jmc

The angles of a regular $n$-gon have measure ${\left(\frac{180(n-2)}{n}\right)}^{\circ }$. Therefore the angles in a regular decagon measure $$y=\frac{180\cdot 8}{10}=144$$degrees.

We also note that since the larger angles of the quadrilateral are equal, and the three corresponding sides are equal, this is an isosceles trapezoid. Therefore we get the following angles:



The sum of the angle measures in a quadrilateral is always $360^{\circ }$, so we have $$360=x+x+y+y=x+x+144+144.$$Therefore $$x+x=360-144-144=72$$degrees, so $x=\boxed{36}$ degrees.