Quadrilateral
A quadrilateral is a polygon with four sides, four vertices, and interior angles that always sum to 360°. Squares, rectangles, parallelograms, rhombuses, trapezoids, and kites are all quadrilaterals.
A quadrilateral is a four-sided polygon — a closed figure made of four straight line segments meeting at four vertices. No matter its shape, every quadrilateral's interior angles sum to 360°, a fact that follows from splitting the figure into two triangles of 180° each.
Quadrilaterals form a family tree based on their properties. A parallelogram has two pairs of parallel sides, which forces opposite sides to be equal and opposite angles to be congruent. A rectangle is a parallelogram with four right angles; a rhombus is a parallelogram with four equal sides; a square is both at once. A trapezoid has exactly one pair of parallel sides (in US usage), and a kite has two pairs of adjacent equal sides. Diagonal properties distinguish them too: a parallelogram's diagonals bisect each other, a rhombus's diagonals are perpendicular, and a rectangle's diagonals are equal in length.
A special case worth knowing is the cyclic quadrilateral, whose four vertices all lie on a circle. Its opposite angles are supplementary — they sum to 180° — a property that unlocks many competition geometry problems.
Quadrilaterals appear on nearly every math-based standardized test. The GRE tests area, perimeter, and the properties of parallelograms and trapezoids, while AMC competitions push further into cyclic quadrilaterals, diagonal relationships, and creative applications of the 360° angle sum. Knowing which properties belong to which shape — and which don't — is the key to avoiding trap answers.
Key takeaways
- A quadrilateral is a polygon with four sides and four vertices; its interior angles always sum to 360°.
- Parallelograms have two pairs of parallel sides; rectangles, rhombuses, and squares are special parallelograms.
- Diagonals distinguish the shapes: they bisect each other in parallelograms, are perpendicular in rhombuses, and are equal in rectangles.
- A cyclic quadrilateral is inscribed in a circle, and its opposite angles sum to 180°.
- The GRE and AMC exams test quadrilateral properties, area formulas, and angle relationships.