1. **Stating the problem:** We are given multiple angle relationships involving expressions with $x$ and constants, mostly pairs of angles that are either opposite (vertical) angles or adjacent angles, some forming right angles. 2. **Key formulas and rules:** - Vertical angles are equal. - Adjacent angles on a straight line sum to 180°. - Adjacent angles forming a right angle sum to 90°. 3. **Solve the first problem (top-left):** Angles $(9x - 2)^6$ and $52^6$ are vertical angles, so they are equal. $$9x - 2 = 52$$ 4. **Solve for $x$:** $$9x = 52 + 2$$ $$9x = 54$$ Intermediate step showing cancellation: $$\cancel{9}x = \cancel{9}6$$ Divide both sides by 9: $$x = 6$$ 5. **Final answer:** $$x = 6$$ This completes the solution for the first angle relationship.