#### Answer

There are two possible values for the angle $A$
$A =54^{\circ}20'$
$A = 125^{\circ}40'$

#### Work Step by Step

We can use the law of sines to find angle $A$:
$\frac{a}{sin~A} = \frac{b}{sin~B}$
$sin~A = \frac{a~sin~B}{b}$
$A = arcsin(\frac{a~sin~B}{b})$
$A = arcsin(\frac{(340~m)~sin~39^{\circ}50'}{268~m})$
$A = arcsin(\frac{(340~m)~sin~39.83}{268~m})$
$A = arcsin(0.81259)$
$A = 54.35^{\circ} = 54^{\circ}20'$
Note that another possible value for angle $A$ is $180^{\circ}- 54^{\circ}20'$ which is $125^{\circ}40'$