#### Answer

$m \angle E \approx 88.5^{\circ}$

#### Work Step by Step

First, we want to know what angle is opposite the side in question. The side that is opposite to the angle we are looking for is $\overline{FD}$, so let's plug in what we know into the formula for the law of cosines:
$27^2 = 13^2 + 24^2 - 2(13)(24)$ cos $\angle E$
Evaluate exponents first, according to order of operations:
$729 = 169 + 576 - 2(13)(24)$ cos $m \angle E$
Add to simplify on the right side of the equation:
$729 = 745 - 2(13)(24)$ cos $m \angle E$
Multiply to simplify:
$729 = 745 - 624$ cos $m \angle E$
Subtract $745$ from each side of the equation to move constants to the left side of the equation:
$-16 = -624$ cos $m \angle E$
Divide each side by $-624$:
cos $m \angle E \approx 0.02564$
Take $cos^{-1}$ to solve for $\angle E$:
$m \angle E \approx 88.5^{\circ}$