To find the degree measure of angle $A$ in a triangle with sides $a = 22.9$ m, $b = 13.5$ m, and $c = 21.6$ m, we can use the Law of Cosines. The Law of Cosines states:

$a^2 = b^2 + c^2 - 2bc \cos A$

We can rearrange this to solve for $\cos A$:

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

Let's substitute the given values into this equation:

$\cos A = \frac{13.5^2 + 21.6^2 - 22.9^2}{2 \cdot 13.5 \cdot 21.6}$

First, calculate each term:

$13.5^2 = 182.25$ $21.6^2 = 466.56$ $22.9^2 = 524.41$

Substitute these values back into the equation:

$\cos A = \frac{182.25 + 466.56 - 524.41}{2 \cdot 13.5 \cdot 21.6}$ $\cos A = \frac{124.4}{583.2}$ $\cos A = 0.2133$

Next, find the angle $A$ by taking the inverse cosine:

$A = \cos^{-1}(0.2133)$

Using a calculator:

$A \approx 77.7^\circ$

Thus, the degree measure of angle $A$ is approximately $77.7^\circ$.

Would you like more details or have any questions?

Here are some related questions:

1. How would you find the measures of the other two angles in the triangle?
2. What is the Law of Cosines and when is it used?
3. How can you verify the accuracy of your calculated angles?
4. What are some practical applications of the Law of Cosines?
5. How can the Law of Sines be used in conjunction with the Law of Cosines?

Tip: Always double-check your calculations to ensure accuracy, especially when dealing with trigonometric functions and their inverses.