jmc

The angles of a regular $n$-gon have measure ${\left(\frac{180(n-2)}{n}\right)}^{\circ }$. Therefore the angles in a regular heptagon measure $$y=\frac{180\cdot 5}{7}=\frac{900}{7}$$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 $$360=x+x+y+y=x+x+\frac{900}{7}+\frac{900}{7}.$$Therefore $$\begin{array}{rl}x+x& =\frac{360\cdot 7-900-900}{7}\\ & =\frac{180\cdot 14-180\cdot 5-180\cdot 5}{7}\\ & =\frac{180\cdot 4}{7}\\ & =\frac{720}{7}.\end{array}$$Since $x$ is half of that, $x=\boxed{\frac{360}{7}}$ degrees.