PQRSTUVW is a regular octagon, then the measure of  $\angle VPQ$ is

Sum of interior angles of a regular octagon

$\text{=}(n-2)180°$

$=(8-2)180°$

$=6\times 180°$

$=1080°$

$\therefore \text{each\hspace{0.17em}interior\hspace{0.17em}angle}=\frac{1080°}{8}=135°$

$\text{In\hspace{0.17em}}\Delta VWP,\angle WVP+\angle VWP+\angle WPV=180°$

$x°+135°+x°=180°$

$2x°+135°=180°$

$2x°=180°-135°=45°$

$x=\frac{45°}{2}=22.5°$

$\therefore \angle VPQ=135°-22.5°=112.5°$