1. **State the problem:** We have a rectangle ABCD with diagonals AC and BD. The lengths of the diagonals are given as expressions: AC = $9x + 20$ and BD = $10x + 8$. 2. **Important property:** In a rectangle, the diagonals are equal in length. Therefore, we set the expressions equal: $$9x + 20 = 10x + 8$$ 3. **Solve for $x$:** Subtract $9x$ from both sides: $$\cancel{9x} + 20 = \cancel{9x} + 10x + 8 \Rightarrow 20 = x + 8$$ Subtract 8 from both sides: $$20 - 8 = x + \cancel{8} - \cancel{8} \Rightarrow 12 = x$$ 4. **Conclusion:** The value of $x$ is 12. This satisfies the property that the diagonals of a rectangle are equal in length.