1. **Stating the problem:** We have two circle-related angle problems. **Problem 1:** Given angles on a circle with points S, T, U, and W, where the exterior angle at S is 34°, and the arcs UT and SV measure $(x+6)°$ and $(3x-2)°$ respectively. **Problem 2:** A circle tangent to lines FG and GH at point G, with angle $75°$ between FG and GH, and an interior angle of $17x°$ inside the circle. We will solve only the first problem as per instructions. 2. **Formula and rules for Problem 1:** The exterior angle formed outside a circle by two secants equals half the difference of the intercepted arcs. Mathematically: $$\text{Exterior angle} = \frac{1}{2} |\text{arc}_1 - \text{arc}_2|$$ 3. **Apply the formula:** Given exterior angle at S is $34°$, arcs are $(x+6)°$ and $(3x-2)°$. Set up the equation: $$34 = \frac{1}{2} |(3x - 2) - (x + 6)|$$ 4. **Simplify inside the absolute value:** $$(3x - 2) - (x + 6) = 3x - 2 - x - 6 = 2x - 8$$ So: $$34 = \frac{1}{2} |2x - 8|$$ 5. **Multiply both sides by 2:** $$2 \times 34 = |2x - 8|$$ $$68 = |2x - 8|$$ 6. **Solve the absolute value equation:** Two cases: Case 1: $$2x - 8 = 68$$ $$2x = 76$$ $$x = 38$$ Case 2: $$2x - 8 = -68$$ $$2x = -60$$ $$x = -30$$ 7. **Check for valid solution:** Since angles must be positive, $x = -30$ is invalid. 8. **Final answer:** $$\boxed{x = 38}$$