Yes a trapezoid is a quadrilateral. Quadrilateral means "four sided", and trapezoids have four sides.
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No, not every quadrilateral is a trapezoid, although a trapezoid is also a quadrilateral because it has four sides.
No. A quadrilateral is any four-sided figure. A trapezoid is one example of a quadrilateral.
For the same reason that a cat is an animal but an animal is not always a cat. Because a trapezoid is a special kind of quadrilateral. A quadrilateral is anything with four sides. A trapezoid has four sides, and also two of them are parallel to each other. Therefore some quadrilaterals are trapezoids, but most of them aren't.
All the names to classify a trapezoid are a trapezoid and a quadrilateral.
A trapezoid will always be a 4 sided quadrilateral.