1. **State the problem:** We have a rectangle ABCD with diagonals intersecting at point E. The diagonal segment AE is given as 20, and the diagonal segment DE is given as $m - 5$. We need to find the value of $m$. 2. **Recall properties of rectangles:** In a rectangle, the diagonals are equal in length and they bisect each other. This means that the diagonals intersect at their midpoints. 3. **Apply the property:** Since E is the midpoint of diagonal AC and diagonal BD, the segments AE and DE must be equal in length. 4. **Set up the equation:** $$AE = DE$$ $$20 = m - 5$$ 5. **Solve for $m$:** $$m - 5 = 20$$ $$m = 20 + 5$$ $$m = 25$$ 6. **Conclusion:** The value of $m$ is 25.