1. **State the problem:** We are given a circle with a chord and three angles: an outside angle labeled $74^\circ$, an inside angle adjacent to the chord labeled $81^\circ$, and another outside angle labeled $(17x - 23)^\circ$. We need to find the value of $x$. 2. **Recall the circle angle rules:** The outside angle formed by two secants, tangents, or a secant and a tangent is equal to half the difference of the intercepted arcs. Here, the two outside angles $74^\circ$ and $(17x - 23)^\circ$ are related to the inside angle $81^\circ$. 3. **Set up the equation:** The sum of the two outside angles equals twice the inside angle adjacent to the chord: $$74 + (17x - 23) = 2 \times 81$$ 4. **Simplify the equation:** $$74 + 17x - 23 = 162$$ $$51 + 17x = 162$$ 5. **Isolate $x$:** $$17x = 162 - 51$$ $$17x = 111$$ 6. **Divide both sides by 17:** $$x = \frac{111}{17}$$ 7. **Simplify the fraction:** $$x = \cancel{\frac{111}{17}} = 6.5294117647$$ (approximate decimal) **Final answer:** $$x = \frac{111}{17}$$ or approximately $6.53$