To solve for the missing angles in this problem, let's analyze the given angles and relationships between them in the triangle and adjacent lines.

Step-by-Step Breakdown:

1. Angle at $J$ (MJK):

   • The angle $\angle MJL$ is given as 132°, and since $MJL$ is a straight line, the angle next to it, $\angle KJL$, can be found by subtracting 132° from 180° (angles on a straight line sum to 180°): $\angle KJL = 180^\circ - 132^\circ = 48^\circ.$

2. Angle at $L$ (KLN):

   • Similarly, for the straight line at point $L$, $\angle KLN$ is supplementary to $\angle MLN = 112^\circ$. Therefore, the angle $\angle KLM$ is: $\angle KLM = 180^\circ - 112^\circ = 68^\circ.$

3. Angle at $K$:

   • To find $\angle JKL$, the third angle in triangle $JKL$, we use the fact that the angles in a triangle sum to 180°: $\angle JKL + \angle KJL + \angle KLM = 180^\circ.$ Substituting the known values: $\angle JKL + 48^\circ + 68^\circ = 180^\circ,$ $\angle JKL = 180^\circ - 116^\circ = 64^\circ.$

4. Finding $x$:

   • The angle at $K$ is expressed as $(2x + 4)$. From the previous calculation, we know that $\angle JKL = 64^\circ$, so: $2x + 4 = 64^\circ.$ Solving for $x$: $2x = 64^\circ - 4^\circ = 60^\circ,$ $x = \frac{60}{2} = 30^\circ.$

Final Answers:

• $\angle KJL = 48^\circ$,
• $\angle KLM = 68^\circ$,
• $\angle JKL = 64^\circ$,
• $x = 30^\circ$.

Let me know if you want further details or have any questions! Here are 5 related questions:

1. How can you verify if a set of angles in a triangle is correct?
2. What is the sum of interior angles for polygons other than triangles?
3. How do you calculate supplementary angles?
4. What is the rule for angles on a straight line?
5. Can you find the missing side of a triangle if only angles are known?

Tip: In any geometric configuration involving straight lines and triangles, always look for supplementary angles to simplify the problem.