jmc

Let the measure of the angle be $x$, so $5^{\circ }$ more than four times the angle is $4x+5^{\circ }$. Since these two measures are complementary, we have $x+(4x+5^{\circ })=90^{\circ }$. Simplifying the left side gives $5x+5^{\circ }=90^{\circ }$, so $5x=85^{\circ }$ and $x=\boxed{17^{\circ }}$.