Answer
The angle opposite the longer diagonal is $163.5^{\circ}$
Work Step by Step
Let $a = 25.9~cm$, let $b = 32.5~cm$, and let $c = 57.8~cm$.
We can use the law of cosines to find $C$, which is the angle opposite the longer diagonal:
$c^2 = a^2+b^2-2ab~cos~C$
$2ab~cos~C = a^2+b^2-c^2$
$cos~C = \frac{a^2+b^2-c^2}{2ab}$
$C = arccos(\frac{a^2+b^2-c^2}{2ab})$
$C = arccos(\frac{25.9^2+32.5^2-57.8^2}{(2)(25.9)(32.5)})$
$C = arccos(-0.9586)$
$C = 163.5^{\circ}$
The angle opposite the longer diagonal is $163.5^{\circ}$
