1. **Problem statement:** We have a pentagon with given angles 30°, 40°, and 90°, and we need to find the angle $x$ at the bottom-left corner. 2. **Key fact:** The sum of interior angles of a pentagon is given by the formula: $$\text{Sum of interior angles} = (5-2) \times 180^\circ = 540^\circ$$ 3. **Given angles:** 30°, 40°, and 90° are three of the pentagon's interior angles. 4. **Unknown angles:** Let the angle at the bottom-left corner be $x$, and the remaining angle be $y$. 5. **Equation setup:** Using the sum of interior angles, $$30^\circ + 40^\circ + 90^\circ + x + y = 540^\circ$$ 6. **Using the tick marks and equal sides:** The diagonal with three tick marks and the sides with equal tick marks imply certain triangles are isosceles, which helps relate $x$ and $y$. 7. **From the isosceles triangles, we find that $y = x$** (because the sides opposite equal angles are equal). 8. **Substitute $y = x$ into the equation:** $$30 + 40 + 90 + x + x = 540$$ $$160 + 2x = 540$$ 9. **Solve for $x$:** $$2x = 540 - 160 = 380$$ $$x = \frac{380}{2} = 190^\circ$$ 10. **Conclusion:** The angle $x$ at the bottom-left corner is $190^\circ$. This large angle is consistent with the pentagon's shape and the given constraints.