Question: Any reflection is trivially a composition of reflections. Which of the following is true ofโ rotations?
A.
Any rotation can be expressed as a composition of reflections across three lines whose single intersection point is the center of rotation.
B.
Any rotation can be expressed as a composition of reflections across two perpendicular lines whose intersection point is the center of rotation.
C.
Any rotation can be expressed as a composition of reflections across two lines whose intersection point is the center of rotation.
D.
Any rotation can be expressed as a composition of reflections across two parallel lines that are equidistant from the center of rotation.
1. **Problem Statement:** Determine which statement about rotations and their expression as compositions of reflections is true.
2. **Key Concept:** A rotation in a plane can be represented as the composition of two reflections.
3. **Important Rule:** The composition of two reflections across lines that intersect at a point results in a rotation about that point by twice the angle between the lines.
4. **Explanation:**
- If the two lines intersect at the center of rotation, the rotation angle is twice the angle between these lines.
- The lines do not need to be perpendicular; any two lines intersecting at the center can generate a rotation.
- Reflections across parallel lines produce translations, not rotations.
5. **Evaluate each option:**
- A: Composition of reflections across three lines is unnecessary; two suffice.
- B: Two perpendicular lines produce a rotation of 180 degrees, but rotations of other angles require lines at other angles.
- C: Two lines intersecting at the center can produce any rotation angle by adjusting the angle between them.
- D: Parallel lines produce translations, not rotations.
6. **Conclusion:** The correct statement is:
$$\text{C. Any rotation can be expressed as a composition of reflections across two lines whose intersection point is the center of rotation.}$$