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Given the equation $C = \frac{5}{9} (F - 32)$, we can analyze the relationship between temperature changes in Fahrenheit and Celsius.

1. A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of $\frac{5}{9}$ degree Celsius.

   To check this, consider the equation for a change in temperature. Let $\Delta F$ represent the change in Fahrenheit and $\Delta C$ represent the change in Celsius. The equation can be differentiated with respect to $F$:

   $\Delta C = \frac{5}{9} \Delta F$

   For $\Delta F = 1$:

   $\Delta C = \frac{5}{9} \times 1 = \frac{5}{9}$

   Therefore, a temperature increase of 1 degree Fahrenheit is indeed equivalent to a temperature increase of $\frac{5}{9}$ degree Celsius.

2. A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.

   To check this, we need to find the change in Fahrenheit corresponding to a 1 degree Celsius change. Rearrange the equation to solve for $F$:

   $F = \frac{9}{5}C + 32$

   Differentiating with respect to $C$:

   $\Delta F = \frac{9}{5} \Delta C$

   For $\Delta C = 1$:

   $\Delta F = \frac{9}{5} \times 1 = 1.8$

   Therefore, a temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.

3. A temperature increase of $\frac{5}{9}$ degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius.

   For $\Delta F = \frac{5}{9}$:

   $\Delta C = \frac{5}{9} \times \frac{5}{9} = \left( \frac{5}{9} \right)^2 = \frac{25}{81}$

   This does not simplify to 1 degree Celsius, so this statement is incorrect.

Therefore, the correct statements are:

• A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of $\frac{5}{9}$ degree Celsius.
• A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.

Would you like more details or have any questions?

Here are some questions you might find interesting:

1. How do you convert 100 degrees Fahrenheit to Celsius using the given equation?
2. Can you explain how to derive the conversion formula from Celsius to Fahrenheit?
3. What is the significance of the constants 5/9 and 9/5 in the temperature conversion formulas?
4. How does the freezing point of water compare in Celsius and Fahrenheit scales?
5. What is the boiling point of water in both Celsius and Fahrenheit?

Tip: When converting between temperature scales, always remember to adjust for the offset (32 degrees) when switching from Fahrenheit to Celsius or vice versa.