This worksheet asks students to identify the order of rotational symmetry and draw lines of reflective symmetry for twelve different polygons, ranging from squares to regular nonagons.

Problem Overview:

The task requires determining the order of rotational symmetry and identifying all the lines of reflective symmetry for each given polygon. Let's solve this step by step.

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Definitions:

1. Rotational Symmetry: The order of rotational symmetry is the number of times a shape looks exactly the same as it rotates through 360°.
2. Reflective Symmetry: A line of reflective symmetry is a line that divides a shape into two identical halves that are mirror images of each other.

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Solution:

#### 1. Square
- Rotational Symmetry: A square looks the same after rotations of 90°, 180°, 270°, and 360°. Therefore, the order of rotational symmetry is 4.
- Reflective Symmetry: A square has 4 lines of symmetry: 2 diagonals and 2 lines passing through the midpoints of opposite sides.
- Answer: Order of rotational symmetry = 4, Lines of reflective symmetry = 4.

#### 2. Kite
- Rotational Symmetry: A kite only looks the same after a rotation of 360°. Therefore, the order of rotational symmetry is 1.
- Reflective Symmetry: A kite has 1 line of symmetry, which is the vertical line passing through the longer diagonal.
- Answer: Order of rotational symmetry = 1, Lines of reflective symmetry = 1.

#### 3. Trapezium
- Rotational Symmetry: A general trapezium does not look the same after any rotation other than 360°. Therefore, the order of rotational symmetry is 1.
- Reflective Symmetry: A general trapezium may or may not have reflective symmetry. If it is an isosceles trapezium (as shown), it has 1 line of symmetry, which is the vertical line passing through the midpoints of the parallel sides.
- Answer: Order of rotational symmetry = 1, Lines of reflective symmetry = 1.

#### 4. Rhombus
- Rotational Symmetry: A rhombus looks the same after rotations of 180° and 360°. Therefore, the order of rotational symmetry is 2.
- Reflective Symmetry: A rhombus has 2 lines of symmetry: the two diagonals.
- Answer: Order of rotational symmetry = 2, Lines of reflective symmetry = 2.

#### 5. Rectangle
- Rotational Symmetry: A rectangle looks the same after rotations of 180° and 360°. Therefore, the order of rotational symmetry is 2.
- Reflective Symmetry: A rectangle has 2 lines of symmetry: the lines passing through the midpoints of opposite sides.
- Answer: Order of rotational symmetry = 2, Lines of reflective symmetry = 2.

#### 6. Parallelogram
- Rotational Symmetry: A parallelogram only looks the same after a rotation of 360°. Therefore, the order of rotational symmetry is 1.
- Reflective Symmetry: A general parallelogram does not have any lines of symmetry.
- Answer: Order of rotational symmetry = 1, Lines of reflective symmetry = 0.

#### 7. Arrow Head
- Rotational Symmetry: An arrow head only looks the same after a rotation of 360°. Therefore, the order of rotational symmetry is 1.
- Reflective Symmetry: An arrow head has 1 line of symmetry, which is the vertical line passing through the vertex.
- Answer: Order of rotational symmetry = 1, Lines of reflective symmetry = 1.

#### 8. Equilateral Triangle
- Rotational Symmetry: An equilateral triangle looks the same after rotations of 120°, 240°, and 360°. Therefore, the order of rotational symmetry is 3.
- Reflective Symmetry: An equilateral triangle has 3 lines of symmetry: the lines passing through each vertex and the midpoint of the opposite side.
- Answer: Order of rotational symmetry = 3, Lines of reflective symmetry = 3.

#### 9. Regular Pentagon
- Rotational Symmetry: A regular pentagon looks the same after rotations of 72°, 144°, 216°, 288°, and 360°. Therefore, the order of rotational symmetry is 5.
- Reflective Symmetry: A regular pentagon has 5 lines of symmetry: the lines passing through each vertex and the midpoint of the opposite side.
- Answer: Order of rotational symmetry = 5, Lines of reflective symmetry = 5.

#### 10. Regular Hexagon
- Rotational Symmetry: A regular hexagon looks the same after rotations of 60°, 120°, 180°, 240°, 300°, and 360°. Therefore, the order of rotational symmetry is 6.
- Reflective Symmetry: A regular hexagon has 6 lines of symmetry: 3 lines passing through opposite vertices and 3 lines passing through the midpoints of opposite sides.
- Answer: Order of rotational symmetry = 6, Lines of reflective symmetry = 6.

#### 11. Regular Octagon
- Rotational Symmetry: A regular octagon looks the same after rotations of 45°, 90°, 135°, 180°, 225°, 270°, 315°, and 360°. Therefore, the order of rotational symmetry is 8.
- Reflective Symmetry: A regular octagon has 8 lines of symmetry: 4 lines passing through opposite vertices and 4 lines passing through the midpoints of opposite sides.
- Answer: Order of rotational symmetry = 8, Lines of reflective symmetry = 8.

#### 12. Regular Nonagon
- Rotational Symmetry: A regular nonagon looks the same after rotations of 40°, 80°, 120°, 160°, 200°, 240°, 280°, 320°, and 360°. Therefore, the order of rotational symmetry is 9.
- Reflective Symmetry: A regular nonagon has 9 lines of symmetry: the lines passing through each vertex and the midpoint of the opposite side.
- Answer: Order of rotational symmetry = 9, Lines of reflective symmetry = 9.

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Final Answers:

\[
\begin{array}{lll}
\text{Square} & \text{Order of rotational symmetry = } \boxed{4}, & \text{Lines of reflective symmetry = } \boxed{4} \\
\text{Kite} & \text{Order of rotational symmetry = } \boxed{1}, & \text{Lines of reflective symmetry = } \boxed{1} \\
\text{Trapezium} & \text{Order of rotational symmetry = } \boxed{1}, & \text{Lines of reflective symmetry = } \boxed{1} \\
\text{Rhombus} & \text{Order of rotational symmetry = } \boxed{2}, & \text{Lines of reflective symmetry = } \boxed{2} \\
\text{Rectangle} & \text{Order of rotational symmetry = } \boxed{2}, & \text{Lines of reflective symmetry = } \boxed{2} \\
\text{Parallelogram} & \text{Order of rotational symmetry = } \boxed{1}, & \text{Lines of reflective symmetry = } \boxed{0} \\
\text{Arrow Head} & \text{Order of rotational symmetry = } \boxed{1}, & \text{Lines of reflective symmetry = } \boxed{1} \\
\text{Equilateral Triangle} & \text{Order of rotational symmetry = } \boxed{3}, & \text{Lines of reflective symmetry = } \boxed{3} \\
\text{Regular Pentagon} & \text{Order of rotational symmetry = } \boxed{5}, & \text{Lines of reflective symmetry = } \boxed{5} \\
\text{Regular Hexagon} & \text{Order of rotational symmetry = } \boxed{6}, & \text{Lines of reflective symmetry = } \boxed{6} \\
\text{Regular Octagon} & \text{Order of rotational symmetry = } \boxed{8}, & \text{Lines of reflective symmetry = } \boxed{8} \\
\text{Regular Nonagon} & \text{Order of rotational symmetry = } \boxed{9}, & \text{Lines of reflective symmetry = } \boxed{9} \\
\end{array}
\]

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Thus, the final boxed answers are:
\[
\boxed{
\begin{array}{lll}
\text{Square} & 4 & 4 \\
\text{Kite} & 1 & 1 \\
\text{Trapezium} & 1 & 1 \\
\text{Rhombus} & 2 & 2 \\
\text{Rectangle} & 2 & 2 \\
\text{Parallelogram} & 1 & 0 \\
\text{Arrow Head} & 1 & 1 \\
\text{Equilateral Triangle} & 3 & 3 \\
\text{Regular Pentagon} & 5 & 5 \\
\text{Regular Hexagon} & 6 & 6 \\
\text{Regular Octagon} & 8 & 8 \\
\text{Regular Nonagon} & 9 & 9 \\
\end{array}
}
\]