https://file.aiquickdraw.com/mathbot/file/1724000814411km5vawql.jpg

To solve for \( x \), I'll analyze the angles in the image. The diagram shows intersecting lines, with angles marked, and a linear pair of angles that are supplementary.

### Analyzing the Diagram:

1. The given angle \( \angle JEF = 127^\circ \).
2. The angle \( \angle KEL \) (which is labeled as \( 5x \)) and \( \angle JEF \) form a linear pair, which means they are supplementary.

### Applying the Supplementary Angles Theorem:

Since \( \angle KEL \) and \( \angle JEF \) are supplementary, we know that:
\[
\angle KEL + \angle JEF = 180^\circ
\]
Substitute the given angle and the expression for \( \angle KEL \):
\[
5x + 127^\circ = 180^\circ
\]

### Solving for \( x \):

\[
5x = 180^\circ - 127^\circ
\]
\[
5x = 53^\circ
\]
\[
x = \frac{53^\circ}{5}
\]
\[
x = 10.6^\circ
\]

So, \( x = 10.6^\circ \).

Would you like a more detailed explanation or have any questions about this? 

Here are five additional questions related to this problem:
1. What are supplementary angles, and how do they relate to linear pairs?
2. How can you identify corresponding angles in parallel line problems?
3. If another angle was given as \( 4x + 3^\circ \), how would you solve for \( x \)?
4. How do vertical angles relate to each other in this context?
5. What are the properties of angles formed by a transversal intersecting parallel lines?

**Tip:** Remember that the sum of angles on a straight line always equals 180 degrees, which is useful in many geometry problems.

Learn how to solve for x using the Supplementary Angles Theorem in geometry. This problem involves analyzing intersecting lines and angles, applying the concept of supplementary angles, and solving a linear equation. Suitable for students in Grades 9-12.

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