We are solving for $\angle K$ in $\Delta KLM$, given the following:

• $m = 460 \, \text{cm}$ (side opposite $\angle M$),
• $k = 360 \, \text{cm}$ (side opposite $\angle K$),
• $\angle L = 142^\circ$ (angle at vertex $L$).

This is a triangle with sides and an angle given (SAS). We will use the Law of Cosines to find the length of the third side $l$ (opposite $\angle L$), then apply the Law of Sines to find $\angle K$.

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Step 1: Use the Law of Cosines to find $l$

The formula for the Law of Cosines is: $l^2 = m^2 + k^2 - 2mk \cdot \cos(\angle L)$ Substitute the known values: $l^2 = 460^2 + 360^2 - 2(460)(360)\cos(142^\circ)$

Calculate: $460^2 = 211,600, \quad 360^2 = 129,600$ $l^2 = 211,600 + 129,600 - 2(460)(360)\cos(142^\circ)$ The cosine of $142^\circ$ is approximately $\cos(142^\circ) = -0.788$: $l^2 = 211,600 + 129,600 + 2(460)(360)(0.788)$ $l^2 = 211,600 + 129,600 + 261,158.4$ $l^2 = 602,358.4 \quad \implies \quad l \approx \sqrt{602,358.4} \approx 776 \, \text{cm}$

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Step 2: Use the Law of Sines to find $\angle K$

The Law of Sines states: $\frac{\sin(\angle K)}{k} = \frac{\sin(\angle L)}{l}$ Rearrange to solve for $\sin(\angle K)$: $\sin(\angle K) = \frac{k \cdot \sin(\angle L)}{l}$ Substitute the known values ($k = 360$, $l \approx 776$, $\angle L = 142^\circ$, and $\sin(142^\circ) = 0.615$): $\sin(\angle K) = \frac{360 \cdot 0.615}{776}$ $\sin(\angle K) \approx \frac{221.4}{776} \approx 0.285$

Find $\angle K$ using the arcsine function: $\angle K = \arcsin(0.285) \approx 17^\circ$

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Final Answer:

$\angle K \approx 17^\circ$

Let me know if you'd like to see detailed calculations for verification!

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Related Questions:

1. How is the Law of Cosines useful in solving SAS triangles?
2. Why is the Law of Sines limited to certain cases in non-right triangles?
3. What is the significance of angle sum in triangles when solving for the last angle?
4. How do we compute trigonometric functions like $\cos(142^\circ)$ without a calculator?
5. What happens if the triangle does not satisfy the triangle inequality?

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Tip:

Always double-check your trigonometric values for obtuse angles, as they can affect the calculations significantly!