1. **State the problem:** We have a six-pointed star made up of 6 identical quadrilaterals. Each quadrilateral has angles labeled $x$ and two angles of $35^\circ$. We need to find the size of angle $x$. 2. **Analyze the quadrilateral:** Each quadrilateral has 4 angles. Since the star is made of 6 identical quadrilaterals, and each has angles $x$, $x$, $35^\circ$, and $35^\circ$, the sum of the angles in one quadrilateral is: $$2x + 2 \times 35^\circ = 2x + 70^\circ$$ 3. **Sum of angles in a quadrilateral:** The sum of interior angles in any quadrilateral is $360^\circ$. Therefore: $$2x + 70^\circ = 360^\circ$$ 4. **Solve for $x$:** $$2x = 360^\circ - 70^\circ = 290^\circ$$ $$x = \frac{290^\circ}{2} = 145^\circ$$ 5. **Conclusion:** The size of angle $x$ is $145^\circ$.