Yes. A very "flat" kite.
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A kite can have 1, 2 or 3 acute angles.
A quadrilateral kite can have two obtuse angles and two acute angles that add up to 360 degrees.
Yes, they can. An example of this is when a kite's opposite angles are both 90°. (90° + 90° = 180°) In the example, the kite is more specifically a square, but because of the Quadrilateral Hierarchy Theorem, this is possible.
Only if the kite is a rhombus or square. For the kite shape (aka deltoid), only the two sides have equal angles, and their sides are equilateral. The top and bottom angles are not equal.
a rhombus?