https://file.aiquickdraw.com/mathbot/file/1725925433737bjcn94l4.jpg

To solve this problem, we need to find the measure of angle \(\angle NKM\). Let's use the information provided:

1. \( m\angle LMN = 116^\circ \)
2. \( m\angle JKM = 122^\circ \)
3. \( m\angle JNM = 103^\circ \)

We want to find \( m\angle NKM \).

### Step-by-Step Solution:

1. **Identify the relevant angles:**
   The angle \(\angle NKM\) is inside triangle \( \triangle KNM\). To find \(\angle NKM\), we need to know the angles of this triangle.

2. **Use angle relationships:**
   In triangle \( \triangle KNM \):
   - The sum of the internal angles of any triangle is \(180^\circ\).
   - We already have \( m\angle JKM = 122^\circ\) which is an external angle to triangle \( \triangle KNM\).

3. **Find the interior angle adjacent to the external angle \(122^\circ\):**
   The angle adjacent to \( m\angle JKM = 122^\circ \) inside the triangle \( \triangle KNM \) is \( m\angle NKM\) because they are supplementary (they form a linear pair).
   
   Thus:
   \[
   m\angle NKM + 122^\circ = 180^\circ
   \]
   Solving for \( m\angle NKM \):
   \[
   m\angle NKM = 180^\circ - 122^\circ = 58^\circ
   \]

Therefore, the measure of \( m\angle NKM \) is \( 58^\circ \).

Let me know if you need further details or have any questions! Here are some related questions you might be interested in:

1. How do you find the measures of angles in a triangle using the exterior angle theorem?
2. What is the sum of the angles in any polygon?
3. How can you use linear pairs to find unknown angles?
4. How do supplementary and complementary angles differ?
5. What are other ways to find angles in a quadrilateral?

**Tip:** Always double-check your angle relationships in a diagram, especially when dealing with external and internal angles, to ensure accuracy.

Look for Relationships In the diagram at the right, m∠LMN = 116°, m∠JKM = 122°, and m∠JNM = 103°. What is m∠NKM?

Learn how to solve for the angle m∠NKM in a triangle using angle relationships and the external angle theorem. This problem provides a step-by-step guide and is suitable for students in Grades 8-10.

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