Final answer:
The counterexample to José's conjecture that the midpoints of the sides of any quadrilateral with two pairs of congruent sides can be connected to form a rectangle is (c) a trapezoid because its midpoints do not form a shape with all sides equal and all angles right angles.
Step-by-step explanation:
José's conjecture states that the midpoints of the sides of any quadrilateral with two pairs of congruent sides can be connected to form a rectangle. A counterexample to this conjecture would be a quadrilateral with two pairs of congruent sides that, when connected at the midpoints, does not form a rectangle. While a square and a rhombus both have two pairs of congruent sides and midpoints that connect to form rectangles, a trapezoid is the counterexample we're looking for.
The midpoints of a trapezoid's sides, when connected, do not form a rectangle, because only one pair of opposite sides are parallel, which is a necessity for a shape to be a rectangle. The other pair of connected midpoints won't generally be parallel nor equal in length, thus the figure formed would not have all right angles and all sides of equal length, which disqualifies it from being a rectangle. Therefore, the correct answer to the counterexample to José's conjecture is (c) a trapezoid.