Skill: Identify rotational symmetry

Rotational symmetry means a figure matches itself when it is turned around a fixed center by an angle less than $360^\circ$. The figure must have the same position and appearance after the turn.

To identify it:

1. Find the center of the figure.
2. Imagine turning the figure around that center.
3. Check whether it matches itself before a full turn.
4. Count how many times it matches during one full turn. This is the order of rotational symmetry.
5. Find the smallest angle using

$$\text{smallest angle}=\frac{360^\circ}{\text{order}}.$$

Example

A parallelogram is shown. Does it have rotational symmetry? If so, find its order and smallest angle.

1. The center is where the diagonals cross.
2. Imagine rotating the parallelogram $180^\circ$ around this center.
3. Each vertex moves to the opposite vertex. The parallelogram matches its original position.
4. A smaller turn, such as $90^\circ$, does not make it match.
5. During one full turn, it matches itself twice: at $180^\circ$ and again at $360^\circ$.

Therefore, the parallelogram has rotational symmetry of order $2$.

Its smallest angle is

$$\frac{360^\circ}{2}=180^\circ.$$

So, the answer is: order $2$ and smallest angle $180^\circ$. This is different from reflection symmetry, which involves flipping a figure across a line.