No.
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There is one pair of parallel sides. A trapezoid can have at most one right angle. Not all trapezoids have right angles.
trapezoid
A trapezoid is a quadrilateral that has one pair of parallel sides. Since a right angle is formed by two perpendicular lines, it would mean that one of the non-parallel sides would have to be perpendicular to one of the parallel sides, which violates the definition of a trapezoid. Therefore, a trapezoid cannot have a right angle.
A rectangle has four right angles, two parallel pairs of line segments, and the lines are perpendicular. A trapezoid has one pair of parallel angles, and the angles do not have to be right, although a trapezoid with a right angle is called a right angle trapezoid, and a trapezoid with no parallel segments is a trapezium.
Yes. Imagine a right-angled triangle with one of its legs forming the base. Cut of the apex (the top angle) using a line parallel to the base. You will then have a trapezoid with a right angle. In fact, since a trapezoid contains a pair of parallel lines, it has two pairs of supplementary angles (add up to 180 deg). So a trapezoid cannot have only one right angle. If there is one, there must be 2 (or 4).