Final Answer:
a. A 180° clockwise rotation and a 180° counterclockwise rotation around the same axis describe only one symmetry because they result in the same orientation of the figure after rotation.
b. A 360° rotation with one axis and a 360° rotation with a different axis describe only one symmetry because a full rotation (360°) brings the figure back to its original position, regardless of the axis.
Step-by-step explanation:
a. When a figure undergoes a 180° clockwise rotation followed by a 180° counterclockwise rotation around the same axis, the net rotation is 180° + (-180°) = 0°. This means the figure returns to its original position, indicating a single symmetry. Mathematically, this can be expressed as R₁ * R₂ = R₂ * R₁, where R₁ and R₂ represent the two rotations.
b. In the case of a 360° rotation, the figure returns to its original position, forming a single symmetry. Whether the rotations are around the same axis or different axes, the net result is the same – a complete restoration of the original orientation. This can be represented as R₃ * R₄ = R₄ * R₃, where R₃ and R₄ denote the two 360° rotations.
In both scenarios, the mathematical expressions confirm the commutative property of rotations, emphasizing that the order in which rotations are applied does not affect the final symmetry. The figures exhibit rotational symmetry, and the pairs of rotations described maintain this symmetry, contributing to the overall understanding of rotational transformations in geometry.