We are given a triangle with a right angle, one of the angles marked as $60^\circ$, and another angle labeled as $(3x - 12)^\circ$.

Since the sum of the angles in any triangle is always $180^\circ$, we can set up the following equation using this information:

$90^\circ + 60^\circ + (3x - 12)^\circ = 180^\circ$

Let's solve for $x$:

1. Add the known angles: $90 + 60 = 150^\circ$

2. Set up the equation: $150^\circ + (3x - 12)^\circ = 180^\circ$

3. Simplify further: $3x - 12 + 150 = 180$

4. Combine like terms: $3x + 138 = 180$

5. Subtract 138 from both sides: $3x = 42$

6. Finally, divide by 3: $x = 14$

The value of $x$ is $14$.

Would you like further details on the steps, or have any additional questions? Here are some related questions to think about:

1. What is the sum of the angles in any triangle?
2. How would the equation change if one of the angles was unknown?
3. What happens if the triangle is not a right triangle?
4. How do you solve for angles in an isosceles triangle?
5. Can you find the other angles if the value of $x$ was different?

Tip: Always remember that the sum of interior angles of a triangle is always $180^\circ$.