Question

Which statements are true? Check all that apply.

If a figure has point symmetry, it also must have rotational symmetry.
If a figure has rotational symmetry, it also must have point symmetry.
If a figure has point symmetry, it also must have 4-fold reflectional symmetry.
All parallelograms have point symmetry.
A triangle never has point symmetry.

Answer 1

True: 1, 4, 5. False: 2, 3.

Brief reasons:
1. True — point symmetry means 180° rotational symmetry, so it is a type of rotational symmetry.
2. False — rotational symmetry might be by e.g. 120° (equilateral triangle), which does not give a 180° rotation.
3. False — point symmetry does not imply reflection axes; a non‑rectangular parallelogram has point symmetry but no reflection symmetry.
4. True — every parallelogram is invariant under a 180° rotation about the intersection of its diagonals.
5. True — a nondegenerate triangle cannot have 180° rotational (point) symmetry (an odd number of vertices cannot pair up under a half‑turn).