Final answer:
To show that quadrilateral ABCD is a rhombus given AB parallel and congruent to CD, the additional statement needed is B. BC=AD, as it confirms all four sides are congruent, which defines a rhombus.
Step-by-step explanation:
The student's question is about determining what additional statement shows that a quadrilateral ABCD, with diagonals intersecting at P and sides AB parallel and congruent to CD, is a rhombus. To prove that ABCD is a rhombus, we need to show that all four sides are equal in length. Given that AB is congruent to CD, we look at the options provided:
- A. AP=CP would imply that P is the midpoint of both diagonals, which is a characteristic of a rhombus.
- B. BC=AD is the statement that directly proves ABCD is a rhombus since this would mean all four sides are equal, satisfying the definition of a rhombus.
- C. △DPA≅△DPC might suggest equal angles but doesn't directly prove all sides equal.
- D. overline BC||overline AD shows opposite sides are parallel, characteristic of a parallelogram but not sufficient alone to prove a rhombus.
Thus, the correct option is B. BC=AD, as it ensures all sides of the quadrilateral are congruent, defining the shape as a rhombus.