How about a rectangle which is a quadrilateral with sides of different lengths
a rectangle (that is NOT a square) has 4 right angles and consecutive sides of unequal lengths.
The quadrilateral you are describing is a scalene trapezoid. It has two sides of unequal lengths (the long sides) and two other sides of equal, shorter lengths (the short sides). This shape does not exhibit reflective symmetry and lacks right angles, fitting your criteria perfectly.
They are unequal in length but one of the bisects the other at right angles.
A trapezoid can have 2 right angles and 1 pair of opposite parallel sides of different lengths. The 4 interior angles will add up to 360 degrees.
rectangle
a rectangle (that is NOT a square) has 4 right angles and consecutive sides of unequal lengths.
That's a triangle in which all three sides have three different lengths.
A trapezium (has only one pair of unequal parallel lines).
Rhoumbus
A rhomboid
A shape with four unequal sides is called an irregular quadrilateral.
A right trapezium or an irregular quadrilateral.
They are unequal in length but one of the bisects the other at right angles.
rectangle
A trapezoid can have 2 right angles and 1 pair of opposite parallel sides of different lengths. The 4 interior angles will add up to 360 degrees.
A right trapezoid.
A rectangle.