Let $ x $ be the number of degrees in the measure of $ \angle A $. Then we have \[ \frac{3x}{4(90-x)}=\frac{3}{14}, \]from the information "the ratio of three times the measure of $ \angle A $ to four times the measure of the complement of $ \angle A $ is $ 3:14 $." Multiplying both sides by $ \frac{2}{3} $ and clearing denominators, we find $$7x=180-2x\implies 9x=180\implies x=20.$$The measure of the complement of 20 degrees is $ \boxed{70} $ degrees. Note: The hypothesis "if all angles are measured in degrees'' is not necessary. The angle is determined uniquely by the given information regardless of the units used.