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.