A quadrilateral is any **four-sided** figure.

In any quadrilateral, the sum of all interior angles is 360ᵒ.

Quadrilateral properties are discussed below, which are used to logically test whether a certain shape is one kind of quadrilateral.

## Parallelogram

A parallelogram is a quadrilateral with parallel opposite sides.

Properties of a parallelogram:

- Opposite sides are equal
- Opposite angles are equal
- Diagonals bisect each other
- Each diagonal bisects the parallelogram into two congruent triangles

(*Bisect* means cut exactly in two halves)

## Rectangle

A rectangle is a parallelogram with all angles forming a 90ᵒ angle.

Properties of a rectangle:

- Opposite sides are equal
- Opposite angles are equal
- Diagonals bisect each other
- Each diagonal bisects the parallelogram into two congruent triangles
- Diagonals are equal

## Rhombus

A rhombus is a parallelogram with a pair of equal adjacent sides.

Properties of a rhombus:

- Opposite sides are equal
- Opposite angles are equal
- Diagonals bisect each other
- Each diagonal bisects the parallelogram into two congruent triangles
- Diagonals bisect at right angles
- Diagonals bisect the angles of the rhombus

## Square

A square is a rectangle with a pair of equal adjacent sides.

- Opposite sides are equal
- Opposite angles are equal
- Diagonals bisect each other
- Each diagonal bisects the parallelogram into two congruent triangles
- Diagonals are equal
- Diagonals are perpendicular
- Diagonals make angles of 45° with the sides

## Trapezium

A trapezium is a quadrilateral with one pair of parallel sides.

## Kite

A kite is a quadrilateral with two pairs of adjacent equal sides.

Properties of a kite:

- Two adjacent sides are equal
- Diagonals bisect at 90° angles
- One diagonal bisects the kite to two congruent triangles
- One pair opposite of angles is equal

### Examples

1. ABCD is a square. Lines are drawn from *C* to *M* and *N*, the midpoints *AD* and *AB* respectively. Prove that *MC* = *NC*.

### Show Answer

Using similarity of triangles, we check if the triangles △CDM and △CBN are congruent.

Since *MD* and *NB* are half of a square sides, therefore *MD = NB*. (S)

∠ADC and ∠ABC are equal right angles property of a square. (A)

While, *DC = BC* because a square has equal sides. (S)

Therefore, by SAS, the two triangles △CDM and △CBN are congruent.

Hence, the sides *MC* and *NC* are equal because they are corresponding sides of congruent triangles.

2. If *A* = (1, 5), *B* = (4, 2), and *C* = (2, -3), find the coordinates of *D* such that *ABCD* is a parallelogram.

### Show Answer

The question is drawn on the above graph, and graphically we could understand where the points are.

A parallelogram has 2 pairs of parallel sides, and therefore the distance *BA* must be equal to the distance *CD*. Therefore,

From point *B* to point *A*, the distance moves 3 to the left and then 3 upwards. The same could be done from point *C* to *D* to complete the parallelogram, which we found to be the point *D*(-1, 0). This method complies with the properties of parallelograms, with 2 pairs of parallelograms and equal opposite angles.