The problem presents a triangle $\triangle EFG$ with the measures of its angles in terms of $x$:

• $m\angle E = (7x + 12)^\circ$
• $m\angle F = (6x - 9)^\circ$
• $m\angle G = (2x + 12)^\circ$

Since the sum of the angles in a triangle is always $180^\circ$, we can set up the following equation: $(7x + 12) + (6x - 9) + (2x + 12) = 180$

Let's solve this equation to find the value of $x$.The value of $x$ in the triangle $\triangle EFG$ is $11$.

Would you like more details or do you have any other questions related to this problem?

Here are some related questions you might consider:

1. What are the individual angle measurements of $\triangle EFG$ when $x = 11$?
2. How would the solution change if one of the angle expressions was altered?
3. What is the importance of understanding angle relationships in geometry?
4. Can you describe how to verify if a given triangle is a right triangle based on its angles?
5. What are some real-life applications of solving equations like this in geometry?

Tip: In geometry problems involving algebra, always verify your solutions by substituting back into the original expressions to ensure consistency and accuracy in the context of the problem.