1. **State the problem:** Given rectangle ABCD with angles at vertices B and C expressed as $(2x + 10)^\circ$ and $(3x - 30)^\circ$ respectively, find the value of $x$. 2. **Recall properties of rectangles:** In a rectangle, all angles are right angles, so each angle measures $90^\circ$. 3. **Set up equations:** Since angles at B and C are given by expressions, and both must equal $90^\circ$, we have: $$2x + 10 = 90$$ $$3x - 30 = 90$$ 4. **Solve the first equation:** $$2x + 10 = 90$$ Subtract 10 from both sides: $$2x + \cancel{10} - \cancel{10} = 90 - 10$$ $$2x = 80$$ Divide both sides by 2: $$\frac{\cancel{2}x}{\cancel{2}} = \frac{80}{2}$$ $$x = 40$$ 5. **Solve the second equation:** $$3x - 30 = 90$$ Add 30 to both sides: $$3x - \cancel{30} + \cancel{30} = 90 + 30$$ $$3x = 120$$ Divide both sides by 3: $$\frac{\cancel{3}x}{\cancel{3}} = \frac{120}{3}$$ $$x = 40$$ 6. **Check consistency:** Both equations give $x = 40$, confirming the solution. **Final answer:** $$\boxed{40}$$