. Conclusion: The sum of the interior angles of ABCD is 360.
A. valid
B. not valid
C. cannot be determined
Example 2

19. Judge Conclusions Using Venn Diagrams
Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain your reasoning using a Venn diagram. Given: If a triangle is equilateral, then it is an acute triangle. The triangle is equilateral. Conclusion: The triangle is acute.
Analyze Draw a Venn diagram. According to the conditional, if a triangle is equilateral, then it is acute. Draw a circle for acute triangles. Draw a circle for equilateral triangles inside the circle for acute triangles.
Example 3

20. Formulate We know that the given triangle is equilateral.
Judge Conclusions Using Venn Diagrams
Formulate We know that the given triangle is equilateral.
Determine Since all equilateral triangles are acute, the conclusion is valid.
Answer: From the information given, all equilateral triangles are acute, so the conclusion is valid.
Justify No matter where you draw a dot for an equilateral triangle, it will always be inside the circle for acute triangles. Therefore, the conclusion is valid.
Example 3

21. Judge Conclusions Using Venn Diagrams
Evaluate Venn diagrams are a great way to visualize groups and subgroups. By representing the conditional as a Venn diagram, it becomes clear that all equilateral triangles are acute triangles. Our conclusion is valid.
Example 3

22. Determine whether the conclusion is valid based on the given information. If not, write invalid. Use a Venn diagram to help you. Given: If a figure is a square, then it has 4 right angles. A figure has 4 right angles. Conclusion: The figure is a square.
A. valid
B. invalid
Example 3

23. Concept

24. Determine which statement follows logically from the given statements.
Law of Syllogism
Determine which statement follows logically from the given statements.
(1) If Jamal finishes his homework, he will go out with his friends. (2) If Jamal goes out with his friends, he will go to the movies.
A If Jamal goes out with his friends, then he finishes his homework. B If Jamal finishes his homework, he will go to the movies. C If Jamal does not go to the movies, he does not go out with his friends. D There is no valid conclusion.
Example 4

25. p: Jamal finishes his homework. q: He goes out with his friends.
Law of Syllogism
Read the Item
Let p, q, and r represent the parts of the given conditional statements.
p: Jamal finishes his homework.
q: He goes out with his friends.
r: He goes to the movies.
Solve the Item
Analyze the logic of the given conditional statements using symbols.
Statement (1): p → q Statement (2): q → r
Example 4

26. Answer: The correct choice is B.
Law of Syllogism
Both statements are considered true. By the Law of Syllogism, p → r is also true. Write the statements for p → r. If Jamal finishes his homework, then he will go to the movies.
Answer: The correct choice is B.
Example 4

27. A. If you ride a bus, then you get out of bed.
Determine which statement follows logically from the given statements. (1) If your alarm clock goes off in the morning, then you will get out of bed. (2) If you ride a bus, then you go to work.
A. If you ride a bus, then you get out of bed.
B. If your alarm clock goes off, then you go to work.
C. If you go to work, then you get of out bed.
D. There is no valid conclusion.
Example 4

28. p: It snows more than 5 inches. q: School will be closed.
Apply Laws of Deductive Reasoning
Draw a valid conclusion from the given statements, if possible. Then state whether your conclusion was drawn using the Law of Detachment or the Law of Syllogism. If no valid conclusion can be drawn, write no conclusion and explain your reasoning. Given: If it snows more than 5 inches, school will be closed. It snows 7 inches.
p: It snows more than 5 inches.
q: School will be closed.
Answer: Since It snows 7 inches satisfies the hypothesis, p is true. By the Law of Detachment, a valid conclusion is School is closed.
Example 5

29. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, select invalid. (1) If a children’s movie is playing on Saturday, Janine will take her little sister Jill to the movie. (2) Janine always buys Jill popcorn at the movies. (3) If a children’s movie is playing on Saturday, Jill will get popcorn.
A. Law of Detachment
B. Law of Syllogism
C. invalid
Example 5

30. LESSON 2–4
Deductive Reasoning