Given that ∠ABC and ∠DEF are complementary, m∠ABC=10x+7, and m∠DEF=11x−11.5, find m∠FED.

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°.

Learn how to solve a problem involving complementary angles with expressions 10x + 7 and 11x - 11.5. This problem explores the concept of complementary angles in geometry and provides a step-by-step solution. Suitable for high school students.

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