Problem 91

In a quadrilateral $ABCD$, given that $AD$ is parallel to $BC$ and $ABCD$ is a parallelogram, we analyze the properties of a parallelogram to determine which of the given conditions must also be satisfied.

1. In a parallelogram, opposite sides are parallel. Given that $AD$ is parallel to $BC$, for $ABCD$ to be a parallelogram, $AB$ must also be parallel to $CD$.

2. In a parallelogram, consecutive angles are supplementary. This means that the sum of the degrees of any two consecutive angles equals $180^{\circ}$.

3. Since $AB$ is parallel to $CD$ and $AD$ is parallel to $BC$, by the property of a parallelogram, we have:
- $\angle A + \angle B = 180^{\circ}$ (consecutive angles on one side)
- $\angle C + \angle D = 180^{\circ}$ (consecutive angles on the opposite side)
- $\angle A + \angle D = 180^{\circ}$ (consecutive angles on one side)
- $\angle B + \angle C = 180^{\circ}$ (consecutive angles on the opposite side)

4. The question specifically asks which condition must also be satisfied for $ABCD$ to be a parallelogram, given that $AD$ is parallel to $BC$. The condition that directly results from $ABCD$ being a parallelogram and not mentioned in the given conditions is $\angle B + \angle C = 180^{\circ}$, which is a property of consecutive angles in a parallelogram.

Therefore, the correct answer, based on the properties of a parallelogram and the given information, is:

$$\boxed{C}$$