To solve this problem, we need to determine the smallest angle of rotation for which the shape looks the same as it initially did.

Analysis:

The shape shown is symmetrical along its vertical axis and also has a reflection symmetry horizontally. Additionally, it appears that if you rotate the shape by 180 degrees, it looks identical to its original position.

Conclusion:

The smallest angle at which this shape can be rotated to look exactly the same as it did initially is 180 degrees.

Would you like more details on symmetry and rotational transformations? Here are related questions you might find interesting:

1. What is rotational symmetry, and how is it determined?
2. How do reflectional and rotational symmetries differ?
3. Can a shape have both reflectional and rotational symmetries?
4. How would you calculate the angle of rotation for regular polygons?
5. What is the order of symmetry for common geometric shapes?

Tip: For any shape, if rotating it by an angle of 180 degrees makes it look the same, it has rotational symmetry of order 2.