Answer
$A=22.62^{\circ}$.
$B=67.38^{\circ}$.
$a=30\;mi$.
Work Step by Step
The given values are
$b=72.0\;mi$
$c=78.0\;mi$.
By using trigonometric ratios.
$\frac{b}{c}=\cos{A}$
Isolate $A$.
$A=\cos^{-1}\left(\frac{b}{c} \right )$
Plug all values.
$A=\cos^{-1}\left(\frac{72.0\;mi}{78.0\;mi} \right )$
By using degree calculator simplify.
$A=22.62^{\circ}$ (rounded value).
In a right angle triangle sum of other two angles is $90^{\circ}$.
$A+B=90^{\circ}$
Isolate $B$.
$B=90^{\circ}-A$
Plug value of $A$.
$B=90^{\circ}-22.62^{\circ}$
Simplify.
$B=67.38^{\circ}$.
By using Pythagorean theorem.
$a=\sqrt{c^2-b^2}$
Plug all values.
$a=\sqrt{(78.0\;mi)^2-(72.0\;mi)^2}$
Simplify.
$a=\sqrt{6084\;mi^2-5184\;mi^2}$
$a=\sqrt{(900\;mi^2)}$
$a=30\;mi$.
