## Geometry: Common Core (15th Edition)

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