In ΔKLM, l = 92 inches, k = 72 inches and ∠K=131°. Find all possible values of ∠L, to the nearest degree.

To find the possible values of \(\angle L\) in triangle \( \Delta KLM \) where \( l = 92 \) inches, \( k = 72 \) inches, and \( \angle K = 131^\circ \), we can apply the Law of Sines. The Law of Sines states:

\[
\frac{\sin \angle K}{k} = \frac{\sin \angle L}{l} = \frac{\sin \angle M}{m}
\]

First, we'll find \(\sin \angle L\) using the known values.

### Step 1: Apply the Law of Sines

\[
\frac{\sin \angle L}{l} = \frac{\sin \angle K}{k}
\]

Substituting the given values:

\[
\frac{\sin \angle L}{92} = \frac{\sin 131^\circ}{72}
\]

### Step 2: Solve for \(\sin \angle L\)

Calculate \(\sin 131^\circ\):

\[
\sin 131^\circ \approx 0.7547
\]

So,

\[
\frac{\sin \angle L}{92} = \frac{0.7547}{72}
\]

\[
\sin \angle L = \frac{92 \times 0.7547}{72} \approx 0.9646
\]

### Step 3: Find \(\angle L\)

Now, solve for \(\angle L\):

\[
\angle L = \arcsin(0.9646)
\]

\[
\angle L \approx 75.1^\circ
\]

To the nearest degree:

\[
\angle L \approx 75^\circ
\]

### Step 4: Consider the Ambiguity

In a triangle, \(\sin(\theta)\) has two possible angle values, \(\theta\) and \(180^\circ - \theta\), as long as both are valid within the triangle.

Given that \(\angle K = 131^\circ\), and the sum of angles in a triangle must equal \(180^\circ\):

1. If \(\angle L \approx 75^\circ\), then \(\angle M = 180^\circ - 131^\circ - 75^\circ = 180^\circ - 206^\circ\), which is not possible.

2. If \(\angle L \approx 180^\circ - 75^\circ = 105^\circ\), then \(\angle M = 180^\circ - 131^\circ - 105^\circ = 44^\circ\), which is valid.

### Conclusion

The only possible value for \(\angle L\) in \(\Delta KLM\) is \( \boxed{75^\circ} \).

Would you like more details on any steps, or do you have any questions?

### Relative Questions:
1. What is the Law of Sines, and when is it used?
2. How do you find the sine of an angle using a calculator?
3. What is the importance of considering the angle ambiguity in trigonometry?
4. How can we verify if a triangle is valid based on given side lengths and angles?
5. How do you find the third angle of a triangle if two angles are known?

### Tip:
Always remember to check for possible ambiguous cases when using the Law of Sines, especially in non-right triangles.

Learn how to find angle L in triangle KLM using the Law of Sines. Given side lengths and one angle, solve for the possible values of angle L with a step-by-step solution. Suitable for high school students studying trigonometry.

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