Not necessarily. Both may be right angles.
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A rhombus has a pair of opposite equal acute angles and a pair of opposite equal obtuse angles and the four angles add up to 360 degrees.
Not necessarily. A linear pair of angles must be supplementary but supplementary angles need not form a linear pair. For example, the opposite angles of a cyclic quadrilateral are supplementary but they are (by definition) not next to one another.
Angles are usually illustrated as two acute and two obtuse, but there can be two right, one acute and one obtuse. Angles cannot be parallel since that is a characteristic of lines, not angles!
Ordinarily, yes: one is acute (less than 90 degrees) and one is obtuse (more than 90 degrees), such that their sum is 180 degrees. The exception is if both angles are right angles (2 x 90 degrees).
No, a trapezoid does not have four obtuse angles. A trapezoid has one pair of parallel sides and the other pair of non-parallel sides. The angles of a trapezoid can be a combination of acute, obtuse, and right angles, but it cannot have four obtuse angles.