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.