1. **State the problem:** We need to find the value of $x$ given four angles in a quadrilateral-like shape: $40^\circ$, $2x$, $x + 100^\circ$, and $x$. 2. **Recall the rule:** The sum of the interior angles of any quadrilateral is $360^\circ$. 3. **Set up the equation:** $$40 + 2x + (x + 100) + x = 360$$ 4. **Simplify the equation:** $$40 + 2x + x + 100 + x = 360$$ $$40 + 100 + 2x + x + x = 360$$ $$140 + 4x = 360$$ 5. **Isolate $x$:** $$4x = 360 - 140$$ $$4x = 220$$ 6. **Divide both sides by 4:** $$x = \frac{220}{4}$$ $$x = \cancel{\frac{220}{4}} = 55$$ 7. **Final answer:** $$\boxed{55}$$ So, the value of $x$ is 55 degrees.