A parallelogram has sides of length $12.7 \mathrm{~cm}$ and $14.2 \mathrm{~cm}$. The longer diagonal has length $25.1 \mathrm{~cm}$. Find the angle opposite the longer diagonal.
What is the degree measure of the angle opposite the longer diagonal?
(Round to the nearest tenth as needed.)

Step 1 ：We are given a parallelogram with sides of length \(12.7 \mathrm{~cm}\) and \(14.2 \mathrm{~cm}\), and the longer diagonal has length \(25.1 \mathrm{~cm}\). We are asked to find the angle opposite the longer diagonal.

Step 2 ：We can use the Law of Cosines to solve this problem. The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and an angle \(\gamma\) opposite side c, the following equation holds: \(c^2 = a^2 + b^2 - 2ab\cos(\gamma)\).

Step 3 ：In this case, we know the lengths of all three sides of the triangle formed by one side of the parallelogram, the longer diagonal, and the line segment connecting the endpoints of these two lines. We can use these lengths to solve for the cosine of the angle opposite the longer diagonal.

Step 4 ：Let's denote the sides as follows: \(a = 12.7\), \(b = 14.2\), and \(c = 25.1\).

Step 5 ：Substituting these values into the Law of Cosines, we get \(\cos(\gamma) = -0.7404901852057231\).

Step 6 ：We can then use the arccosine function to find the angle itself. The result is \(\gamma = 2.4045957617284923\) radians.

Step 7 ：Finally, we convert this angle from radians to degrees to get \(\gamma = 137.8\) degrees.

Step 8 ：Final Answer: The degree measure of the angle opposite the longer diagonal is \(\boxed{137.8}\) degrees.