Properties of Quadrilaterals

Why This Matters

Quadrilaterals aren't just shapes you memorize—they're a hierarchy of relationships that tests your ability to think logically about geometric properties. In Honors Geometry, you're being tested on how properties inherit from one shape to another: why every square is a rectangle, but not every rectangle is a square. Understanding this classification system helps you tackle proofs, identify which theorems apply in a given problem, and avoid common traps on tests.

The key concepts here are parallel sides, congruent segments, diagonal behavior, and angle relationships. When you see a quadrilateral problem, your first job is classification—because once you know what type you're dealing with, you unlock all its properties. Don't just memorize that a rhombus has perpendicular diagonals; understand why that property matters and how it connects to the rhombus being a special parallelogram. That's what separates strong geometry students from those who struggle on proofs and applications.

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The Parallelogram Family

Most quadrilateral problems center on parallelograms and their special cases. The defining feature is two pairs of parallel sides, which creates a cascade of other properties through parallel line theorems.

Parallelogram

• Opposite sides are parallel and congruent—this is the defining property that generates all others
• Opposite angles are congruent and consecutive angles are supplementary (sum to $180°$)
• Diagonals bisect each other—they cut each other in half but aren't necessarily equal or perpendicular

Rectangle

• Four right angles ($90°$ each)—this is what distinguishes a rectangle from a generic parallelogram
• Diagonals are congruent—equal in length, which parallelograms don't guarantee
• Inherits all parallelogram properties—opposite sides parallel and congruent, diagonals still bisect each other

Rhombus

• All four sides are congruent—equal side lengths define the rhombus within the parallelogram family
• Diagonals are perpendicular—they intersect at $90°$ angles, creating four congruent right triangles
• Diagonals bisect the vertex angles—each diagonal cuts its corner angles in half

Square

• Combines rectangle AND rhombus properties—four right angles plus four congruent sides
• Diagonals are congruent, perpendicular, and bisect each other—the most "special" diagonal behavior possible
• Most restrictive quadrilateral—every square is a rectangle, rhombus, AND parallelogram

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Quadrilaterals with One Pair of Parallel Sides

Trapezoids break from the parallelogram family by having only one pair of parallel sides (the bases). This changes everything about their angle relationships and diagonal behavior.

Trapezoid

• Exactly one pair of parallel sides—called the bases; the non-parallel sides are called legs
• Same-side interior angles are supplementary—angles on the same leg sum to $180°$ (parallel line theorem)
• Midsegment connects leg midpoints—parallel to both bases with length equal to $\frac{b_1 + b_2}{2}$

Isosceles Trapezoid

• Legs are congruent—this creates symmetry that generic trapezoids lack
• Base angles are congruent—both angles touching the same base are equal
• Diagonals are congruent—like rectangles, but without the parallelogram properties

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Quadrilaterals with No Parallel Sides

Kites have a completely different structure—defined by adjacent congruent sides rather than parallel sides. Their symmetry runs along one diagonal instead of through the center.

Kite

• Two pairs of consecutive (adjacent) sides are congruent—not opposite sides like parallelograms
• One pair of opposite angles are congruent—the angles between the unequal sides (the "vertex angles")
• Diagonals are perpendicular—one diagonal bisects the other, but they don't bisect each other

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Fundamental Angle Properties

Every quadrilateral—regardless of type—follows the same interior angle rule. This comes from the polygon angle formula and applies universally.

Interior Angle Sum

• All quadrilaterals have interior angles summing to $360°$—derived from $(n-2) \times 180°$ where $n = 4$
• Divide any quadrilateral into two triangles—draw one diagonal to see why $2 \times 180° = 360°$
• Use this to find missing angles—if three angles are known, subtract their sum from $360°$

Exterior Angles

• Exterior angles of any convex quadrilateral sum to $360°$—same as all convex polygons
• Each exterior angle is supplementary to its interior angle—they form a linear pair
• Useful for problems involving turns or rotations—walking around a quadrilateral means turning $360°$ total

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Diagonal Properties

Diagonals reveal a quadrilateral's hidden structure. How they interact—whether they bisect, are congruent, or are perpendicular—is often the key to classification and proofs.

Diagonal Behavior by Shape

• Parallelogram diagonals bisect each other—this is actually an "if and only if" condition for proving parallelograms
• Rectangle diagonals are congruent—add this to bisecting, but they're not perpendicular
• Rhombus diagonals are perpendicular bisectors of each other—creating four congruent right triangles

Using Diagonals in Proofs

• To prove a parallelogram: show diagonals bisect each other
• To prove a rectangle: show a parallelogram has congruent diagonals (or one right angle)
• To prove a rhombus: show a parallelogram has perpendicular diagonals (or two consecutive congruent sides)

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Area Formulas

Area formulas reflect each shape's structure. Parallelograms use base times height; shapes with perpendicular diagonals use the diagonal product formula.

Base-Height Formulas

• Parallelogram: $A = bh$ where $h$ is the perpendicular height, not the side length
• Rectangle: $A = lw$—a special case where the side IS the height
• Square: $A = s^2$—all sides equal, so length times width becomes side squared

Diagonal-Based Formulas

• Rhombus: $A = \frac{d_1 \cdot d_2}{2}$—works because diagonals are perpendicular
• Kite: $A = \frac{d_1 \cdot d_2}{2}$—same formula, same reason (perpendicular diagonals)
• Trapezoid: $A = \frac{(b_1 + b_2)}{2} \cdot h$—average of bases times height

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Cyclic and Inscribed Quadrilaterals

When quadrilaterals interact with circles, new angle relationships emerge. These properties connect quadrilateral geometry to circle theorems.

Cyclic Quadrilaterals

• All four vertices lie on a single circle—the quadrilateral is inscribed in the circle
• Opposite angles are supplementary—they sum to $180°$ (this is both necessary and sufficient)
• Brahmagupta's formula gives area: $A = \sqrt{(s-a)(s-b)(s-c)(s-d)}$ where $s$ is the semi-perimeter

Circumscribed Quadrilaterals

• All four sides are tangent to a circle—the circle is inscribed in the quadrilateral
• Opposite sides sum to equal values: $a + c = b + d$—this is the key property
• Useful for problems combining quadrilateral and circle properties—look for tangent segments

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Quick Reference Table

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Self-Check Questions

1. Which two quadrilaterals share the property of perpendicular diagonals but differ in whether those diagonals bisect each other?

2. A quadrilateral has diagonals that are both congruent AND perpendicular. What type of quadrilateral must it be, and why?

3. Compare and contrast the properties you would use to prove a parallelogram is a rectangle versus proving it's a rhombus.

4. If a cyclic quadrilateral has one angle measuring $70°$, what is the measure of the opposite angle? What property justifies your answer?

5. You're given a quadrilateral with vertices on a coordinate plane. Describe the steps you would take to classify it as specifically as possible (parallelogram, rectangle, rhombus, or square).