#### Answer

$a = 0.4832~m$
$b = 0.3934~m$
$A = 50^{\circ}51'$

#### Work Step by Step

We can convert angle B to degrees:
$B = 39^{\circ}09' = (39+\frac{9}{60})^{\circ} = 39.15^{\circ}$
We can use angle B and angle C to find angle A:
$A = 180^{\circ}-90^{\circ}-39.15^{\circ} = 50.85^{\circ}$
$A = 50.85^{\circ}$ which is $50^{\circ}51'$
We can use angle B and $c$ to find $a$:
$cos~B = \frac{a}{c}$
$a = (c)~cos~B$
$a = (0.6231~m)~cos(39.15^{\circ})$
$a = 0.4832~m$
We can use the Pythagorean theorem to find $b$:
$b^2 = c^2-a^2$
$b = \sqrt{c^2-a^2}$
$b = \sqrt{(0.6231~m)^2-(0.4832~m)^2}$
$b = 0.3934~m$