jmc

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 $$7x=180-2x⟹9x=180⟹x=20.$$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.