To find the measure of the angle opposite the longest side of the triangle with sides 17 ft, 19 ft, and 15 ft, we can use the Law of Cosines. Here, the longest side is 19 ft.
According to the Law of Cosines, the formula is:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
where \(c\) is the length of the side opposite angle \(C\), and \(a\) and \(b\) are the lengths of the other two sides.
In our case:
- \(a = 17\) ft
- \(b = 15\) ft
- \(c = 19\) ft
We want to find angle \(C\):
\[ 19^2 = 17^2 + 15^2 - 2 \cdot 17 \cdot 15 \cdot \cos(C) \]
Calculating \(a^2\), \(b^2\), and \(c^2\):
\[ 19^2 = 361,\quad 17^2 = 289,\quad 15^2 = 225 \]
Substituting these values into the formula:
\[ 361 = 289 + 225 - 2 \cdot 17 \cdot 15 \cdot \cos(C) \]
Calculating \(289 + 225\):
\[ 289 + 225 = 514 \]
So our equation becomes:
\[ 361 = 514 - 2 \cdot 17 \cdot 15 \cdot \cos(C) \]
Rearranging gives us:
\[ 2 \cdot 17 \cdot 15 \cdot \cos(C) = 514 - 361 \]
Calculating the right-hand side:
\[ 514 - 361 = 153 \]
Now, calculating \(2 \cdot 17 \cdot 15\):
\[ 2 \cdot 17 \cdot 15 = 510 \]
So we have:
\[ 510 \cdot \cos(C) = 153 \]
Now, solving for \(\cos(C)\):
\[ \cos(C) = \frac{153}{510} = \frac{51}{170} \]
Using a calculator to find the angle:
\[ C = \cos^{-1}\left(\frac{51}{170}\right) \]
Calculating \(C\):
\[ C \approx 107.0^\circ \]
Thus, rounding to the nearest degree, the measure of the angle opposite the longest side is:
\[ \boxed{107^\circ} \]