The image asks to find the measure of angle $\angle EFG$ in the triangle $\triangle EFG$. The given angles and expressions are:

• $\angle FEG = 38^\circ$
• $\angle EFG = 6x - 10$
• $\angle FGN = 7x + 18$

The sum of angles in any triangle is always 180°, so the equation for the triangle $\triangle EFG$ is:

$\angle FEG + \angle EFG + \angle FGN = 180^\circ$

Substituting the given values:

$38^\circ + (6x - 10) + (7x + 18) = 180^\circ$

Now, let's solve this equation to find the value of $x$. Once we know $x$, we can calculate $m \angle EFG$.The value of $x$ is $\frac{134}{13}$, which is approximately $10.31$.

Now, we substitute this value of $x$ into the expression for $\angle EFG = 6x - 10$ to find its measure. Let’s calculate that next.The measure of $\angle EFG$ is approximately $51.85^\circ$.

Since the closest option provided in the choices is 50°, the correct answer is 50°.

Would you like more details or further clarification?

Here are five related questions you might find helpful:

1. How do you find the sum of angles in a triangle?
2. Can you solve for the other angles in the triangle based on this solution?
3. How do you verify the correctness of $x$ in this context?
4. What if the angles were in a quadrilateral—how would the process differ?
5. Can you explain how to approach other triangle problems involving unknown angles?

Tip: In any triangle, the sum of the interior angles will always equal 180°, which is a useful property for solving for unknown angles.