In trapezoid $ABCD$, the diagonals intersect at $E$, the area of $\triangle ABE$ is 72 and the area of $\triangle CDE$ is 50. What is the area of trapezoid $ABCD$?
1. **Identify the given areas and the relationship between the triangles:**
   - The area of $\triangle ABE$ is given as 72.
   - The area of $\triangle CDE$ is given as 50.
   - Let the area of $\triangle AED$ be $x$.
   - Let the area of $\triangle CEB$ be $x$.

2. **Set up the proportion based on the areas of the triangles:**
   Since $\triangle ABE$ and $\triangle CDE$ share the same height from $E$ to the bases $AB$ and $CD$ respectively, the areas of $\triangle AED$ and $\triangle CEB$ must be equal. Therefore, we can write:
   \[
   \frac{\text{Area of } \triangle CDE}{\text{Area of } \triangle AED} = \frac{\text{Area of } \triangle CEB}{\text{Area of } \triangle ABE}
   \]
   Given that the areas of $\triangle CDE$ and $\triangle ABE$ are 50 and 72 respectively, we have:
   \[
   \frac{50}{x} = \frac{x}{72}
   \]

3. **Solve the proportion for $x$:**
   \[
   \frac{50}{x} = \frac{x}{72}
   \]
   Cross-multiplying gives:
   \[
   50 \cdot 72 = x^2
   \]
   \[
   3600 = x^2
   \]
   Taking the square root of both sides, we get:
   \[
   x = \sqrt{3600} = 60
   \]

4. **Calculate the total area of trapezoid $ABCD$:**
   The total area of trapezoid $ABCD$ is the sum of the areas of the four triangles:
   \[
   \text{Area of } \triangle ABE + \text{Area of } \triangle CDE + \text{Area of } \triangle AED + \text{Area of } \triangle CEB
   \]
   Substituting the known areas:
   \[
   72 + 50 + 60 + 60 = 242
   \]

The final answer is $\boxed{242}$