The problem involves using properties of parallel lines and angles. From the diagram, we can see that two lines are parallel (indicated by the arrows), and they intersect with a transversal that creates angles.

The key relationship we need is the property of corresponding angles. Corresponding angles are equal when two lines are parallel and intersected by a transversal.

We have the following:

• Angle at $J$ is represented as $7x - 14^\circ$.
• Angle at $K$ is represented as $4x + 7^\circ$.

Since the lines are parallel, the angles at $J$ and $K$ are corresponding angles. Therefore, we can set up the equation:

$7x - 14 = 4x + 7$

Now, let's solve for $x$:

1. Subtract $4x$ from both sides: $3x - 14 = 7$

2. Add 14 to both sides: $3x = 21$

3. Divide by 3: $x = 7$

Now that we have $x = 7$, we can substitute this value back into either angle expression to find the measure of angle $\angle JKL$. Let's use the expression $4x + 7^\circ$ for $\angle K$:

$m \angle JKL = 4(7) + 7 = 28 + 7 = 35^\circ$

So, $m \angle JKL = 35^\circ$.

Summary:

• $x = 7$
• $m \angle JKL = 35^\circ$

Would you like further explanations or help with any other questions? Here are a few related questions:

1. How do we identify corresponding angles in other geometric figures?
2. What is the difference between corresponding and alternate interior angles?
3. What would happen if the lines were not parallel in this situation?
4. How do properties of parallel lines help us solve for unknowns in geometry?
5. Can we apply this method to solve for angles in polygons?

Tip: Always check if the lines are parallel before assuming angle relationships.