Which geometric shape has one pair of parallel sides, congruent non-parallel sides, and two pairs of congruent base angles?

The geometric shape that satisfies the conditions of having one pair of parallel sides, congruent non-parallel sides, and two pairs of congruent base angles is the isosceles trapezoid. In an isosceles trapezoid, the definition dictates that it has one pair of opposite sides that are parallel and the other pair of sides (the non-parallel sides) are congruent (meaning they have the same length). Furthermore, the angles adjacent to each base are congruent, resulting in two pairs of congruent base angles.

This combination of properties is unique to isosceles trapezoids and distinctly sets them apart from the other listed shapes. A rectangle and a square have two pairs of parallel sides and do not fit the definition of having only one pair of parallel sides. A rhombus, while it has congruent sides, also does not have parallel sides in the same way as an isosceles trapezoid, as both pairs of opposite sides are parallel. Therefore, the isosceles trapezoid is the only shape that fulfills all the criteria mentioned in the question.