Find the measure of angle F in a triangle given the lengths of its sides.

1. Identify the problem type: This is a geometry problem involving a triangle where we need to find an angle given the lengths of all three sides. This suggests using the Law of Cosines.

1. Recall the Law of Cosines: The Law of Cosines states that for any triangle with sides of length $a$, $b$, and $c$, and with angle $C$ opposite side $c$, the following relationship holds: $c^2 = a^2 + b^2 - 2ab \cos(C)$.

3. Assign variables to the triangle: In triangle $FDE$, let $f$ be the length of the side opposite angle $F$, $d$ be the length of the side opposite angle $D$, and $e$ be the length of the side opposite angle $E$. From the diagram, we have:
- Side $DE$ has length $6$, so $f = 6$.
- Side $FD$ has length $11$, so $e = 11$.
- Side $FE$ has length $16$, so $d = 16$.
We want to find the measure of angle $F$, denoted as $m\angle F$.

4. Rearrange the Law of Cosines to solve for the angle: We need to find angle $F$. The Law of Cosines can be written in terms of angle $F$ as: $f^2 = d^2 + e^2 - 2de \cos(F)$.
To solve for $\cos(F)$, we rearrange the formula:
$2de \cos(F) = d^2 + e^2 - f^2$
$\cos(F) = \frac{d^2 + e^2 - f^2}{2de}$

5. Substitute the given side lengths into the formula:
- $d = 16$
- $e = 11$
- $f = 6$

$$\cos(F) = \frac{16^2 + 11^2 - 6^2}{2 \cdot 16 \cdot 11}$$