Final answer:
In a parallelogram, alternate interior angles formed by one transversal cutting through parallel opposite sides are supplementary, which is why ∠a and ∠d add up to 180 degrees.
Step-by-step explanation:
To prove that ∠a and ∠d are supplementary in a parallelogram abcd, we must show that their measures add up to 180 degrees. In any parallelogram, opposite sides are parallel and equal in length, and consecutive angles are supplementary. This is a consequence of the properties of parallel lines being intersected by a transversal. In parallelogram abcd, since ab is parallel to dc, when the transversal ad cuts these parallel lines it creates alternate interior angles ∠a and ∠d. Thus, these angles are supplementary by definition. Therefore, ∠a + ∠d = 180°.