If a figure has a rotational symmetry of 180^(∘), it means that we can flip it upside down and it will map onto itself. For example, a rectangle has 180^(∘) rotational symmetry.
If the rectangle also had a 90^(∘) rotational symmetry, we would be able to rotate it a quarter of a full rotation about the center and have it map onto itself as well. But this is not possible for a rectangle, because a pair of sides is longer than the other.
Therefore, it's possible for a figure to have 180^(∘) rotational symmetry but not 90^(∘).
