1. **Problem Statement:** We have a circle with center $O$, a tangent line $Pi$ touching the circle at point $i$, and a chord $\overline{iQ}$. The chord $\overline{iQ}$ subtends an angle $3x$ at the center $O$ and an angle $x + 120^\circ$ at a point $S$ on the circumference below the chord. 2. **Relevant Theorem:** The angle subtended by a chord at the center of a circle is twice the angle subtended at any point on the circumference on the same side of the chord. This means: $$\text{Angle at center} = 2 \times \text{Angle at circumference}$$ 3. **Apply the theorem:** Given the angle at center is $3x$ and the angle at circumference is $x + 120^\circ$, we have: $$3x = 2(x + 120^\circ)$$ 4. **Solve for $x$:** $$3x = 2x + 240^\circ$$ $$3x - 2x = 240^\circ$$ $$x = 240^\circ$$ 5. **Find the angle between the chord $\overline{iQ}$ and the tangent $Pi$:** The angle between a tangent and a chord through the point of contact equals the angle subtended by the chord in the alternate segment. This angle is the angle at $S$ on the circumference, which is: $$x + 120^\circ = 240^\circ + 120^\circ = 360^\circ$$ However, an angle of $360^\circ$ is a full rotation, so the effective angle is $0^\circ$. This suggests a reconsideration: since $x=240^\circ$ is greater than $180^\circ$, the problem likely expects the smaller angle. The angle at $S$ should be interpreted modulo $180^\circ$: $$x + 120^\circ = 240^\circ + 120^\circ = 360^\circ \equiv 0^\circ$$ Thus, the angle between the chord and tangent is $0^\circ$, meaning they are collinear at point $i$. **Final answers:** - $x = 240^\circ$ - Angle between chord $\overline{iQ}$ and tangent $Pi$ is $0^\circ$ (they are tangent at point $i$).