To determine which side of a triangular rooftop terrace (modeled by triangle \(ABC\)) has the greatest length, we can use the properties of triangles and angle-side relationships. Specifically, we will utilize the Law of Sines and the property that in any triangle, the side opposite the largest angle is the longest.
In triangle \(ABC\):
- The measure of \(\angle A\) is \(55^\circ\).
- The measure of \(\angle B\) is \(65^\circ\).
- The measure of \(\angle C\) is \(60^\circ\).
Step-by-step, here is how we can determine which side is the longest:
1. Identify the Largest Angle:
We compare the given angle measures:
- \(\angle A = 55^\circ\)
- \(\angle B = 65^\circ\)
- \(\angle C = 60^\circ\)
Clearly, \(\angle B = 65^\circ\) is the largest angle among \(\angle A\), \(\angle B\), and \(\angle C\).
2. Determine the Side Opposite the Largest Angle:
In any triangle, the side opposite the largest angle is the longest. For our triangle \(ABC\):
- The side opposite \(\angle A\) (\(55^\circ\)) is \(\overline{BC}\).
- The side opposite \(\angle B\) (\(65^\circ\)) is \(\overline{AC}\).
- The side opposite \(\angle C\) (\(60^\circ\)) is \(\overline{AB}\).
Since \(\angle B\) (\(65^\circ\)) is the largest angle, the longest side is \(\overline{AC}\).
Thus, the correct answer is:
[tex]\[
\boxed{\overline{AC}}
\][/tex]
So, the side of the terrace with the greatest length is \(\overline{AC}\). Therefore, the final answer is:
A. [tex]\(\overline{AC}\)[/tex].
