Answer
$\theta = 54^{\circ}$
Work Step by Step
Consider the right triangle with an angle of $\frac{\theta}{2}$.
Let $h$ be the hypotenuse, and let $a$ be the adjacent side.
Note that $h = a+b$.
We can find $a$:
$h^2 = a^2+50^2$
$(a+b)^2 = a^2+50^2$
$a^2+2ab+b^2 = a^2+50^2$
$2ab+b^2 = 50^2$
$a = \frac{50^2-b^2}{2b}$
$a = \frac{50^2-(12)^2}{(2)(12)}$
$a = 98.17$
We can find $\frac{\theta}{2}$:
$tan~\frac{\theta}{2} = \frac{50}{98.17}$
$\frac{\theta}{2} = arctan(\frac{50}{98.17})$
$\frac{\theta}{2} = 27^{\circ}$
Therefore, $\theta = 54^{\circ}$
