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We are given that ray \( EF \) bisects \( \angle AED \), \( m \angle AEF = 51^\circ \), and \( m \angle AED = (11x - 30)^\circ \). We need to find the value of \( x \).

### Step-by-step solution:

1. Since ray \( EF \) bisects \( \angle AED \), it means that:
   \[
   m \angle AEF = m \angle FED
   \]
   Therefore, we can express \( m \angle FED \) as:
   \[
   m \angle FED = m \angle AEF = 51^\circ
   \]

2. The sum of \( m \angle AEF \) and \( m \angle FED \) gives the measure of the entire angle \( m \angle AED \):
   \[
   m \angle AED = m \angle AEF + m \angle FED
   \]
   Substituting the values:
   \[
   m \angle AED = 51^\circ + 51^\circ = 102^\circ
   \]

3. We are also given that \( m \angle AED = (11x - 30)^\circ \). So we set up the equation:
   \[
   11x - 30 = 102
   \]

4. Solving for \( x \):
   \[
   11x = 102 + 30 = 132
   \]
   \[
   x = \frac{132}{11} = 12
   \]

Thus, the value of \( x \) is \( \boxed{12} \).

Would you like further details on any of the steps?

Here are some related questions you could explore:
1. What is the geometric definition of an angle bisector?
2. How do you solve equations involving angles?
3. What are some properties of supplementary angles?
4. Can we always solve for unknown variables using angle relationships?
5. How does bisecting an angle impact adjacent angles?

**Tip:** When solving for variables in geometry, always double-check by substituting the value back into the original equation to verify consistency.

If ray EF bisects ∠AED, m∠AEF = 51°, and m∠AED = (11x - 30)°, find the value of x.

Learn how to solve for the value of x in an angle bisector problem where ray EF bisects ∠AED. Given m∠AEF = 51° and m∠AED = (11x - 30)°, this detailed step-by-step solution walks you through the process. Suitable for students in Grades 9-10.

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