A scalene triangle, a parallelogram, and a trapezium are three examples of shapes that have no line of symmetry.
Line symmetry (or reflectional symmetry) exists in a shape if it can be folded along a line, called the line of symmetry, such that the two halves exactly match. Some geometric shapes possess one or more lines of symmetry, while others do not.
Based on geometric principles, and as referenced in materials discussing symmetry, certain common shapes notably lack this property. These shapes cannot be divided by any straight line into two identical mirror-image parts.
Here are three specific examples of shapes that do not have any lines of symmetry:
Scalene Triangle
A scalene triangle is a triangle where all three sides are of different lengths, and consequently, all three angles are of different measures. Because there are no equal sides or angles, there is no line through the triangle that can serve as a mirror, reflecting one half onto the other perfectly.
Parallelogram
A parallelogram is a quadrilateral with two pairs of parallel sides. While a rectangle, rhombus, or square (which are special types of parallelograms) may have lines of symmetry, a general parallelogram with unequal adjacent sides and non-right angles does not. No line can be drawn through a parallelogram to divide it into two congruent mirror images. (Note: A parallelogram does have rotational symmetry of order 2 about the intersection of its diagonals, but the question is specifically about line symmetry).
Trapezium (or Trapezoid)
A trapezium (known as a trapezoid in North America) is a quadrilateral with at least one pair of parallel sides. A general trapezium, which does not have equal non-parallel sides or right angles, possesses no lines of symmetry. An isosceles trapezium, which has equal non-parallel sides, does have one line of symmetry, but the general trapezium does not.
In summary, these three shapes—the scalene triangle, the parallelogram, and the trapezium—are classic examples of geometric figures that lack line symmetry because no line can divide them into two perfectly reflective halves.
Here's a quick overview:
| Shape | Key Feature (relevant to symmetry) | Lines of Symmetry |
|---|---|---|
| Scalene Triangle | All sides and angles are different. | 0 |
| Parallelogram | Opposite sides parallel; adjacent sides unequal/non-right angles. | 0 |
| Trapezium | At least one pair of parallel sides (general case). | 0 |
