To solve this problem, we need to analyze the given conditions and determine the value of angle $\alpha$.

Problem Translation: In a 2022 Jiangxi Province mid-term exam question, it's given that the vertices of angle $\alpha$ and $180^\circ - \alpha$ are at the origin. If the terminal sides of these angles are on the non-negative x-axis and the terminal sides coincide, and $450^\circ < \alpha < 720^\circ$, then $\alpha =$ (in degrees).

Steps to Solve:

1. Angles on the Non-negative X-axis:

   • The angles whose terminal sides lie on the non-negative x-axis are multiples of $360^\circ$. Specifically, for $\alpha$ within the range $450^\circ < \alpha < 720^\circ$, we need to check the angles that are multiples of $360^\circ$ plus some multiple of $180^\circ$.

2. Range Adjustment:

   • The range $450^\circ < \alpha < 720^\circ$ can be simplified by subtracting $360^\circ$ from $\alpha$ since angles are periodic with a period of $360^\circ$. So we look at the effective angle within the range $90^\circ < \alpha < 360^\circ$.

3. Checking Values:

   • If $\alpha$ is between $450^\circ$ and $720^\circ$, the angles to check would be:
     • $450^\circ - 360^\circ = 90^\circ$
     • $720^\circ - 360^\circ = 360^\circ$

4. Verification:

   • The angle $90^\circ$ does not satisfy as its terminal side does not lie on the non-negative x-axis.
   • The angle $360^\circ$ satisfies the condition since its terminal side lies on the non-negative x-axis.

Therefore, we need to adjust back for the given range:

Conclusion:

• Since $360^\circ$ corresponds to $450^\circ$ to $720^\circ$, we need to account for a multiple of $360^\circ$:
  • So, $\alpha = 540^\circ$ as it is within the given range and the terminal side lies on the non-negative x-axis.

Hence, $\alpha = 540^\circ$.

Would you like more details on this solution or have any questions?

Here are some questions you might find interesting:

1. How do you determine the terminal side of an angle?
2. Why are angles considered periodic?
3. How can you convert degrees to radians?
4. What are the characteristics of angles in different quadrants?
5. How do you find the reference angle of a given angle?

Tip: Always remember to simplify the range of the angle by using the periodicity of $360^\circ$ when dealing with problems involving large angles.