To solve the problem, let's start by understanding the relationship between the given angles.

Step 1: Identify the key relationship

We know that ∠ABC and ∠DEF are complementary. This means:

$m\angle ABC + m\angle DEF = 90^\circ$

Step 2: Substitute the given expressions

We are given:

$m\angle ABC = 10x + 7$

$m\angle DEF = 11x - 11.5$

Substituting these into the complementary angle equation:

$(10x + 7) + (11x - 11.5) = 90^\circ$

Step 3: Simplify the equation

Combine like terms:

$10x + 11x + 7 - 11.5 = 90^\circ$

$21x - 4.5 = 90^\circ$

Step 4: Solve for $x$

Add 4.5 to both sides:

$21x = 94.5$

Divide by 21:

$x = \frac{94.5}{21} \approx 4.5$

Step 5: Find $m\angle DEF$

Substitute $x = 4.5$ back into the expression for $m\angle DEF$:

$m\angle DEF = 11(4.5) - 11.5 = 49.5 - 11.5 = 38^\circ$

Step 6: Determine $m\angle FED$

Since ∠FED is the same angle as ∠DEF in this context (due to symmetry or identical labeling), we conclude:

$m\angle FED = 38^\circ$

Final Answer: $m\angle FED = 38^\circ$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How do you prove that two angles are complementary using their measures?
2. What would happen if the angles were supplementary instead of complementary?
3. How would you find $x$ if the angles were in a different ratio?
4. Can complementary angles ever be greater than 90°? Why or why not?
5. What is the measure of each angle if they were both equal and complementary?

Tip: Remember, complementary angles always add up to 90°, while supplementary angles add up to 180°.