**Solution**

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

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

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

$\Rightarrow 540-3 x=x$

$\Rightarrow 4 x=540 \Rightarrow x=135$.

Hence, the measure of the given angle is $135^{\circ}$ and the measure of its supplement is $(180-135)^{\circ}=45^{\circ}$.