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The problem asks to solve for the measure of \(\angle K\) that proves \(JK \parallel MN\).

Let's break this down with the information given:

### Step-by-step Analysis:

- We have two intersecting lines \(JN\) and \(KM\) at point \(L\).
- \(\angle JLK = 25^\circ\) and \(\angle MNL = 72^\circ\).
  
We want to prove that \(JK \parallel MN\), which implies that corresponding angles or alternate interior angles between these two lines should be equal according to the properties of parallel lines.

### Use of Alternate Interior Angles:
For \(JK \parallel MN\), the alternate interior angles formed by the transversal should be equal. Here, the transversal is the line \(JN\), and the alternate interior angles would be:

- \(\angle JLK\) (which is given as \(25^\circ\))
- \(\angle MNK\)

Since alternate interior angles are equal when lines are parallel, we set:
\[
\angle MNK = 25^\circ
\]

Now, we can find \(\angle K\) (specifically \(\angle NKM\)):

- The sum of the angles in triangle \(MKN\) is \(180^\circ\).
- We already know that \(\angle MNK = 25^\circ\) and \(\angle MKN = 72^\circ\).

Thus, \(\angle NKM = 180^\circ - 25^\circ - 72^\circ = 83^\circ\).

### Final Answer:
Therefore, the measure of \(\angle K\) is \(83^\circ\).

Would you like more details or have any questions?

### Follow-up Questions:
1. How do alternate interior angles prove parallelism?
2. What are the properties of parallel lines cut by a transversal?
3. Can you explain how the sum of angles in a triangle works?
4. How can we identify alternate interior angles in different configurations?
5. Why are the sum of angles in a triangle always 180°?

### Tip:
Remember, when solving geometry problems involving parallel lines, focus on corresponding angles, alternate interior angles, or consecutive interior angles for proofs!

The figure shows JN and KM intersecting at point L. Solve for the measure of ∠K that proves JK || MN.

Learn how to solve for the measure of ∠K that proves JK is parallel to MN using alternate interior angles and the triangle sum theorem. This problem explores key geometry concepts and is suitable for students in Grades 8-10.

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