A parallelogram
a rectangle
A kite is a quadrilateral with two pairs of adjacent congruent sides. The angles between the unequal sides are equal.
This is a rectangle.
You have described a rectangle.
A General Quadrilateral
A rectangle
A rectangle (that is not a square).
This is a parallelogram. The first requirement is 2 pairs of congruent sides where the congruent sides are not adjacent. This is like a rectangle (excluding a square) that has two pairs of congruent sides where the congruent sides are not adjacent. But the angles are not all congruent (as set in the question) which pushes the shape into the "next less regular" shape, the parallelogram. The angles will not all be congruent, but it will have 2 pairs of congruent angles. There is no way to avoid the 2 pairs of congruent angles because of the requirement that the shape must have 2 pairs of congruent sides (the first requirement).
your mom, and your cousin
One pair of equal angles can be found in an isosceles triangle, where the angles opposite the equal sides are congruent. For the equal adjacent sides, consider a rectangle, where each pair of adjacent sides (length and width) is equal to the corresponding sides on the opposite side. Thus, in a rectangle, we have two pairs of equal adjacent sides, while the opposite angles are also equal.
Adjacent angles
An isosceles trapezoid