We are given a triangle $\Delta KLM$ where:

• $k = 72$ inches (side opposite angle $K$),
• $l = 92$ inches (side opposite angle $L$),
• $\angle K = 131^\circ$.

To find the possible values of $\angle L$, we can use the Law of Sines:

$\frac{k}{\sin(\angle K)} = \frac{l}{\sin(\angle L)}.$

Substituting the known values:

$\frac{72}{\sin(131^\circ)} = \frac{92}{\sin(\angle L)}.$

First, let's calculate $\sin(131^\circ)$, then solve for $\sin(\angle L)$.

$\sin(\angle L) = \frac{92 \times \sin(131^\circ)}{72}.$

Now we compute this step.We have calculated that:

$\sin(\angle L) \approx 0.9644,$ which gives one possible value for $\angle L$ as:

$\angle L \approx 75^\circ.$

Since the sine function can have two possible angles (one acute and one obtuse) that give the same sine value, we need to consider the second possible solution. The second possible value for $\angle L$ would be:

$\angle L' = 180^\circ - 75^\circ = 105^\circ.$

Thus, the possible values for $\angle L$ are approximately $75^\circ$ and $105^\circ$.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How would the Law of Cosines help if all angles and sides were unknown?
2. How would you solve for the remaining angle $\angle M$?
3. What would happen if $\sin(\angle L)$ were greater than 1?
4. How would rounding affect the precision of $\angle L$?
5. Can a triangle have two obtuse angles?

Tip: In non-right triangles, the Law of Sines is useful when two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) are known.