No. A trapezoid has two parallel sides. This is not possible in a concave quadrilateral.
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Yes, it is possible to draw a quadrilateral that is neither a trapezoid nor a parallelogram. One example is a kite, which has two pairs of adjacent sides that are equal in length but does not have parallel sides. Another example is a concave quadrilateral, where at least one of the interior angles is greater than 180 degrees, making it neither a trapezoid nor a parallelogram.
No, not every trapezoid is an isosceles trapezoid.
The altitude of a trapezoid bisects the bases of the trapezoid.
All the names to classify a trapezoid are a trapezoid and a quadrilateral.
A trapezoid can also take the form of an isosceles trapezoid