1. **Problem statement:** We have a star made of 6 identical quadrilaterals. Each quadrilateral has an angle $x$ at the star's points and two adjacent angles of $35^\circ$. We need to find the size of angle $x$. 2. **Key fact:** The star is formed by 6 identical quadrilaterals arranged around a point. The angles $x$ are at the tips of the star, meeting at the center. 3. **Sum of angles around a point:** The sum of all angles meeting at a point is $360^\circ$. Since there are 6 identical angles $x$ meeting at the center, $$6x = 360^\circ$$ 4. **Solve for $x$:** $$x = \frac{360^\circ}{6} = 60^\circ$$ 5. **Check with quadrilateral angle sum:** Each quadrilateral has 4 angles summing to $360^\circ$. Two angles are $35^\circ$, one is $x=60^\circ$, so the fourth angle is $$360^\circ - (35^\circ + 35^\circ + 60^\circ) = 360^\circ - 130^\circ = 230^\circ$$ This is consistent with a concave quadrilateral forming the star. **Final answer:** $$\boxed{60^\circ}$$