A clock currently shows the time $10:10$. The obtuse angle between the hands measures $x$ degrees. What is the next time that the angle between the hands will be $x$ degrees? Round your answer to the nearest minute.
1. **Calculate the initial angle at 10:10:**
   - The hour hand at 10:00 is at \(10 \times 30 = 300\) degrees.
   - The hour hand moves \(0.5\) degrees per minute, so at 10:10, it has moved \(10 \times 0.5 = 5\) degrees.
   - Therefore, the hour hand is at \(300 + 5 = 305\) degrees at 10:10.
   - The minute hand at 10:10 is at \(10 \times 6 = 60\) degrees.
   - The angle between the hour and minute hands is \(|305 - 60| = 245\) degrees.
   - Since we need the obtuse angle, we take \(360 - 245 = 115\) degrees.

2. **Determine the rate of change of the angle between the hands:**
   - The minute hand moves at \(6\) degrees per minute.
   - The hour hand moves at \(0.5\) degrees per minute.
   - The relative speed between the minute and hour hands is \(6 - 0.5 = 5.5\) degrees per minute.

3. **Calculate the time it takes for the hands to form the same angle again:**
   - The hands need to form an angle of \(115\) degrees again.
   - Since the hands differ by \(5.5\) degrees per minute, we need to find the time \(t\) such that \(5.5t = 360\) degrees (a full circle).
   - Solving for \(t\), we get:
     \[
     t = \frac{360}{5.5} \approx 65.454545 \text{ minutes}
     \]

4. **Determine the next time:**
   - Adding approximately \(65\) minutes to 10:10 gives us:
     \[
     10:10 + 65 \text{ minutes} = 11:15
     \]

The final answer is \(\boxed{11:15}\).