#### Answer

$m \angle F = 135^{\circ}$

#### Work Step by Step

Let's find the sum of the measures of the interior angles of the hexagon, which is given by the following formula:
sum of the measures of the interior angles = $(n - 2)180^{\circ}$, where $n$ is the number of sides in the polygon.
Let's plug in $6$ for $n$ because a hexagon has $6$ sides:
sum of the measures of the interior angles = $(6 - 2)180^{\circ}$
Evaluate what is in parentheses first:
sum of the measures of the interior angles = $(4)180^{\circ}$
Multiply to solve:
sum of the measures of the interior angles = $720^{\circ}$
Let's use what we are given to write an equation to find $m \angle F$:
$m \angle A + m \angle B + m \angle C + m \angle D + m \angle E + m \angle F = 720^{\circ}$
Plug in what we know:
$90^{\circ} + 90^{\circ} + m \angle C + m \angle D + m \angle E + m \angle F = 720^{\circ}$
Combine like terms:
$180^{\circ} + m \angle C + m \angle D + m \angle E + m \angle F = 720^{\circ}$
Subtract $180^{\circ}$ from each side of the equation to simplify:
$m \angle C + m \angle D + m \angle E + m \angle F = 540^{\circ}$
The sum of the measures of the four remaining angles equals $540^{\circ}$. We also know the four remaining angles are congruent to one another, so if we divide $540^{\circ}$ by $4$, then we will get the measure of each of the remaining angles, including $\angle F$:
$m \angle F = 540^{\circ}/4$
Divide to solve:
$m \angle F = 135^{\circ}$