To solve this problem, we'll use the Law of Sines. Given:

• $k = 840$ cm,
• $l = 950$ cm,
• $\angle J = 12^\circ$,

we need to find $\angle K$ to the nearest degree in triangle $\triangle JKL$.

The Law of Sines states:

$\frac{\sin \angle J}{l} = \frac{\sin \angle K}{j} = \frac{\sin \angle L}{k}$

Given $k$, $l$, and $\angle J$:

$\frac{\sin 12^\circ}{950} = \frac{\sin K}{840}$

Let's solve for $\sin K$:

$\sin K = \frac{840 \times \sin 12^\circ}{950}$

Now, we will calculate this value and then find $\angle K$ using the inverse sine function.The measure of $\angle K$ is approximately $11^\circ$ when rounded to the nearest degree.

Would you like further details or have any questions about this solution?

Here are some related questions you might find interesting:

1. How does the Law of Sines help solve for unknown angles in a triangle?
2. What is the difference between the Law of Sines and the Law of Cosines?
3. How can you determine whether a triangle is acute, right, or obtuse given its side lengths and angles?
4. What is the process for finding the third angle in a triangle when two angles are known?
5. How do you use the Law of Sines to find missing side lengths in a triangle?

Tip: Always check that the sum of angles in a triangle is $180^\circ$ to verify the accuracy of your calculations.