The Central Idea
Quadrilaterals are classified by properties. A more specialized class satisfies all the defining properties of the more general classes that contain it.
For example, a rectangle is a parallelogram with right angles. Therefore, every rectangle automatically inherits the basic properties of a parallelogram.
A square is even more specialized: it is both a rectangle and a rhombus. That means a square inherits the properties of both families.
What Is a Quadrilateral?
A quadrilateral is a polygon with four sides.
If the vertices are named in order \(A,B,C,D\), then the quadrilateral may be written
\[ \boxed{ABCD}. \]Its sides are
\[ \overline{AB},\qquad \overline{BC},\qquad \overline{CD},\qquad \overline{DA}. \]Its diagonals connect opposite vertices:
\[ \overline{AC} \qquad\text{and}\qquad \overline{BD}. \]Interior Angle Sum of a Quadrilateral
The interior angles of every simple quadrilateral sum to
\[ \boxed{360^\circ}. \]One way to see this is to draw a diagonal. The diagonal divides the quadrilateral into two triangles, and each triangle has angle sum \(180^\circ\):
\[ 180^\circ+180^\circ=360^\circ. \]Therefore, for quadrilateral \(ABCD\),
\[ \boxed{ m\angle A+m\angle B+m\angle C+m\angle D=360^\circ }. \]Example: Finding a Missing Angle
Suppose a quadrilateral has interior angles
\[ 82^\circ,\qquad 97^\circ,\qquad 104^\circ,\qquad x. \]Use the quadrilateral angle sum:
\[ 82+97+104+x=360. \]The known angles total \(283^\circ\), so
\[ x=360-283=77. \]Therefore,
\[ \boxed{x=77^\circ}. \]Convex and Concave Quadrilaterals
A convex quadrilateral has all interior angles less than \(180^\circ\), and each diagonal lies inside the figure.
A concave quadrilateral has one interior angle greater than \(180^\circ\). One of its diagonals passes partly outside the figure.
Most of the familiar special quadrilateral families in this lesson are treated as convex unless stated otherwise.
A Classification Hierarchy
Quadrilateral names are not always mutually exclusive. A figure can belong to several classes at once.
A useful hierarchy is:
- every square is a rectangle;
- every square is a rhombus;
- every rectangle is a parallelogram;
- every rhombus is a parallelogram;
- every parallelogram is a quadrilateral.
The reverse statements are not generally true. A parallelogram need not be a rectangle, and a rectangle need not be a square.
Property Versus Converse
Geometry frequently uses statements of the form:
If a quadrilateral belongs to a certain class, then it has certain properties.
A converse test goes in the opposite direction:
If a quadrilateral has certain properties, then it belongs to that class.
The converse must be justified separately. A true property does not automatically make its converse true.
Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
\[ \boxed{ \overline{AB}\parallel\overline{CD} \qquad\text{and}\qquad \overline{BC}\parallel\overline{AD} }. \]The parallel structure creates several additional properties.
Opposite Sides of a Parallelogram
Opposite sides of a parallelogram are congruent:
\[ \boxed{ AB=CD \qquad\text{and}\qquad BC=AD }. \]Thus, a parallelogram has two pairs of opposite sides that are both parallel and equal in length.
Angles of a Parallelogram
Opposite angles of a parallelogram are congruent:
\[ \boxed{ \angle A\cong\angle C \qquad\text{and}\qquad \angle B\cong\angle D }. \]Consecutive angles are supplementary:
\[ \boxed{ m\angle A+m\angle B=180^\circ }. \]The same relationship holds for every pair of neighboring angles.
Example: Angles in a Parallelogram
Suppose one angle of a parallelogram measures \(68^\circ\).
Its opposite angle also measures
\[ 68^\circ. \]Each adjacent angle measures
\[ 180^\circ-68^\circ=112^\circ. \]Therefore, the four angles are
\[ \boxed{ 68^\circ,\ 112^\circ,\ 68^\circ,\ 112^\circ }. \]Diagonals of a Parallelogram
The diagonals of a parallelogram bisect each other.
If diagonals \(\overline{AC}\) and \(\overline{BD}\) intersect at \(M\), then
\[ \boxed{ AM=MC \qquad\text{and}\qquad BM=MD }. \]The diagonals do not have to be congruent, perpendicular, or angle bisectors in a general parallelogram.
Tests for a Parallelogram
Several useful converse tests can prove that a quadrilateral is a parallelogram. A quadrilateral is a parallelogram if any one of the following sufficient conditions is established:
- both pairs of opposite sides are parallel;
- both pairs of opposite sides are congruent;
- one pair of opposite sides is both parallel and congruent;
- both pairs of opposite angles are congruent;
- the diagonals bisect each other.
These are tests, not merely descriptive observations.
Rectangles
A rectangle is a parallelogram with four right angles.
\[ \boxed{ m\angle A=m\angle B=m\angle C=m\angle D=90^\circ }. \]Because a rectangle is a parallelogram, it also has congruent opposite sides and diagonals that bisect each other.
Diagonals of a Rectangle
A rectangle has an additional diagonal property:
\[ \boxed{AC=BD}. \]Its diagonals are congruent as well as mutually bisecting.
In a general rectangle, however, the diagonals are not necessarily perpendicular and do not necessarily bisect the vertex angles.
A Rectangle Test
If a quadrilateral is already known to be a parallelogram, then proving that its diagonals are congruent is enough to prove that the parallelogram is a rectangle.
Similarly, a parallelogram with one right angle is a rectangle, because the parallelogram angle relationships force all four angles to be right angles.
Rhombi
A rhombus is a parallelogram with four congruent sides.
\[ \boxed{ AB=BC=CD=DA }. \]Because a rhombus is a parallelogram, opposite angles are congruent, consecutive angles are supplementary, and the diagonals bisect each other.
Diagonals of a Rhombus
The diagonals of a rhombus are perpendicular:
\[ \boxed{ \overline{AC}\perp\overline{BD} }. \]Each diagonal also bisects a pair of opposite vertex angles.
The diagonals are not generally congruent. Congruent diagonals occur in the special case where the rhombus is a square.
A Rhombus Test
If a quadrilateral is already known to be a parallelogram, then perpendicular diagonals are sufficient to prove that it is a rhombus.
Another useful test is that a parallelogram in which one diagonal bisects a pair of opposite angles is a rhombus.
Squares
A square is a quadrilateral with four congruent sides and four right angles.
Equivalently, a square is both a rectangle and a rhombus.
Therefore, a square has all of the following properties:
- both pairs of opposite sides are parallel;
- all four sides are congruent;
- all four angles are right angles;
- the diagonals bisect each other;
- the diagonals are congruent;
- the diagonals are perpendicular;
- the diagonals bisect the vertex angles.
Why a Square Belongs to Several Families
A square satisfies the definition of a rectangle because it is a parallelogram with four right angles.
It also satisfies the definition of a rhombus because it is a parallelogram with four congruent sides.
Therefore,
\[ \boxed{ \text{square}\subset\text{rectangle} \qquad\text{and}\qquad \text{square}\subset\text{rhombus} }. \]Classification becomes more precise when the most specific justified name is used while still recognizing all valid broader classes.
Trapezoids
In this lesson, a trapezoid is defined using the inclusive convention: it is a quadrilateral with at least one pair of opposite sides parallel.
Under this convention, every parallelogram is also a trapezoid because it has two pairs of parallel sides.
Some textbooks use an exclusive convention requiring exactly one pair of parallel sides. When comparing sources, always check which definition is being used.
Bases and Legs of a Trapezoid
A chosen pair of parallel sides of a trapezoid are called the bases. The remaining sides are called the legs.
If
\[ \overline{AB}\parallel\overline{CD}, \]then \(\overline{AB}\) and \(\overline{CD}\) may be treated as the bases, while \(\overline{BC}\) and \(\overline{DA}\) are the legs.
Angles Along a Leg of a Trapezoid
Because the bases are parallel, the two interior angles along either leg are supplementary.
For example, if \(\overline{AB}\parallel\overline{CD}\), then
\[ \boxed{ m\angle A+m\angle D=180^\circ } \]and
\[ \boxed{ m\angle B+m\angle C=180^\circ }. \]These are same-side interior angle relationships created by the parallel bases.
Isosceles Trapezoids
An isosceles trapezoid is a trapezoid whose legs are congruent.
In the usual non-parallelogram case, if
\[ \overline{AB}\parallel\overline{CD} \]and
\[ AD=BC, \]then each pair of base angles is congruent:
\[ \boxed{ \angle A\cong\angle B \qquad\text{and}\qquad \angle D\cong\angle C }. \]Its diagonals are also congruent:
\[ \boxed{AC=BD}. \]The Trapezoid Midsegment
The segment joining the midpoints of the legs of a trapezoid is called the midsegment.
The midsegment is parallel to the bases, and its length is the average of the base lengths.
\[ \boxed{ m=\frac{b_1+b_2}{2} }. \]For bases of lengths \(8\) and \(14\),
\[ m=\frac{8+14}{2}=11. \]Therefore, the midsegment length is
\[ \boxed{11}. \]Kites
A kite is a quadrilateral with two pairs of adjacent congruent sides.
For example, one common labeling has
\[ AB=AD \qquad\text{and}\qquad BC=CD. \]Some definitions require the two congruent pairs to have different lengths, while inclusive definitions allow a rhombus to count as a kite. The structural properties below describe the usual kite configuration.
Angles and Diagonals of a Kite
In a kite with
\[ AB=AD \qquad\text{and}\qquad BC=CD, \]the opposite angles between the unequal sides are congruent:
\[ \boxed{\angle B\cong\angle D}. \]The diagonal joining the vertices where the congruent side pairs meet, \(\overline{AC}\), is an axis of symmetry in the standard kite configuration. It is perpendicular to \(\overline{BD}\), bisects \(\overline{BD}\), and bisects \(\angle A\) and \(\angle C\).
The other diagonal does not generally bisect \(\overline{AC}\).
Comparing Diagonal Properties
Diagonals are especially useful for distinguishing quadrilateral families.
| Figure | Diagonals Bisect Each Other | Diagonals Congruent | Diagonals Perpendicular |
|---|---|---|---|
| Parallelogram | Yes | Not necessarily | Not necessarily |
| Rectangle | Yes | Yes | Not necessarily |
| Rhombus | Yes | Not necessarily | Yes |
| Square | Yes | Yes | Yes |
| Isosceles trapezoid | Not generally | Yes | Not generally |
| Kite | Only one diagonal generally bisects the other | Not generally | Yes |
Comparing Side and Angle Properties
| Figure | Parallel Sides | Congruent Sides | Angle Structure |
|---|---|---|---|
| Parallelogram | Two opposite pairs | Opposite pairs | Opposite angles congruent |
| Rectangle | Two opposite pairs | Opposite pairs | Four right angles |
| Rhombus | Two opposite pairs | All four | Opposite angles congruent |
| Square | Two opposite pairs | All four | Four right angles |
| Isosceles trapezoid | At least one opposite pair | Legs congruent | Each pair of base angles congruent |
| Kite | Not required | Two adjacent pairs | One opposite pair congruent |
Coordinate Geometry as a Classification Tool
When a quadrilateral is given by coordinates, its properties can be tested numerically.
Useful tools include:
- slope to test parallel and perpendicular sides;
- the distance formula to compare side lengths and diagonals;
- midpoints to test whether diagonals bisect each other.
Combining several tests usually produces a stronger classification than relying on appearance.
Example: Coordinate Rectangle
Consider
\[ A(0,0),\quad B(6,0),\quad C(6,3),\quad D(0,3). \]Segment \(\overline{AB}\) is horizontal and \(\overline{BC}\) is vertical, so they are perpendicular.
Opposite sides are parallel, and all four corners are right angles. Therefore,
\[ \boxed{ABCD\text{ is a rectangle}}. \]Since the side lengths are \(6\) and \(3\), it is not a square.
Example: Coordinate Rhombus
Consider
\[ A(0,3),\quad B(4,0),\quad C(8,3),\quad D(4,6). \]The four side lengths are equal:
\[ AB=BC=CD=DA=5. \]The diagonals are horizontal and vertical:
\[ \overline{AC}\perp\overline{BD}. \]Therefore, the quadrilateral is a rhombus.
Its diagonals have lengths \(8\) and \(6\), so they are not congruent. Therefore, it is not a square.
Example: Using Midpoints to Prove a Parallelogram
Suppose the diagonals of a quadrilateral have the same midpoint.
If
\[ M_{AC}=M_{BD}, \]then the diagonals bisect each other.
By the converse diagonal test,
\[ \boxed{\text{the quadrilateral is a parallelogram}}. \]This method is often faster than computing the slopes of all four sides.
Choosing the Most Specific Classification
Suppose a quadrilateral is shown to have:
- four congruent sides;
- four right angles.
It is correctly described as a quadrilateral, parallelogram, rectangle, and rhombus.
The most specific familiar classification is
\[ \boxed{\text{square}}. \]A complete mathematical description may mention all classes it belongs to, but classification questions often ask for the most specific justified name.
Common Error: Treating the Classes as Separate Boxes
A square is not excluded from being a rectangle merely because it has extra properties.
Definitions work by inclusion. If a figure satisfies the defining conditions of a class, it belongs to that class even if it satisfies additional conditions.
Thus,
\[ \boxed{\text{every square is a rectangle and a rhombus}}. \]Common Error: Assuming the Converse
Opposite sides of every parallelogram are congruent.
It would be incorrect to conclude from only one congruent pair of opposite sides that a quadrilateral must be a parallelogram.
One pair of opposite sides must be both parallel and congruent for that particular converse test to work.
Common Error: Confusing Diagonal Properties
The diagonals of a rectangle are congruent, but they are not generally perpendicular.
The diagonals of a rhombus are perpendicular, but they are not generally congruent.
A square has both properties.
Keeping these distinctions clear makes diagonal-based classification much more reliable.
Common Error: Assuming a Drawing Is to Scale
A quadrilateral may look like a rectangle, rhombus, or trapezoid without actually satisfying the necessary conditions.
Geometric classification must rely on stated markings, measurements, coordinates, or proven relationships rather than visual appearance alone.
Common Error: Ignoring Definition Conventions
The terms trapezoid and kite are not defined identically in every textbook.
This lesson uses an inclusive trapezoid definition: at least one pair of opposite sides is parallel.
When using another source, check the source definition before deciding whether parallelograms count as trapezoids or rhombi count as kites.
A Reliable Classification Procedure
- Confirm that the figure is a quadrilateral.
- Identify any parallel side relationships.
- Compare side lengths.
- Identify right angles or other angle relationships.
- Analyze the diagonals if needed.
- Use converse theorems only when their full hypotheses are satisfied.
- Choose the most specific classification justified by the evidence.
- Check whether the terminology convention affects the answer.
A Reliable Coordinate Procedure
- Plot or list the vertices in boundary order.
- Compute slopes to test parallel and perpendicular sides.
- Compute side lengths when congruence matters.
- Compute diagonal lengths when rectangle or isosceles-trapezoid properties matter.
- Compute diagonal midpoints when testing for a parallelogram.
- Combine the results rather than relying on one isolated measurement.
- State the most specific classification supported by the calculations.
Practice 1: Missing Quadrilateral Angle
Find \(x\):
\[ 91^\circ+84^\circ+103^\circ+x=360^\circ. \]Practice 2: Parallelogram Angles
One angle of a parallelogram measures \(127^\circ\). Find the measures of the other three angles.
Practice 3: Parallelogram Diagonals
Diagonals \(\overline{AC}\) and \(\overline{BD}\) of parallelogram \(ABCD\) intersect at \(M\).
If
\[ AM=3x+2 \qquad\text{and}\qquad MC=5x-8, \]find \(x\).
Practice 4: Rectangle or Square?
A parallelogram has four right angles. Its adjacent side lengths are \(9\) and \(5\).
Give the most specific classification.
Practice 5: Rhombus or Square?
A parallelogram has four congruent sides. Its diagonals are perpendicular but have lengths \(10\) and \(24\).
Give the most specific classification.
Practice 6: Square Properties
List three diagonal properties that a square has.
Practice 7: Trapezoid Midsegment
A trapezoid has bases of lengths \(17\) and \(29\). Find the length of its midsegment.
Practice 8: Isosceles Trapezoid Angles
In an isosceles trapezoid, one lower base angle measures \(72^\circ\).
Find the other three angles.
Practice 9: Kite Angle Relationship
In kite \(ABCD\),
\[ AB=AD \qquad\text{and}\qquad BC=CD. \]If \(m\angle B=118^\circ\), what is \(m\angle D\)?
Practice 10: Parallelogram Test
A quadrilateral has diagonals that intersect at their common midpoint. What can be concluded?
Practice 11: Coordinate Classification
Classify the quadrilateral with vertices
\[ A(0,0),\quad B(4,0),\quad C(4,4),\quad D(0,4). \]Give the most specific classification.
Practice 12: Identify the Error
A student says:
“The diagonals of this quadrilateral are congruent, so the quadrilateral must be a rectangle.”
Explain why the conclusion is not justified from that fact alone.
Practice Answers
Practice 1
\[ x=360-(91+84+103)=82. \]Therefore,
\[ \boxed{x=82^\circ}. \]Practice 2
The opposite angle is also \(127^\circ\). Each adjacent angle is
\[ 180^\circ-127^\circ=53^\circ. \]The other three angles are
\[ \boxed{53^\circ,\ 127^\circ,\ 53^\circ}. \]Practice 3
Diagonals of a parallelogram bisect each other, so
\[ 3x+2=5x-8. \] \[ 10=2x. \] \[ \boxed{x=5}. \]Practice 4
Four right angles make the parallelogram a rectangle. The adjacent side lengths are unequal, so it is not a square.
\[ \boxed{\text{rectangle}}. \]Practice 5
Four congruent sides make the parallelogram a rhombus. The diagonals are not congruent, so it is not a square.
\[ \boxed{\text{rhombus}}. \]Practice 6
A square's diagonals:
- bisect each other;
- are congruent;
- are perpendicular.
They also bisect the vertex angles.
Practice 7
\[ m=\frac{17+29}{2}=\frac{46}{2}=23. \] \[ \boxed{23}. \]Practice 8
The other angle on the same base is also \(72^\circ\). Each angle on the opposite base is supplementary to \(72^\circ\):
\[ 180^\circ-72^\circ=108^\circ. \]The other three angles are
\[ \boxed{72^\circ,\ 108^\circ,\ 108^\circ}. \]Practice 9
The opposite angles between the unequal sides of the kite are congruent. Therefore,
\[ \boxed{m\angle D=118^\circ}. \]Practice 10
If the diagonals bisect each other, then the quadrilateral satisfies a converse test for a parallelogram.
\[ \boxed{\text{The quadrilateral is a parallelogram.}} \]Practice 11
The sides are horizontal or vertical, so adjacent sides are perpendicular. Each side has length \(4\).
The quadrilateral has four right angles and four congruent sides.
\[ \boxed{\text{square}}. \]Practice 12
Congruent diagonals alone do not force a general quadrilateral to be a rectangle. For example, an isosceles trapezoid can also have congruent diagonals.
The congruent-diagonal test proves a rectangle only when the quadrilateral is already known to be a parallelogram.
What to Remember
- a quadrilateral has four sides and interior angle sum \(360^\circ\);
- a parallelogram has two pairs of opposite parallel sides;
- a rectangle is a parallelogram with four right angles;
- a rhombus is a parallelogram with four congruent sides;
- a square is both a rectangle and a rhombus;
- diagonal properties help distinguish quadrilateral families;
- an isosceles trapezoid has congruent legs, congruent base angles, and congruent diagonals;
- a kite has two pairs of adjacent congruent sides and perpendicular diagonals in the standard configuration;
- properties and converse tests must not be confused;
- coordinate geometry can verify classifications using slopes, distances, and midpoints.
The most useful organizing principle is that special quadrilaterals form a hierarchy. More specialized figures inherit the properties of the broader classes to which they belong.
Where We Go Next
Quadrilateral properties describe relationships built into specific four-sided figures. Euclidean geometry also asks a broader question: which points satisfy a given geometric condition?
The next lesson, Loci and Distance Relationships, develops that idea through sets of points defined by equal distances, fixed distances, perpendicular bisectors, angle bisectors, circles, and related constructions.