A rhombus is a parallelogram that has all 4 sides congruent. If a parallelogram is a rhombus then it’s diagonals are perpendicular. In a rhombus it’s diagonals bisects a pair of opposite angles. If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus. Also if one diagonal of a parallelogram bisects a pair of opposite angles then it’s a rhombus.
A trapezoid is a quadrilateral with exactly one pair of parallel sides. The side on a trapezoid are called based and the non parallel sides are called legs. If a trapezoid is isosceles then each pair of base angles is congruent. If a trapezoid has one pair of congruent base angles then it is an isosceles trapezoid. It is isosceles only if it’s diagonals are congruent.
Alexis Andrews Geometry Blog 