Symmetries of Polygons: Rotations and Reflections - Geometry

1. What is the smallest positive rotation that carries a general trapezoid onto itself?

   $360^\circ$. General trapezoid has no rotational symmetry except the identity.

2. A regular polygon has $15$ reflection axes. What is $n$?

   $15$. Regular polygon has $n$ reflection axes, so $n = 15$.

3. In a regular polygon with odd $n$, what does each reflection axis pass through?

   A vertex and the midpoint of the opposite side. For odd $n$, reflection axes pass through vertices and opposite edge midpoints.

4. In a regular polygon with even $n$, how many reflection axes go through opposite vertices?

   $\frac{n}{2}$. Half the reflection axes connect pairs of opposite vertices.

5. How many lines of reflection symmetry does a rhombus (non-square) have?

   $2$. Rhombus has reflection symmetry across both diagonal lines only.

6. How many rotational symmetries does a regular $n$-gon have (including $0^\circ$)?

   $n$. Includes identity rotation plus $n-1$ non-trivial rotations.

7. Identify the order of rotational symmetry of an isosceles trapezoid.

   $1$. Only the identity rotation maps isosceles trapezoid to itself.

8. Identify the reflection axis of an isosceles trapezoid.

   The line through midpoints of the bases, perpendicular to the bases. This line is equidistant from corresponding points on the legs.

9. Which option is a symmetry of an isosceles trapezoid: rotation $180^\circ$ or one reflection?

   One reflection. Isosceles trapezoid has reflection symmetry but not rotational.

10. Which rotation angles carry any rectangle onto itself (include $0^\circ$ and $360^\circ$)?

    $0^\circ,\ 180^\circ,\ 360^\circ$. These are multiples of the fundamental $180^\circ$ symmetry rotation.

11. Identify the reflection axes of a non-square rectangle in relation to its sides.

    The two midlines through the center, parallel to the sides. These lines bisect opposite sides and pass through the center.

12. Which is a symmetry of a non-square rectangle: rotation $90^\circ$ or $180^\circ$?

    $180^\circ$. Rectangle has $180^\circ$ rotational symmetry but not $90^\circ$.

13. Which option is a reflection symmetry of a non-square rhombus: diagonal or midline?

    Diagonal. Non-square rhombus reflects across diagonals, not midlines.

14. How many lines of reflection symmetry does a kite (non-rhombus) have?

    $1$. Kite has reflection symmetry across one diagonal only.

15. Identify the order of rotational symmetry of a general trapezoid.

    $1$. Only the identity rotation maps general trapezoid to itself.

16. What is the smallest positive rotation that carries a non-square rectangle onto itself?

    $180^\circ$. Rectangle maps to itself by rotation about its center every half-turn.

17. What is the smallest positive rotation that carries a regular $12$-gon onto itself?

    $30^\circ$. For $n = 12$, smallest rotation is $\frac{360^\circ}{12} = 30^\circ$.

18. Which rotation angles carry a square onto itself (include $0^\circ$ and $360^\circ$)?

    $0^\circ,\ 90^\circ,\ 180^\circ,\ 270^\circ,\ 360^\circ$. These are multiples of the fundamental $90^\circ$ symmetry rotation.

19. What is the center of rotation for a rectangle, parallelogram, or regular polygon?

    The intersection point of its diagonals (its center). The center is equidistant from all vertices of the polygon.

20. What is the smallest positive rotation that carries a kite (non-rhombus) onto itself?

    $360^\circ$. Non-rhombus kite has only one reflection symmetry, no rotations.

21. Which rotation angles carry any parallelogram onto itself (include $0^\circ$ and $360^\circ$)?

    $0^\circ,\ 180^\circ,\ 360^\circ$. These are multiples of the fundamental $180^\circ$ symmetry rotation.

22. What is the smallest positive rotation that carries a rhombus (non-square) onto itself?

    $180^\circ$. Non-square rhombus has 2-fold rotational symmetry like other parallelograms.

23. How many lines of reflection symmetry does a non-square rectangle have?

    $2$. Rectangle has reflection symmetry across horizontal and vertical midlines only.

24. What is the smallest positive rotation that carries any parallelogram onto itself?

    $180^\circ$. All parallelograms have 2-fold rotational symmetry about their center.

25. For an isosceles trapezoid, how many lines of reflection symmetry are there?

    $1$. Isosceles trapezoid has reflection symmetry across its perpendicular bisector.

26. What is the order of rotational symmetry of a regular $n$-gon?

    $n$. Regular $n$-gon has $n$ rotational symmetries including the identity.

27. Identify the reflection axes of a rhombus (non-square).

    The two diagonals. The diagonals of a rhombus are perpendicular bisectors of each other.

28. What is the smallest positive rotation that carries an isosceles trapezoid onto itself?

    $360^\circ$. Isosceles trapezoid has only reflection symmetry, no rotational symmetry.

29. For a general trapezoid (not isosceles), how many reflection symmetries are there?

    $0$. General trapezoid has no symmetries except the identity.

30. Which is a symmetry of a general parallelogram: reflection across a diagonal or rotation $180^\circ$?

    Rotation $180^\circ$. Parallelograms have $180^\circ$ rotation but no diagonal reflections.

31. A regular polygon has smallest rotational symmetry $45^\circ$. What is $n$?

    $8$. Since $45^\circ = \frac{360^\circ}{8}$, we have $n = 8$.

32. Which option is a reflection symmetry of a non-square rectangle: diagonal or midline?

    Midline. Rectangle reflects across midlines, not diagonals.

33. Identify the order of rotational symmetry of a non-square rectangle.

    $2$. Rectangle maps to itself under identity and $180^\circ$ rotation.

34. A regular $n$-gon maps to itself by a $120^\circ$ rotation. What is the least possible $n$?

    $3$. For $120^\circ$ rotation, we need $\frac{360^\circ}{n} = 120^\circ$, so $n = 3$.

35. In a regular polygon with even $n$, how many reflection axes go through midpoints of opposite sides?

    $\frac{n}{2}$. Half the reflection axes connect midpoints of opposite edges.

36. What is the smallest positive rotation that carries a regular $n$-gon onto itself?

    $\frac{360^\circ}{n}$. Regular polygon divides full rotation equally among $n$ symmetries.

37. What is the smallest positive rotation that carries a regular hexagon onto itself?

    $60^\circ$. Regular hexagon has 6-fold symmetry, so $\frac{360^\circ}{6} = 60^\circ$.

38. Identify the four reflection axes of a square.

    Two diagonals and two midlines through the center. Diagonals connect opposite vertices, midlines bisect opposite sides.

39. What is the smallest positive rotation that carries a square onto itself?

    $90^\circ$. Square has 4-fold rotational symmetry, rotating by quarter-turns.

40. Which rotation angles carry a regular $n$-gon onto itself?

    $k\cdot\frac{360^\circ}{n}$ for $k=0,1,\dots,n-1$. These are all multiples of the fundamental rotation $\frac{360^\circ}{n}$.

41. Identify the order of rotational symmetry of a general parallelogram.

    $2$. Parallelogram maps to itself under identity and $180^\circ$ rotation.

42. How many lines of reflection symmetry does a square have?

    $4$. Square has reflections across two diagonals and two midlines.

43. Which rotation is always a symmetry of any parallelogram about its center?

    $180^\circ$. Half-turn swaps opposite vertices in any parallelogram.

44. Identify the line(s) of symmetry of a non-square rectangle using the word "midline."

    Exactly $2$ midlines through the center. The midlines pass through the rectangle's center parallel to its sides.

45. How many lines of reflection symmetry does a general parallelogram have?

    $0$. General parallelograms have no reflection symmetry lines.

46. Which rotation carries every figure onto itself, regardless of shape?

    $360^\circ$. Full rotation brings every point back to its original position.

47. How many rotation symmetries does a regular heptagon have (including $0^\circ$)?

    $7$. Heptagon has $n = 7$ rotational symmetries including identity.

48. How many lines of reflection symmetry does a regular $n$-gon have?

    $n$. Regular $n$-gon has one reflection axis through each vertex or edge.

49. A regular polygon maps to itself by a $40^\circ$ rotation. What is $n$?

    $9$. For $40^\circ$ rotation, we need $\frac{360^\circ}{n} = 40^\circ$, so $n = 9$.

50. How many reflection symmetries does a regular nonagon have?

    $9$. Regular nonagon has $n = 9$ reflection symmetries.

51. Which rotation angles carry a regular pentagon onto itself (exclude $360^\circ$)?

    $0^\circ,\ 72^\circ,\ 144^\circ,\ 216^\circ,\ 288^\circ$. Pentagon rotates by multiples of $72^\circ$ to map onto itself.

Study these flashcards