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Let's take a look at the questions in the image you uploaded. Here is the text as best I can extract:

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7. **Adjacent angles share one common ray.**  
In Exercises 6 and 7, use the Law of Detachment to determine what you can conclude from the given information, if possible.

8. **If a polygon has three sides, then it is a triangle. If a triangle has two congruent sides, then it is an isosceles triangle.**  
In Exercises 8 and 9, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements, if possible.

9. **If it is Tuesday, then you mow the grass. If you mow the grass, then you water the flowers.**  
In Exercises 10 and 11, decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning.

10. **All mammals have hair. Cats are mammals. So, all cats have hair.**

11. **Each time you go to school, you walk. You went to school today, so you walked.**

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### Solutions and Reasoning:

1. **Exercise 7 (Law of Detachment)**  
- **Statement:** Adjacent angles share one common ray.
- **Conclusion:** Based on the Law of Detachment, if we know that two angles are adjacent, we can conclude that they must share one common ray.

2. **Exercise 8 (Law of Syllogism)**  
- **Statements:**
  - If a polygon has three sides, then it is a triangle.
  - If a triangle has two congruent sides, then it is an isosceles triangle.
- **New conditional statement:** Using the Law of Syllogism, we can combine these into a new conditional:  
  **"If a polygon has three sides and two congruent sides, then it is an isosceles triangle."**

3. **Exercise 9 (Law of Syllogism)**  
- **Statements:**
  - If it is Tuesday, then you mow the grass.
  - If you mow the grass, then you water the flowers.
- **New conditional statement:**  
  **"If it is Tuesday, then you water the flowers."**

4. **Exercise 10 (Inductive vs. Deductive Reasoning)**  
- **Statement:**  
  "All mammals have hair. Cats are mammals. So, all cats have hair."
- **Type of reasoning:** This is **deductive reasoning**, as the conclusion follows logically from the premises.

5. **Exercise 11 (Inductive vs. Deductive Reasoning)**  
- **Statement:**  
  "Each time you go to school, you walk. You went to school today, so you walked."
- **Type of reasoning:** This is also **deductive reasoning**, because it follows a set pattern or rule.

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Do you have any questions or would you like further explanation on any of these? 

Here are 5 related questions to extend your understanding:

1. How does the Law of Syllogism work when connecting two true statements?
2. What is the difference between inductive and deductive reasoning?
3. Can the Law of Detachment always be applied when given a conditional statement?
4. What makes a triangle isosceles, and how can you identify it in a problem?
5. How can you find counterexamples to test conditional statements?

### Tip:
When solving logic problems, it helps to carefully break down the premises and use the right logical law, whether it's detachment or syllogism.

7. Adjacent angles share one common ray. 8. If a polygon has three sides, then it is a triangle. If a triangle has two congruent sides, then it is an isosceles triangle. 9. If it is Tuesday, then you mow the grass. If you mow the grass, then you water the flowers. 10. All mammals have hair. Cats are mammals. So, all cats have hair. 11. Each time you go to school, you walk. You went to school today, so you walked.

This exercise covers logical reasoning and geometry principles such as the Law of Detachment, Law of Syllogism, and inductive vs. deductive reasoning. Students will learn how to apply these concepts to adjacent angles, isosceles triangles, and conditional statements. Suitable for Grades 9-12.

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