Regular hexagons have all obtuse angles
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No.
Four exactly? A regular hexagon has 6. But a hexagon can have four obtuse angles. A regular hexagon can be made a lot flatter so that the two angles at the sides become acute, while the remaining 4 are obtuse.
If 1 or 2 of the obtuse angles are 'bent into' the hexagon then you can do it.A hexagon has 720° sum of interior angles. If you have 2 right angles, then that's 180°, leaving 540° to be distributed over four angles. If the two acute angles are 89°, then that leaves 362° to be divided between two angles, so at least one of these two angles will need to be greater than 180°. Below is my text graphics attempt of one possibility (ignore the 'dot'). Note that both of the obtuse angles are greater than 180°:|\_/||__.|Look at the related link to play with different angle possibilities for polygons.
If the triangle and hexagon are equilateral, then the trapezoid is not like the others because all of its angles are not equal. A rectangle will have 4 equal right angles. An equilateral triangle will have 3 equal acute 60 degree angles. An equilateral hexagon will have 6 equal obtuse 120 degreeangles.
A hexagon has no right angles.