1. **Problem statement:** We need to find the value of $x$ given a geometric figure with angles $\alpha$, $2\alpha$, and $x + \alpha$. 2. **Understanding the problem:** The figure shows a large triangle with angles $\alpha$, $2\alpha$, and a smaller triangle inside with two angles marked $2\alpha$. One side is labeled $x + \alpha$. We want to find $x$ in terms of $\alpha$. 3. **Key geometric rule:** The sum of angles in any triangle is $180^\circ$. So for any triangle with angles $A$, $B$, and $C$, we have: $$A + B + C = 180^\circ$$ 4. **Apply the rule to the smaller triangle:** The smaller triangle has two angles $2\alpha$ and the third angle is $x$ (since the side labeled $x + \alpha$ corresponds to angle $x + \alpha$, the angle inside the triangle is $x$). So, $$2\alpha + 2\alpha + x = 180^\circ$$ 5. **Simplify:** $$4\alpha + x = 180^\circ$$ 6. **Solve for $x$:** $$x = 180^\circ - 4\alpha$$ 7. **Final answer:** $$\boxed{x = 180^\circ - 4\alpha}$$ This means $x$ depends on $\alpha$ and is the difference between $180^\circ$ and $4\alpha$.