1. **Problem statement:** We have a six-pointed star made up of 6 identical quadrilaterals. Each quadrilateral has an angle $x$ inside the star and two opposite angles of $35^\circ$ at the tips. We need to find the size of angle $x$. 2. **Key idea:** The star is formed by 6 identical quadrilaterals arranged around a point. The angles around that point sum to $360^\circ$. 3. **Step 1: Understand the angles at the center.** Each quadrilateral contributes one angle $x$ at the center where all 6 meet. Since there are 6 such angles around the point, their sum is: $$6x = 360^\circ$$ 4. **Step 2: Solve for $x$.** $$x = \frac{360^\circ}{6} = 60^\circ$$ 5. **Step 3: Verify with the given $35^\circ$ angles.** Each quadrilateral has two $35^\circ$ angles at the tips, and the remaining angle is $x=60^\circ$. The sum of interior angles of a quadrilateral is $360^\circ$: $$x + 35^\circ + 35^\circ + \text{other angle} = 360^\circ$$ $$60^\circ + 70^\circ + \text{other angle} = 360^\circ$$ $$\text{other angle} = 360^\circ - 130^\circ = 230^\circ$$ This is consistent with a concave quadrilateral forming the star. **Final answer:** $$\boxed{60^\circ}$$