1. **State the problem:** We are given an isosceles triangle with two equal sides and two angles: one angle is $ (8x - 23)^\circ $ and the opposite angle is $ 34^\circ $. We need to solve for $ x $. 2. **Recall the properties of an isosceles triangle:** In an isosceles triangle, the angles opposite the equal sides are equal. Since the two sides are equal, the angles opposite them must be equal. Therefore, the angle $ (8x - 23)^\circ $ is equal to $ 34^\circ $. 3. **Set up the equation:** $$ 8x - 23 = 34 $$ 4. **Solve for $ x $:** Add 23 to both sides: $$ 8x - 23 + 23 = 34 + 23 $$ $$ 8x = 57 $$ 5. **Divide both sides by 8:** $$ \frac{\cancel{8}x}{\cancel{8}} = \frac{57}{8} $$ $$ x = \frac{57}{8} = 7.125 $$ 6. **Final answer:** $$ x = 7.125 $$