A quadrilateral that lacks parallel opposite sides is not a parallelogram. This includes irregular quadrilaterals, trapezoids with no parallel sides, kites with nonperpendicular diagonals, rhombuses with unequal diagonals, oblique rectangles with unequal diagonals, and squares with unequal diagonals (which do not exist). In general, parallelograms are defined by the presence of parallel opposite sides, and quadrilaterals that lack this property fall outside this category.
Trapezoid with No Parallel Sides: Unveiling the Irregular Quadrilateral
In the realm of geometry, we encounter a curious shape that defies the norm of parallelism: the irregular quadrilateral. Unlike its trapezoidal brethren, which boast at least one pair of parallel sides, this geometric enigma lacks such a defining feature.
An irregular quadrilateral, also known as a trapezoid with no parallel sides, possesses four sides of varying lengths. Crucially, its opposite sides do not run parallel to each other, giving it an asymmetrical appearance. This unique characteristic distinguishes it from its more familiar counterparts, such as rectangles and parallelograms.
Characteristics of an Irregular Quadrilateral
 Four sides: An irregular quadrilateral has four sides, each with a distinct length.
 No parallel sides: Unlike trapezoids, irregular quadrilaterals lack parallel opposite sides. This absence of parallelism sets them apart from other quadrilateral shapes.
 Irregular shape: The lack of parallel sides results in an irregular shape, devoid of the symmetry found in other quadrilaterals.
Irregular quadrilaterals may be encountered in various realworld applications, such as architectural designs, art installations, and even geological formations. They serve as a reminder that geometric shapes can deviate from the familiar and still embody mathematical principles.
The Enigmatic Deltoid: A Kite with No Perpendicular Diagonals
In the world of geometry, shapes often conform to specific rules and relationships. However, there are a few exceptions that defy conventions, such as the peculiar deltoid. This intriguing quadrilateral possesses a unique combination of features that sets it apart from its more familiar cousins.
Unlike ordinary kites, a deltoid does not boast perpendicular diagonals. Instead, its two diagonals intersect at an angle that is not 90 degrees. This nonperpendicularity is what distinguishes a deltoid from a true kite and earns it the alternative name of “diamond.”
Characteristics of a Deltoid:

Four Equal Sides: Like a kite, a deltoid has four sides of equal length. This symmetry gives it a pleasing visual appearance, often resembling a rhombus.

Adjacent Sides: While the sides are equal, they are not all parallel to each other. The deltoid’s adjacent sides are parallel, but its opposite sides are not.

NonPerpendicular Diagonals: The most striking characteristic of a deltoid is its two diagonals. They are not perpendicular to each other, unlike a square or a parallelogram. This lack of perpendicularity is what gives the deltoid its unique shape and geometric classification.
The deltoid’s existence challenges the notion that all quadrilaterals with equal sides must have perpendicular diagonals. It stands as a testament to the diverse and fascinating world of geometric forms, where the unexpected can sometimes be found.
Rhombus: A Quadrilateral Enigma with Unequal Diagonals
Imagine a quadrilateral, a shape with four sides, that boasts the alluring characteristic of equal sides. Meet the rhombus, a captivating figure renowned for its uniform sides. However, beneath its graceful exterior lies a hidden twist: unequal diagonals.
A rhombus is more than just a quadrilateral with equal sides. It also possesses two defining diagonals that intersect at its heart. These diagonals, like dutiful servants, cut the rhombus into four congruent triangles, making it a true marvel of geometric symmetry.
But not all rhombi are created equal. Some defy the convention of having equal diagonals, resulting in an intriguing departure from the norm. These unorthodox rhombi, devoid of perpendicular diagonals, embark on a journey to a realm of their own, where they challenge the traditional boundaries of their kind.
This deviation from the norm transforms the rhombus from a mere quadrilateral into a captivating entity, unyielding to the rules of parallelogramhood. It refuses to be confined to the rigid constraints of equal diagonals, setting it apart as a unique and enigmatic figure in the world of quadrilaterals.
Oblique Rectangle: Unraveling the Parallel Sides and Unequal Diagonals
In the captivating world of geometry, there lies a fascinating shape known as an oblique rectangle. Unlike its ordinary counterpart, this rectangle possesses a unique characteristic that sets it apart and sparks curiosity.
Parallel Sides: The Cornerstone of a Rectangle
The essence of a rectangle lies in its parallel and congruent opposite sides. Just like the tracks of a railway, these sides run alongside each other, creating a sense of symmetry and balance. This fundamental property distinguishes rectangles from other quadrilaterals and grants them their esteemed status in the geometric realm.
Unequal Diagonals: A Twist in the Tale
The defining feature that distinguishes an oblique rectangle from its ordinary counterpart is its unequal diagonals. Diagonals, those enchanting lines that connect opposite vertices, typically meet at a perfect right angle in a rectangle. However, in the case of an oblique rectangle, this harmonious intersection is lost. Its diagonals dance across the shape at an angle other than 90 degrees, creating a subtle yet noticeable deviation from the norm.
A Duality of Parallelism and Asymmetry
This intriguing duality, where parallel sides intertwine with unequal diagonals, gives the oblique rectangle a peculiar charm. It is a testament to the boundless diversity that exists within the realm of quadrilaterals. While it shares the parallel sides of its rectangular cousins, the asymmetry introduced by its unequal diagonals opens up a new dimension of geometric exploration.
Breaking the Mold: Why an Oblique Rectangle Cannot Be a Parallelogram
Despite its rectangular attributes, an oblique rectangle falls outside the boundaries of parallelograms. Parallelograms, by their very nature, demand that their opposite sides be congruent and parallel. The oblique rectangle, with its unequal diagonals, defies this requirement and carves out its own unique place in the vast geometric tapestry.
The oblique rectangle, with its parallel sides and unequal diagonals, is a captivating shape that challenges our preconceived notions of rectangles. It showcases the infinite possibilities that geometry has to offer and invites us to delve deeper into the intricacies of shapes and their fascinating properties.
Square with Unequal Diagonals
 Explanation of why a square with unequal diagonals does not exist, citing its definition as a parallelogram with equal sides and perpendicular diagonals.
The Nonexistent Paradox: Squares with Unequal Diagonals
In the realm of geometry, the square stands as an emblem of symmetry and perfection, its four equal sides and perpendicular diagonals forming an elegant dance of precise angles. However, the allure of perfection can sometimes lead us to question its limits, prompting us to ponder the existence of squares that defy their fundamental characteristics.
Consider the notion of a square with unequal diagonals. Such a concept seems to contradict the very definition of a square, which stipulates that it is a parallelogram with equal sides and perpendicular diagonals. A parallelogram, by its nature, possesses opposite sides that are parallel and equal in length, and its diagonals intersect at perpendicular angles.
To envision a square with unequal diagonals is to challenge these defining traits. If the diagonals are unequal, then the parallelogram property is violated, as the shape would no longer have opposite sides of equal length. Additionally, if the diagonals are not perpendicular, then the square would no longer possess perpendicular bisectors, further deviating from its geometric definition.
In essence, the existence of a square with unequal diagonals would be an oxymoron, a geometric paradox that defies the very essence of what makes a square a square. The symmetry, the precision, the inherent order that we associate with squares would all be shattered, leaving us with a shape that fails to meet the criteria of its own definition.
Therefore, we can conclude that the concept of a square with unequal diagonals is a mathematical impossibility, a footnote in the annals of geometry that serves to remind us of the inviolable principles that govern the world of shapes and forms.