The problem involves solving for $x$ in a triangle, where the angles are given as follows:

• One angle is $60^\circ$,
• One angle is a right angle ( $90^\circ$ ),
• The third angle is $(3x - 12)^\circ$.

We know that the sum of the interior angles of a triangle is always $180^\circ$. Therefore, we can set up the equation:

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

Simplifying this equation:

$150^\circ + (3x - 12)^\circ = 180^\circ$

$3x - 12 = 180^\circ - 150^\circ$

$3x - 12 = 30^\circ$

Now, add $12$ to both sides:

$3x = 42^\circ$

Finally, divide by 3:

$x = 14^\circ$

So, the value of $x$ is $14$.

Would you like a more detailed explanation or have any other questions?

Here are five questions to deepen your understanding:

1. What is the sum of the angles in a triangle?
2. How does knowing one angle in a right triangle help solve for the others?
3. Can you use this method to solve for angles in non-right triangles?
4. What are other common equations involving angles in geometry?
5. How can algebraic expressions like $3x - 12$ be used to represent angles?

Tip: Always remember that the sum of the interior angles in a triangle is $180^\circ$, which is helpful for solving many geometry problems.