Find the angle whose complement is one-third of its supplement.

Let the measure of the required angle be $x^{\circ}$

Then, the measure of its complement $=(90-x)^{\circ}$.

And the measure of its supplement $=(180-x)^{\circ}$.

Therefore,

$(90-x)=\frac{1}{3}(180-x)$

$\Rightarrow 3(90-x)=(180-x)$

$\Rightarrow 270-3 x=180-x$

$\Rightarrow 2 x=90$

$\Rightarrow x=45$

Hence, the measure of the required angle is $45^{\circ}$.