https://file.aiquickdraw.com/mathbot/file/17270694842385zhnbm02.jpg

Let's go through the exercises you have in the image.

### Directions:
You are asked to apply the **Law of Detachment (LD)**, **Law of Syllogism (LS)**, or conclude **Not Valid (NV)** if neither law applies.

### Exercise Breakdown:

#### 8. If a student gets an A on a final exam, then the student will pass the course.
   - *Premise*: Tracey got an A on the final exam.
   - **Conclusion**: Tracey will pass the course.
   - **Law Used**: **LD** (Law of Detachment) since we have the first condition met (getting an A).

#### 9. If a rectangle has side lengths 3 cm and 4 cm, then it has an area of 12 cm².
   - *Premise*: Rectangle ABCD has an area of 12 cm².
   - **Conclusion**: **Not Valid (NV)**. The fact that the area is 12 cm² does not necessarily imply the specific dimensions of 3 cm and 4 cm.

#### 10. If a whole number ends in 6, then it is divisible by 2.
   - *Premise*: If a whole number ends in 4, then it is divisible by 2.
   - **Conclusion**: **Not Valid (NV)**. Both statements are true, but they are separate; one does not follow from the other.

#### 11. If you are studying botany, then you are studying biology.
   - *Premise*: If you are studying biology, then you are studying a science.
   - **Conclusion**: If you are studying botany, then you are studying a science.
   - **Law Used**: **LS** (Law of Syllogism) as the first conclusion (studying biology) is linked to the second (studying a science).

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### Label as LD, LS, or NV:

#### 12. If Professor Jones wrote a paper about the binomial theorem, then he is familiar with Pascal's triangle.
   - *Premise*: Professor Jones wrote a paper about the binomial theorem.
   - **Conclusion**: Professor Jones is familiar with Pascal's triangle.
   - **Law Used**: **LD** (Law of Detachment) since the first condition is directly met.

#### 13. If you are open-minded, you will listen to both sides of a story.
   - *Premise*: If you listen to both sides of a story, then you will make more intelligent decisions.
   - **Conclusion**: If you are open-minded, then you will make more intelligent decisions.
   - **Law Used**: **LS** (Law of Syllogism) since the first conclusion links to the second.

#### 14. If two base angles of a triangle are congruent, then the triangle has two congruent sides.
   - *Premise*: Triangle ABC has two congruent base angles.
   - **Conclusion**: Triangle ABC has two congruent sides.
   - **Law Used**: **LD** (Law of Detachment) since the condition is directly satisfied.

#### 15. If you use Shining Smile toothpaste, then you will be successful.
   - *Premise*: Jack did not use Shining Smile toothpaste.
   - **Conclusion**: Jack is not successful.
   - **Law Used**: **NV**. This conclusion is invalid because the premise only tells us what happens if you use the toothpaste, but it doesn’t necessarily mean not using it leads to failure.

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Would you like further clarification on any of the exercises, or do you want to dive deeper into these concepts?

### 5 Related Questions:
1. What are the differences between the Law of Detachment and the Law of Syllogism?
2. Can a situation involve both the Law of Detachment and the Law of Syllogism simultaneously?
3. How can you identify when a conclusion is **Not Valid (NV)**?
4. In what real-life scenarios can the Law of Syllogism be applied outside of mathematical logic?
5. How does the Law of Detachment ensure logical consistency?

### Tip:
Always identify the premises first and see if the conditions are satisfied before jumping to a conclusion.

Use the Law of Detachment (LD) and the Law of Syllogism (LS) to make conclusions from the following statements. State which law was used. Write Not Valid (NV) if it is not possible to make a conclusion.

Learn how to apply the Law of Detachment and the Law of Syllogism to make logical conclusions. This example explores conditional logic and is suitable for students in Grades 8-10, covering statements and conclusions using the laws of logic.

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