Key Points

Symmetry exists when a figure is composed of parts that repeat in a definite pattern. Symmetrical figures appear balanced and harmonious.

A line of symmetry is a line that divides a figure into two identical parts. If the figure is folded along this line, the two halves, called mirror halves, will perfectly overlap.

A figure that has one or more lines of symmetry is said to have reflection symmetry. The line of symmetry is also known as the axis of symmetry.

Figures like a scalene triangle or a parallelogram have no lines of symmetry. They cannot be divided into two mirror halves by any straight line.

An isosceles triangle has one line of symmetry. A rectangle has two, an equilateral triangle has three, and a square has four lines of symmetry.

A regular polygon with $n$ sides has exactly $n$ lines of symmetry. For instance, a regular pentagon has 5 lines of symmetry, and a regular hexagon has 6.

A circle has an infinite number of lines of symmetry. Every line passing through the center of the circle (every diameter) is a line of symmetry.

A figure has rotational symmetry if it looks identical to its original position after being rotated by an angle less than a full turn ($360^\circ$) around a fixed point.

The fixed point about which a figure is rotated is called the centre of rotation. The angle of turn for which the figure looks the same is the angle of rotational symmetry.

The order of rotational symmetry is the number of times a figure fits onto itself in one complete $360^\circ$ rotation. An order of 1 indicates no rotational symmetry.

The order of rotational symmetry can be found by dividing $360^\circ$ by the smallest angle of rotation, $\theta$. The formula is: Order $= \frac{360^\circ}{\theta}$.

A square has rotational symmetry of order 4 (smallest angle $90^\circ$). An equilateral triangle has order 3 (smallest angle $120^\circ$). A rectangle has order 2 (smallest angle $180^\circ$).

A circle has rotational symmetry of infinite order. It looks the same after any angle of rotation about its center.

Shapes like squares, equilateral triangles, and circles have both line symmetry and rotational symmetry. The number of lines of symmetry in a regular polygon is equal to its order of rotational symmetry.

A parallelogram has rotational symmetry of order 2 but it does not have any lines of symmetry. The blades of a fan also show only rotational symmetry.