A Parallelogram Looks Symmetrical but Has No Line of Symmetry

None at all. Fold a parallelogram along either diagonal, or down the middle, and the halves never match. Yet turn it through half a turn and it lands exactly on itself — so it has rotational symmetry without line symmetry.

The two kinds of symmetry are independent, and that shape proves it. This page covers everything in the ICSE Class 7 Mathematics chapter on symmetry: lines of symmetry, completing figures, reflection in the axes, and rotational symmetry.

A line of symmetry is a line that divides a figure into two halves that fold exactly onto each other.

Triangles:

- Equilateral — 3
- Isosceles — 1
- Scalene — 0

Quadrilaterals:

- Square — 4 (two diagonals and two through opposite side midpoints)
- Rectangle — 2 (through opposite side midpoints, not the diagonals)
- Rhombus — 2 (both diagonals)
- Parallelogram — 0
- Kite — 1
- Isosceles trapezium — 1

Others: a regular pentagon has 5, a regular hexagon 6, and a circle has infinitely many, since every diameter is a line of symmetry. In general a **regular polygon of $n$ sides has $n$ lines of symmetry.

Capital letters with line symmetry include A, B, C, D, E, K, M, T, U, V, W and Y with one, while H, I, O and X have two. Letters such as F, G, J, L, N, P, Q, R, S and Z have none.

Rangoli patterns and the Ashoka Chakra on a flag are everyday figures built around many lines of symmetry.

The rectangle is the case most often got wrong. Its diagonals divide it into two triangles of equal area, but folding along a diagonal does not make the halves coincide — so a diagonal is not a line of symmetry. A rhombus is the opposite: its diagonals are** its two lines.

Reflect every important point across the axis of symmetry, keeping each one the same perpendicular distance on the other side, then join them in the same order.

The method, point by point:

1. Pick the corners and key points of the given half.
2. From each, measure the perpendicular distance to the axis.
3. Mark a point the same distance on the far side, along that same perpendicular.
4. Join the new points in the order matching the original.

So if a corner lies 3 cm from a vertical axis, its image lies 3 cm on the other side, level with it — not 3 cm measured diagonally.

A point on the axis is its own image and does not move, which is why figures completed this way stay joined along the fold.

Folding a painted paper in half while the paint is wet produces exactly this, and it is worth doing once to see that distances are preserved.

The measurement must be perpendicular to the axis, and that is the step students rush. Measuring at a slant stretches the second half, so the completed figure looks lopsided even though every distance was copied faithfully.

Three rules cover every reflection in the syllabus:

- In the x-axis: $(x,y)$ becomes $(x,-y)$ — the sign of y flips.
- In the y-axis: $(x,y)$ becomes $(-x,y)$ — the sign of x flips.
- In the origin: $(x,y)$ becomes $(-x,-y)$ — both signs flip.

Worked example with the point $P(3,4)$:

- In the x-axis: $(3,-4)$
- In the y-axis: $(-3,4)$
- In the origin: $(-3,-4)$

Another, with $Q(-2,5)$:

- In the x-axis: $(-2,-5)$
- In the y-axis: $(2,5)$
- In the origin: $(2,-5)$

For a line segment, reflect both endpoints and join them. The segment AB with $A(1,2)$ and $B(4,2)$ reflects in the x-axis to $A^{\prime }(1,-2)$ and $B^{\prime }(4,-2)$ — a segment of the same length, since reflection never changes size.

A neat way to remember which sign changes: reflecting in the x-axis moves the point across the x-axis, so it is the $y$-coordinate — the one measuring distance from that axis — that reverses.

The boundary cases are worth noting. A point on the x-axis, such as $(5,0)$, is unchanged by reflection in the x-axis. And the origin $(0,0)$ is unchanged by all three reflections.

A figure has rotational symmetry if it looks unchanged after being turned through less than a full turn about its centre. The order is the number of positions in a full turn where it matches itself.

The angle of each step is

$$\text{angle of rotation}=\frac{360°}{\text{order}}$$

Worked examples:

- Square — order 4, turning through $\frac{360}{4}=90°$ each time.
- Equilateral triangle — order 3, at $\frac{360}{3}=120°$.
- Rectangle — order 2, at $180°$.
- Parallelogram — order 2, at $180°$.
- Regular hexagon — order 6, at $60°$.
- Circle — infinite order, since every angle works.
- Scalene triangle — order 1, meaning no rotational symmetry beyond a full turn.

A regular polygon of $n$ sides has order $n$, matching its $n$ lines of symmetry.

Comparing the two symmetries shows they are independent:

- Both — square (4 lines, order 4), equilateral triangle (3 and 3), rectangle (2 lines, order 2), circle.
- Rotational only — parallelogram (0 lines, order 2).
- Line only — kite (1 line, order 1), isosceles triangle (1 and 1).

A ceiling fan with three blades has order 3 and no line of symmetry, because each blade is angled.

Order 1 counts as no rotational symmetry, since every figure returns to itself after a full $360°$ turn. So an answer of "order 1" means the figure has none.