https://file.aiquickdraw.com/mathbot/file/1728269138599v3wurjyq.jpg

Let's analyze the conjectures from the image one by one:

### 7. "All birds can fly."
- **False.** Not all birds can fly. A counterexample is the penguin, which is a bird but cannot fly.

### 8. "All vegetables are green."
- **False.** Vegetables come in various colors. A counterexample is a carrot, which is orange.

### 9. "All USA state names have at least 2 syllables."
- **False.** Some state names have only one syllable. A counterexample is Maine.

### 10. "The product of two even numbers is always divisible by 4."
- **True.** The product of any two even numbers is divisible by 4. For example, multiplying 2 and 6 gives 12, which is divisible by 4.

### 11. "The square of a number is always greater than the number itself."
- **False.** For numbers between 0 and 1, the square is smaller than the number itself. A counterexample is \( \left( \frac{1}{2} \right)^2 = \frac{1}{4} \), which is less than \( \frac{1}{2} \).

### 12. "The A-C systems in our school have never broken down this year."
- **Unverifiable.** This is a specific statement that cannot be universally verified or disproved without local knowledge.

### 13. "If \( A \times B = 0 \), then either \( A = 0 \) or \( B = 0 \)."
- **True.** This is a property of multiplication in real numbers. If the product of two numbers is 0, then at least one of the numbers must be 0.

### 14. "Vertical angles are always congruent."
- **True.** This is a known fact from geometry. Vertical angles, formed by two intersecting lines, are always equal in measure.

### 15. "The difference (subtraction) between two negative numbers is always negative."
- **False.** The difference between two negative numbers can be positive. For example, \( -2 - (-5) = 3 \), which is positive.

Would you like further details on any of these? 

Here are five related questions for further exploration:
1. Can you list other examples of birds that cannot fly?
2. How can the divisibility rule for 4 be extended to larger sets of even numbers?
3. What are other single-syllable states in the USA besides Maine?
4. Why do vertical angles always have equal measure in geometry?
5. Can you explain why subtracting two negative numbers can result in a positive number?

**Tip:** When working with numbers less than 1, their squares will always be smaller than the original number.

Determine if each conjecture is true and false. If false, provide a counterexample.

Analyze and determine the truth or falsity of various mathematical conjectures. This problem covers key concepts like multiplication, vertical angles, number properties, and the zero product property. Suitable for Grades 6-9 with counterexamples provided for false conjectures.

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