Properties of Quadrilaterals and Triangles

Labelling Conventions in Geometry

Consistent labelling is fundamental for communicating geometric ideas accurately. In the triangle $ABC$, capital letters $A$, $B$, and $C$ denote the angles at the respective vertices. The lowercase letters $a$, $b$, and $c$ represent the lengths of the sides opposite those angles.

For quadrilaterals and other plane figures, vertices must be labelled in a consistent order, either clockwise or anti-clockwise around the perimeter. The choice of the starting vertex does not matter as long as the sequence is maintained.

Properties of Special Quadrilaterals

Square

A square is defined as a regular quadrilateral. It possesses two pairs of parallel sides where all four sides are equal in length. Every interior angle in a square is exactly $90^\circ$. A square has $4$ lines of symmetry and rotational symmetry of order $4$.

In geometric notation, single arrows indicate that sides $AB$ and $DC$ are parallel, while double arrows indicate that $DA$ and $CB$ are parallel. Single marks on each side indicate that all four sides are equal in length. Dotted lines represent the lines of symmetry.

Rectangle

A rectangle features two pairs of parallel sides and four interior angles of $90^\circ$. Typically, a rectangle has $2$ lines of symmetry and rotational symmetry of order $2$. However, if the rectangle is a square, it will possess the higher symmetries of that specific type.

Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides. Its opposite sides are equal in length, and its opposite angles are equal. Adjacent angles in a parallelogram are supplementary, meaning they sum to $180^\circ$. Unless it is a special case like a square, rhombus, or rectangle, a parallelogram generally has no lines of symmetry and rotational symmetry of order $2$.

Trapezium

A trapezium is defined by having exactly one pair of parallel sides. In most instances, it does not have any lines of symmetry.

Kite

A kite has two pairs of equal sides that are adjacent to each other. It possesses exactly $1$ line of symmetry and one pair of equal opposite angles. Crucially, the diagonals of a kite always intersect at right angles ($90^\circ$).

Rhombus

A rhombus has two pairs of parallel sides and all four sides are equal in length. Its diagonals bisect each other at right angles. It typically has $2$ lines of symmetry and rotational symmetry of order $2$, unless it is a square.

Types of Triangle

Triangles can be classified by their angles. An acute angled triangle has all interior angles less than $90^\circ$. A right angled triangle has exactly one angle of $90^\circ$, and an obtuse angled triangle has one angle greater than $90^\circ$.

Scalene Triangles

A scalene triangle has no sides of equal length and contains no right angles. It has no lines of symmetry.

Isosceles Triangles

An isosceles triangle has at least $2$ equal sides. It can be acute, right angled, or obtuse. It features one line of symmetry and no rotational symmetry.

Equilateral Triangles

An equilateral triangle has $3$ equal angles (each $60^\circ$) and $3$ equal sides. It has $3$ lines of symmetry and rotational symmetry of order $3$.

Worked Examples and Applications

Example 1: Diagonal Intersections

Name three types of quadrilaterals whose diagonals intersect at right angles.

As shown in the following diagrams, the diagonals of a rhombus, a square, and a kite intersect at right angles. A delta (or dart) is also a valid answer, as it is a special case of a kite.

Example 2: Combining Triangles

Two identical equilateral triangles are joined along one edge to form a quadrilateral. Name the quadrilateral and describe its symmetries.

When two identical equilateral triangles are joined, the resulting quadrilateral has four equal sides. This means it must be either a square or a rhombus. Since each triangle has angles of $60^\circ$, the resulting quadrilateral has two opposite angles of $60^\circ$ and two opposite angles of $120^\circ$ (from $60^\circ + 60^\circ$). Because the angles are not $90^\circ$, it is a rhombus.

This rhombus has $2$ lines of symmetry and rotational symmetry of order $2$, as it fits onto itself twice during a $360^\circ$ rotation.