We are given a diagram with parallel lines $HJ \parallel KM$, and we are asked to find the measure of $\angle KLI$ given that $\angle JIL = 50^\circ$.

Step-by-step solution:

1. Identify the parallel lines:

   • $HJ \parallel KM$ are the two parallel lines, and $LI$ acts as a transversal intersecting these lines.

2. Recognize the corresponding angles:

   • Since $HJ \parallel KM$ and $LI$ is a transversal, we know that the corresponding angles formed by these parallel lines and transversal are equal.
   • Therefore, $\angle JIL$ (which is $50^\circ$) is congruent to $\angle KLI$ by the corresponding angles theorem.

3. Conclusion:

   • Since $\angle JIL = 50^\circ$, we have $\angle KLI = 50^\circ$.

Thus, the measure of $\angle KLI$ is $\boxed{50^\circ}$.

Would you like more details on any part of the solution, or do you have further questions?

Here are 5 related questions:

1. What are alternate interior angles in the context of parallel lines and transversals?
2. How do you determine when angles are supplementary using parallel lines?
3. Can you explain why corresponding angles are congruent in parallel lines?
4. How would the problem change if $\angle JIL$ were an obtuse angle instead?
5. What happens to the angles if the lines are not parallel?

Tip: When working with parallel lines, always check for angle relationships such as corresponding, alternate interior, or alternate exterior angles.