Quadrilaterals

Understanding Quadrilaterals

A quadrilateral is any polygon with four sides (and therefore four vertices and four interior angles). In this chapter we focus on the main types of quadrilaterals, how to recognize them, and the key properties that make each type special.

Throughout, we will assume quadrilaterals are in a flat (plane) geometry and their sides are straight line segments.

Basic Definitions and Angle Sum

A quadrilateral has:

• 4 sides
• 4 vertices (corners)
• 4 interior angles

If you draw both diagonals (segments joining opposite vertices), they divide the quadrilateral into triangles. Any quadrilateral can be divided into exactly two triangles by drawing one diagonal. Using the fact that the sum of angles in a triangle is $180^\circ$, the sum of the interior angles in any quadrilateral is
$$
180^\circ + 180^\circ = 360^\circ.
$$

So for every quadrilateral:
$$
\text{Angle}_1 + \text{Angle}_2 + \text{Angle}_3 + \text{Angle}_4 = 360^\circ.
$$

This is true whether the quadrilateral is regular or irregular, convex or concave (as long as the sides are straight and it lies in a plane).

Classifying Quadrilaterals

Quadrilaterals are often classified by:

• Relationships between sides (parallel, equal length)
• Relationships between angles (right angles, equal angles)
• Properties of diagonals (equal length, perpendicular, they bisect each other, etc.)

The main special types we will study are:

• Parallelogram
• Rectangle
• Rhombus
• Square
• Trapezoid (and isosceles trapezoid)
• Kite

Every square is also a rectangle and a rhombus, and every rectangle and rhombus is a parallelogram. We will describe these relationships as we go.

When we say a quadrilateral $ABCD$, we list the vertices in order around the shape: $A$–$B$–$C$–$D$–back to $A$.

Parallelograms

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

So in $ABCD$:

• $AB \parallel CD$
• $BC \parallel AD$

Key Properties of Parallelograms

If $ABCD$ is a parallelogram, then:

1. Opposite sides are equal in length:
   $$
   AB = CD,\quad BC = AD.
   $$
2. Opposite angles are equal:
   $$
   \angle A = \angle C,\quad \angle B = \angle D.
   $$
3. Consecutive angles are supplementary (they add to $180^\circ$):
   $$
   \angle A + \angle B = 180^\circ,\quad
   \angle B + \angle C = 180^\circ,\ \text{etc.}
   $$
4. Diagonals bisect each other:
   Let the diagonals $AC$ and $BD$ intersect at $E$. Then
   $$
   AE = EC,\quad BE = ED.
   $$
   The diagonals are not necessarily equal in length.

Ways to Prove a Quadrilateral Is a Parallelogram

You do not always start with “both pairs of opposite sides parallel.” Sometimes you are given coordinates or lengths and asked to show the shape is a parallelogram. The following conditions are each sufficient (any one is enough):

A quadrilateral is a parallelogram if:

• Both pairs of opposite sides are parallel, or
• Both pairs of opposite sides are equal in length, or
• One pair of opposite sides is both parallel and equal in length, or
• The diagonals bisect each other, or
• Both pairs of opposite angles are equal.

In coordinate geometry, you often use slope (for parallel) and distance (for equal lengths) to apply these tests.

Rectangles

A rectangle is a parallelogram with four right angles.

In $ABCD$:

• $\angle A = \angle B = \angle C = \angle D = 90^\circ$.

Because every rectangle is a parallelogram, it inherits all parallelogram properties, plus extra ones.

Key Properties of Rectangles

In a rectangle:

1. All angles are $90^\circ$ (by definition).
2. Opposite sides are parallel and equal:
   $$
   AB = CD,\quad BC = AD,\quad AB \parallel CD,\quad BC \parallel AD.
   $$
3. Consecutive sides meet at right angles:
   Adjacent sides are perpendicular.
4. Diagonals are equal in length:
   $$
   AC = BD.
   $$
   As a parallelogram, the diagonals also bisect each other.

So in a rectangle, diagonals are both equal and bisect each other.

Recognizing Rectangles

A quadrilateral is a rectangle if:

• It is a parallelogram with one right angle, or
• It is a parallelogram with equal diagonals, or
• All four angles are right angles.

In coordinate form, a common check is to show that adjacent sides are perpendicular (slopes are negative reciprocals) and opposite sides are parallel.

Rhombuses

A rhombus is a parallelogram with all four sides equal in length.

So in $ABCD$:

• $AB = BC = CD = DA$.

A rhombus is like a “tilted square,” but the angles need not be $90^\circ$.

Key Properties of Rhombuses

In a rhombus:

1. All sides are equal (definition).
2. Opposite sides are parallel (since it is a parallelogram).
3. Opposite angles are equal, and consecutive angles are supplementary (parallelogram properties).
4. Diagonals bisect each other (parallelogram property).
5. Diagonals are perpendicular:
   $$
   AC \perp BD.
   $$
6. Each diagonal bisects a pair of opposite angles:
• Diagonal $AC$ cuts $\angle A$ and $\angle C$ into two equal angles.
• Diagonal $BD$ cuts $\angle B$ and $\angle D$ into two equal angles.

Recognizing Rhombuses

A quadrilateral is a rhombus if:

• It is a parallelogram with two adjacent sides equal, or
• It is a parallelogram with diagonals that are perpendicular, or
• All four sides are equal in length.

In coordinate geometry, you might detect a rhombus by showing all four sides have the same length using the distance formula.

Squares

A square is a quadrilateral that is both a rectangle and a rhombus.

Equivalently, a square has:

• Four equal sides, and
• Four right angles.

So squares combine all the properties of parallelograms, rectangles, and rhombuses.

Key Properties of Squares

In a square:

1. All sides are equal, and all angles are $90^\circ$.
2. Opposite sides are parallel.
3. Diagonals are equal, perpendicular, and bisect each other.
4. Diagonals bisect the angles at each vertex.

Because of these many symmetries, squares often appear in problems about symmetry, transformations, and coordinate geometry.

Recognizing Squares

A quadrilateral is a square if:

• It is a rectangle with all sides equal, or
• It is a rhombus with one right angle, or
• It is a rhombus with equal diagonals, or
• All four sides are equal and one angle is $90^\circ$.

Trapezoids

A trapezoid (in many curricula; also called a trapezium in some regions) is a quadrilateral with at least one pair of parallel sides. We will use this “at least one pair” definition.

Let $ABCD$ be a trapezoid with $AB \parallel CD$. Then:

• $AB$ and $CD$ are called the bases.
• The other two sides, $BC$ and $DA$, are called the legs.

The bases are the parallel sides; the legs are not necessarily parallel or equal.

Angle Relationships in a Trapezoid

In a trapezoid $ABCD$ with $AB \parallel CD$:

• The angles along each leg are supplementary:
  $$
  \angle A + \angle D = 180^\circ,\quad \angle B + \angle C = 180^\circ.
  $$
  This follows from the fact that each pair is a pair of same-side interior angles formed by a transversal of parallel lines.

There is no requirement that opposite angles or sides be equal, unless we add more conditions (see isosceles trapezoid).

Midsegment of a Trapezoid

The midsegment (or median) of a trapezoid is the segment that joins the midpoints of the legs.

If $M$ is the midpoint of $AD$ and $N$ is the midpoint of $BC$, then $MN$ is the midsegment.

Important facts about the midsegment $MN$:

1. $MN$ is parallel to the bases $AB$ and $CD$.
2. Its length is the average of the lengths of the bases:
   $$
   MN = \frac{AB + CD}{2}.
   $$

This formula is often used in geometry and coordinate problems involving trapezoids.

Isosceles Trapezoids

An isosceles trapezoid is a trapezoid whose legs are equal in length.

So in $ABCD$ with $AB \parallel CD$:

• $AD = BC$.

Key Properties of Isosceles Trapezoids

If $ABCD$ is an isosceles trapezoid:

1. Base angles are equal:
   $$
   \angle A = \angle D,\quad \angle B = \angle C.
   $$
   Each pair of angles next to the same base is equal.
2. Diagonals are equal in length:
   $$
   AC = BD.
   $$
3. Bases are parallel, but not equal unless it happens to be a special case (like a rectangle, which also fits the “at least one pair parallel” definition).

These properties make isosceles trapezoids resemble rectangles in some ways, but only one pair of sides is guaranteed parallel.

Kites

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

For $ABCD$ to be a kite:

• $AB = AD$, and
• $CB = CD$,

with the equal pairs sharing a vertex (the shape looks like a traditional flying kite).

Note that, in general, opposite sides of a kite are not equal and not parallel.

Key Properties of Kites

In a kite $ABCD$ with $AB = AD$ and $CB = CD$:

1. One pair of opposite angles is equal:
   The angles between the unequal sides are equal:
   $$
   \angle B = \angle D.
   $$
2. Diagonals have special properties:
• The diagonals are perpendicular:
  $$
  AC \perp BD.
  $$
• One diagonal bisects the other:
  Typically, the diagonal joining the vertices where the equal sides meet bisects the other diagonal.
• The diagonal that connects the vertices between unequal sides bisects the pair of equal angles.
3. A kite is generally not a parallelogram (no guarantee of parallel opposite sides) unless it becomes a special case like a rhombus.

Kites are useful in problems involving perpendicular diagonals and symmetry around one axis.

Convex and Concave Quadrilaterals

A quadrilateral can be:

• Convex: all interior angles are less than $180^\circ$, and the segment between any two points of the quadrilateral lies entirely inside or on the boundary.
• Concave: one interior angle is greater than $180^\circ$, and the quadrilateral has a “dent.”

All of the standard special quadrilaterals described above (parallelogram, rectangle, rhombus, square, trapezoid, isosceles trapezoid, kite) are convex. Concave quadrilaterals are usually irregular and do not fit these special categories.

Inclusion Relationships Among Special Quadrilaterals

It is useful to see how these types fit inside one another. You can think of them in layers of conditions:

• Parallelogram: both pairs of opposite sides parallel.
• Rectangle: parallelogram + all right angles.
• Rhombus: parallelogram + all sides equal.
• Square: rhombus + all right angles (also a rectangle).

Separately:

• Trapezoid: at least one pair of parallel sides.
• Isosceles trapezoid: trapezoid + legs equal.
• Kite: two pairs of adjacent equal sides.

Notice:

• Every square is a rectangle, a rhombus, and a parallelogram.
• Every rectangle and every rhombus is a parallelogram.
• Not every parallelogram is a rectangle or rhombus.
• Kites and isosceles trapezoids are not usually parallelograms, except in special cases that turn them into rhombuses or rectangles.

Understanding these relationships helps in choosing which properties you are allowed to use in a problem: once you know the exact type (for example, “this is a rectangle, not just any parallelogram”), you can apply all the properties that come with that type.

Using Properties in Problems

In practice, quadrilateral problems often ask you to:

• Prove that a quadrilateral is of a certain type, using side lengths, parallels, angle measures, or diagonals.
• Use known properties (like equal angles or bisected diagonals) to find missing side lengths, angle measures, or coordinates.
• Recognize when a more specific classification applies (for example, a parallelogram with equal diagonals must be a rectangle).

The key skill is to connect the given information (about sides, angles, diagonals, or parallel lines) to the definitions and properties of these special quadrilaterals and then select the most specific classification that fits.