To solve for $x$ in this triangle problem, let’s use the fact that the sum of the interior angles in any triangle is always $180^\circ$.

Given:

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

Since $m\angle E + m\angle F + m\angle G = 180^\circ$, we can set up the equation:

$(7x + 12) + (6x - 9) + (2x + 12) = 180$

Now, let’s simplify and solve this equation.

1. Combine like terms:

$7x + 6x + 2x + 12 - 9 + 12 = 180$

$15x + 15 = 180$

2. Subtract 15 from both sides:

$15x = 165$

3. Divide by 15:

$x = 11$

Answer:

The value of $x$ is $11$.

Would you like further explanation on any part of the solution?

Here are five related questions to explore:

1. What are the actual measures of angles $E$, $F$, and $G$ when $x = 11$?
2. How can you verify that the angles indeed add up to $180^\circ$?
3. What if one of the angle expressions was an exterior angle? How would that change the equation?
4. How can you use the triangle inequality theorem to check the validity of the angle measures?
5. What would happen if the triangle was a right triangle? How would that affect the expressions?

Tip: When solving for variables in angle expressions, always check if the calculated values satisfy the properties of a triangle, such as the angle sum property.