1. **State the problem:** We need to find the unknown interior angle $x$ of a pentagon given the other four interior angles: $84^\circ$, $153^\circ$, $131^\circ$, and $105^\circ$. 2. **Formula for sum of interior angles of a polygon:** The sum of interior angles of an $n$-sided polygon is given by: $$\text{Sum} = (n-2) \times 180^\circ$$ For a pentagon, $n=5$, so: $$\text{Sum} = (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$$ 3. **Set up the equation:** The sum of all interior angles including $x$ must be $540^\circ$: $$84^\circ + 153^\circ + 131^\circ + 105^\circ + x = 540^\circ$$ 4. **Add the known angles:** $$84 + 153 = 237$$ $$237 + 131 = 368$$ $$368 + 105 = 473$$ So, $$473 + x = 540$$ 5. **Solve for $x$:** $$x = 540 - 473$$ $$x = 67$$ **Final answer:** $$\boxed{67^\circ}$$