1. **State the problem:** We are given a circle tangent to a vertical line and a slanted line, with angles labeled 17x° and 75°. We need to solve for $x$. 2. **Identify the relationship:** The angles around a point on a straight line sum to 180°. Since the circle is tangent to the vertical line and the slanted line, the angle sum at the vertex is 180°. 3. **Write the equation:** $$17x + 75 = 180$$ 4. **Solve for $x$:** $$17x = 180 - 75$$ $$17x = 105$$ 5. **Divide both sides by 17:** $$x = \frac{105}{17}$$ 6. **Simplify the fraction:** $$x = \frac{\cancel{105}}{\cancel{17}}$$ (no common factors to cancel) 7. **Final answer:** $$x = \frac{105}{17} \approx 6.18$$ So, the value of $x$ is approximately 6.18 degrees.