1. **State the problem:** We are given three angles at vertex S in a triangle formed by two tangents to a circle: 34°, (x + 6)°, and (3x - 2)°. We need to find the value of $x$. 2. **Recall the rule:** The sum of angles around a point is 360°. Since these three angles meet at vertex S and form a full circle around that point, their sum is 360°. 3. **Set up the equation:** $$34 + (x + 6) + (3x - 2) = 360$$ 4. **Simplify the equation:** $$34 + x + 6 + 3x - 2 = 360$$ $$34 + 6 - 2 + x + 3x = 360$$ $$38 + 4x = 360$$ 5. **Isolate $x$:** $$4x = 360 - 38$$ $$4x = 322$$ 6. **Divide both sides by 4:** $$x = \frac{322}{4}$$ $$x = \cancel{\frac{322}{\cancel{4}}} \Rightarrow x = 80.5$$ 7. **Final answer:** $$\boxed{80.5}$$ This means the value of $x$ that satisfies the angle conditions at vertex S is 80.5 degrees.