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To solve this problem, let's analyze the given information:

1. **Ray KJ and Ray KL are opposite rays**: This implies that \(\angle JKL\) forms a straight angle, which is 180°.
2. **Ray KN bisects \(\angle LKM\)**: This means that \(\angle LKM\) is split into two equal parts: \(\angle LKN\) and \(\angle NKM\).

We are also given the following equations for the angles:

- \(\angle LKM = 7x - 5\)
- \(\angle NKM = 3x + 9\)

Since Ray KN bisects \(\angle LKM\), the two angles formed (\(\angle LKN\) and \(\angle NKM\)) are equal. Therefore,

\[
\angle LKN = \angle NKM = 3x + 9
\]

So, \(\angle LKM\) can also be written as:

\[
\angle LKM = \angle LKN + \angle NKM = 2(3x + 9)
\]

Now, equate this with the original expression for \(\angle LKM\):

\[
7x - 5 = 2(3x + 9)
\]

### Solving for \(x\):

\[
7x - 5 = 6x + 18
\]

Subtract \(6x\) from both sides:

\[
x - 5 = 18
\]

Add 5 to both sides:

\[
x = 23
\]

### Finding \(\angle LKM\):

Substitute \(x = 23\) into the expression for \(\angle LKM\):

\[
\angle LKM = 7(23) - 5 = 161 - 5 = 156^\circ
\]

Thus, \(\angle LKM = 156^\circ\).

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do you determine if two rays are opposite rays?
2. What is the importance of an angle bisector in geometry?
3. How do you solve linear equations involving angles?
4. How does knowing that angles form a straight line help in solving geometry problems?
5. Can you derive a general formula for solving similar angle bisector problems?

**Tip**: When solving geometry problems involving bisectors, always remember that the bisector divides the angle into two equal parts. This can simplify the problem significantly.

If ∠LKM = 7x - 5 and ∠NKM = 3x + 9, find ∠LKM. Ray KN bisects ∠LKM.

Learn how to find the value of ∠LKM using angle bisector properties and algebraic equations. This problem involves solving linear equations and understanding geometric concepts related to angles. Suitable for students in Grades 8-10.

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