Types of Quadrilaterals

Why This Matters

Quadrilaterals aren't just shapes to memorize—they form a hierarchy of properties that geometry exams love to test. When you understand how a square inherits properties from both rectangles and rhombuses, or why a trapezoid sits outside the parallelogram family, you're thinking like a geometer. You'll be tested on property inheritance, diagonal behavior, angle relationships, and the logical connections between these figures.

The key insight is that quadrilaterals are classified by their defining constraints—parallel sides, equal sides, right angles, and diagonal properties. Don't just memorize that a rhombus has equal sides; know that this constraint forces its diagonals to bisect at right angles. When you understand the "why," you can reconstruct facts on test day instead of blanking on a memorized list.

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The Parallelogram Family

These quadrilaterals all share one fundamental property: both pairs of opposite sides are parallel. This single constraint creates a cascade of consequences—opposite sides become equal, opposite angles become equal, and diagonals always bisect each other.

Parallelogram

• Both pairs of opposite sides are parallel and equal—this is the defining property that all other family members inherit
• Diagonals bisect each other but aren't necessarily equal or perpendicular—bisection is guaranteed, nothing else
• Opposite angles are equal; consecutive angles are supplementary—knowing one angle lets you find all four

Rectangle

• All four angles are right angles ($90°$)—this is the added constraint beyond a basic parallelogram
• Diagonals are equal in length and still bisect each other—the right angles force diagonal equality
• Opposite sides are equal (inherited from parallelogram)—a rectangle is not required to have four equal sides

Rhombus

• All four sides are equal in length—this constraint (not right angles) defines a rhombus
• Diagonals bisect each other at right angles ($90°$)—equal sides force perpendicular diagonals, a key testable property
• Opposite angles are equal; consecutive angles are supplementary—a rhombus can be "tilted" with no right angles at vertices

Square

• All four sides equal AND all four angles are $90°$—satisfies both rectangle and rhombus requirements simultaneously
• Diagonals are equal, bisect each other, AND intersect at right angles—inherits diagonal properties from both parent shapes
• Most constrained parallelogram—every square is a rectangle, rhombus, and parallelogram, but not vice versa

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The Trapezoid Family

These quadrilaterals have exactly one pair of parallel sides (called bases), which excludes them from the parallelogram family. The non-parallel sides are called legs.

Trapezoid

• Exactly one pair of parallel sides (the bases)—this is the defining and limiting property
• Consecutive angles between a base and leg are supplementary—same-side interior angle theorem applies to the parallel bases
• Legs can be any length—no equality requirements unless it's a special trapezoid type

Isosceles Trapezoid

• Legs are equal in length—this added constraint creates symmetry about a perpendicular axis
• Base angles are equal—angles adjacent to the same base are congruent, a frequently tested property
• Diagonals are equal in length—like a rectangle, but diagonals don't bisect each other

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Quadrilaterals Outside Both Families

The kite has no parallel sides, placing it outside both the parallelogram and trapezoid families. Its properties come entirely from adjacent side equality.

Kite

• Two distinct pairs of adjacent sides are equal—not opposite sides, but sides that share a vertex
• Diagonals intersect at right angles—the "main diagonal" (connecting vertices between unequal sides) is bisected by the other
• One pair of opposite angles are equal—the angles between the unequal sides are congruent; the other pair typically aren't

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Quick Reference Table

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Self-Check Questions

1. Which two quadrilaterals have perpendicular diagonals but are not parallelograms? What property distinguishes them from each other?

2. A quadrilateral has diagonals that are both equal and perpendicular. What shape must it be, and why can't it be anything else?

3. Compare and contrast a rectangle and an isosceles trapezoid: what diagonal property do they share, and what fundamental difference separates them?

4. If you know a parallelogram has one right angle, what can you conclude about the other three angles and the shape's classification?

5. A student claims "all rhombuses are squares." Explain the error and identify what additional constraint would make a rhombus into a square.