https://file.aiquickdraw.com/mathbot/file/1726331320922dwvvua15.jpg

In the image, you have two parallel lines \( \overleftrightarrow{AB} \parallel \overleftrightarrow{CD} \) and two transversals creating angles involving the expressions \( 3x + 11^\circ \) and \( 10x^\circ \). The goal is to find the value of \( x \) that makes \( \overleftrightarrow{AB} \parallel \overleftrightarrow{CD} \) and then determine the angle measures.

### Step 1: Use the Corresponding Angles Postulate
Since \( \overleftrightarrow{AB} \parallel \overleftrightarrow{CD} \), the angles \( \angle AEF \) (labeled \( 3x + 11^\circ \)) and \( \angle CFE \) (labeled \( 10x^\circ \)) must be equal because they are corresponding angles.

Thus, set the two expressions equal to each other:
\[
3x + 11 = 10x
\]

### Step 2: Solve for \( x \)
Now, solve for \( x \) by isolating it:
\[
10x - 3x = 11
\]
\[
7x = 11
\]
\[
x = \frac{11}{7}
\]

### Step 3: Calculate the Angles
Now that we have \( x = \frac{11}{7} \), we can substitute this value into the expressions for the angles.

- For \( \angle AEF = 3x + 11 \):
  \[
  \angle AEF = 3\left(\frac{11}{7}\right) + 11 = \frac{33}{7} + 11 = \frac{33}{7} + \frac{77}{7} = \frac{110}{7} \approx 15.71^\circ
  \]

- For \( \angle CFE = 10x \):
  \[
  \angle CFE = 10\left(\frac{11}{7}\right) = \frac{110}{7} \approx 15.71^\circ
  \]

### Conclusion:
- The value of \( x \) that makes \( \overleftrightarrow{AB} \parallel \overleftrightarrow{CD} \) is \( \frac{11}{7} \).
- The measure of \( \angle AEF \) is approximately \( 15.71^\circ \).
- The measure of \( \angle CFE \) is approximately \( 15.71^\circ \).

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### Related Questions:
1. What are corresponding angles, and how do they relate to parallel lines?
2. How would the equation change if the angles were supplementary instead of equal?
3. What is the general method to solve for \( x \) in similar parallel line and transversal problems?
4. How can you verify if two lines are parallel using angle relationships?
5. What happens if we find different values of \( x \) for different angle pairs?

### Tip:
In parallel line problems involving transversals, always look for relationships between angles such as corresponding, alternate interior, or supplementary angles to set up equations.

Find the value of x that makes \( \overleftrightarrow{AB} \parallel \overleftrightarrow{CD} \). Then, determine the angle measures of \( \angle AEF \) and \( \angle CFE \).

Solve for the value of x that makes lines \( \overleftrightarrow{AB} \parallel \overleftrightarrow{CD} \) using corresponding angles. Calculate the angle measures of \( \angle AEF \) and \( \angle CFE \) in this detailed problem suitable for Grades 8-10.

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