Every pair of supplementary angles includes one obtuse angle?
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Not unless the parallelogram is a rectangle. In every parallelogram, consecutive angles are supplementary.
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).
-- The sum of all three angles inside every triangle is always 180 degrees. -- An obtuse angle is an angle with more than 90 degrees in it. -- So two obtuse angles have more than 180 degrees between them. -- So they can't both be in the same triangle, unless at least part of one of them is left hanging out.
Every triangle must have at least 2 acute angles, otherwise it is impossible to make the 3 sides connect.
No. Except for the case of a square (a special case of rhombus), a rhombus will have 2 congruent acute angles, and 2 congruent obtuse angles. The square has 4 right angles. In fact, every quadrilateral will have either all 4 angles equal to right angle (square and rectangle), or will have at least 1 obtuse angle (also at least 1 acute).