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Let's break down and solve the problem visible in the image.

### Problem Details:
- In triangle \( \triangle JKL \), the line \( JL \) is extended through point \( L \) to point \( M \).
- We are given three angles:
  - \( m\angle JKL = (3x + 8)^\circ \)
  - \( m\angle LJK = (2x + 20)^\circ \)
  - \( m\angle KLM = (7x - 10)^\circ \)
- We need to find \( m\angle LJK \).

### Solution Approach:

1. **Exterior Angle Theorem**:
   According to the exterior angle theorem, the exterior angle is equal to the sum of the two non-adjacent interior angles. In this case, \( m\angle KLM \) is an exterior angle to \( \triangle JKL \), so:
   \[
   m\angle KLM = m\angle JKL + m\angle LJK
   \]
   Substituting the given expressions:
   \[
   (7x - 10) = (3x + 8) + (2x + 20)
   \]

2. **Solve for \( x \)**:
   Simplifying the equation:
   \[
   7x - 10 = 3x + 8 + 2x + 20
   \]
   Combine like terms:
   \[
   7x - 10 = 5x + 28
   \]
   Subtract \( 5x \) from both sides:
   \[
   2x - 10 = 28
   \]
   Add 10 to both sides:
   \[
   2x = 38
   \]
   Divide by 2:
   \[
   x = 19
   \]

3. **Find \( m\angle LJK \)**:
   Now that we have \( x = 19 \), substitute this value into the expression for \( m\angle LJK \):
   \[
   m\angle LJK = 2x + 20 = 2(19) + 20 = 38 + 20 = 58^\circ
   \]

### Final Answer:
The measure of \( m\angle LJK \) is \( \boxed{58^\circ} \).

Would you like a detailed explanation of any part, or do you have further questions?

### Related Questions:
1. How can we verify the solution using the interior angle sum of a triangle?
2. What is the measure of \( m\angle JKL \) when \( x = 19 \)?
3. What is the measure of \( m\angle KLM \) when \( x = 19 \)?
4. Can we apply the exterior angle theorem in any triangle?
5. How does the exterior angle theorem work with parallel lines?

### Tip:
Remember, the exterior angle theorem applies to any triangle and is a powerful tool for solving angle-related problems!

In triangle JKL, JL is extended through point L to point M, m∠JKL = (3x + 8)°, m∠LJK = (2x + 20)°, and m∠KLM = (7x - 10)°. Find m∠LJK.

Learn how to find the measure of angle LJK in triangle JKL, where JL is extended to point M. The problem uses the exterior angle theorem and algebraic techniques to solve for the unknown variable x and the measure of the angle. Suitable for students in Grades 8-10.

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