No
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Every pair of supplementary angles includes one obtuse angle?
Angles larger than a right angle and smaller than two right angles (between 90° and 180°) are called obtuse angles("obtuse" meaning "blunt").
No, they cannot.
Yes if they are right angles
Two angles are adjacent if they have the same vertex, share a side and do not overlap. It is, therefore, perfectly possible for two obtuse angles to be adjacent. In fact, every pair of adjacent angles in a hexagonal tessellation (a honeycomb, for example), consists of a pair of obtuse angles (120 degrees).