Let's go through the properties you've listed and check if they are correct.

1. (-9 + (4 + x) = (-9 + 4) + x)
   Your answer: Commutative Property
   Correct Answer: Associative Property
   The associative property involves regrouping numbers without changing the order. Here, it's about changing the grouping of terms in the addition.

2. $7(c + 4) = 7c + 28$
   Your answer: Distributive Property
   Correct. The distributive property allows you to distribute the 7 across both terms in the parentheses.

3. $-5 \cdot 3 = 3 \cdot -5$
   Your answer: Commutative Property
   Correct. The commutative property of multiplication states that changing the order of the factors does not affect the product.

4. $12 \cdot 1 = 12$
   Your answer: Identity Property
   Correct. The identity property of multiplication says that any number multiplied by 1 is itself.

5. $\frac{2}{5} \cdot \frac{5}{2} = 1$
   Your answer: Inverse Property
   Correct. This is the multiplicative inverse property, where multiplying a number by its reciprocal equals 1.

6. $\left( \frac{1}{2} \cdot \frac{4}{9} \right) \cdot \frac{2}{3} = \frac{1}{2} \cdot \left( \frac{4}{9} \cdot \frac{2}{3} \right)$
   Your answer: Associative Property
   Correct. The associative property of multiplication allows you to regroup the numbers.

7. $14x \cdot 0 = 0$
   Your answer: Property of Zero
   Correct. This demonstrates the zero property of multiplication, where any number multiplied by 0 equals 0.

8. $x(y + z) = xy + xz$
   Your answer: (Not filled)
   Correct Answer: Distributive Property
   This is the distributive property, where $x$ is distributed to both $y$ and $z$.

9. $7 \cdot (-2 \cdot 8) = 7 \cdot (8 \cdot -2)$
   Your answer: (Not filled)
   Correct Answer: Commutative Property
   This uses the commutative property of multiplication, where the order of multiplication doesn't affect the product.

10. $c^2 + 0 = c^2$
    Your answer: (Not filled)
    Correct Answer: Identity Property
    This is the identity property of addition, where adding 0 to any number leaves the number unchanged.

11. $10 + (-10) = 0$
    Your answer: (Not filled)
    Correct Answer: Inverse Property
    This demonstrates the additive inverse property, where a number plus its opposite equals 0.

12. $5 \cdot (m + n) = (m + n) \cdot 5$
    Your answer: (Not filled)
    Correct Answer: Commutative Property
    This is the commutative property of multiplication, as the order of multiplication doesn't affect the result.

13. $(a + b) \cdot 1 = (a + b)$
    Your answer: (Not filled)
    Correct Answer: Identity Property
    This is the identity property of multiplication, where multiplying by 1 leaves the number unchanged.

14. $(2x - y) \cdot 0 = 0$
    Your answer: (Not filled)
    Correct Answer: Zero Property
    This uses the zero property of multiplication, where any number multiplied by 0 equals 0.

Would you like more details or further explanations on any of these? Here are some related questions:

1. What is the difference between the commutative and associative properties?
2. How does the identity property differ between addition and multiplication?
3. Can you provide examples of the distributive property involving division?
4. Why is the zero property of multiplication important in algebra?
5. How does the inverse property help in solving equations?

Tip: The identity property always involves either 1 (for multiplication) or 0 (for addition).