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You can get 2 triangles by cutting a parallelogram in half
Two isosceles right triangles, if joined hypotenuse-to-hypotenuse will make a square. Two squares, sided by side, make a rectangle which is a parallelogram, and not a square.
If you draw one diagonal across a parallelogram, it will split it into two congruent triangles. A rectangle is a parallelogram, with all four angles equal to 90°.
No, the diagonals of a parallelogram do not necessarily bisect the angles. The diagonals of a parallelogram divide it into four congruent triangles, but they do not necessarily bisect the angles of those triangles.
You would get two scalene triangles.