Quadrilaterals

Four-sided polygons — parallelograms, rectangles, rhombuses, trapezoids — and the relationships between them.

Quadrilateral types

Definition

A quadrilateral is any closed shape with exactly four straight sides and four angles. The four interior angles always add up to $360°$ .

The main types, from most to least symmetrical:

Square: four equal sides, four right angles
Rectangle: opposite sides equal, four right angles
Rhombus: four equal sides, opposite angles equal
Parallelogram: opposite sides parallel and equal, opposite angles equal
Trapezoid (trapezium): exactly one pair of parallel sides
Kite: two pairs of adjacent equal sides

Every square is a rectangle. Every rectangle is a parallelogram. Every rhombus is a parallelogram. The relationships form a hierarchy.

Definition

A diagonal of a quadrilateral is a line segment connecting two non-adjacent vertices. Every quadrilateral has exactly two diagonals. In a parallelogram, the diagonals bisect each other (cut each other in half).

Missing angle in a quadrilateral

A quadrilateral has angles $90°$ , $85°$ , $110°$ . Find the fourth angle.

$\text{fourth angle} = 360° - 90° - 85° - 110° = 75°$

Try it

A parallelogram has one angle of $65°$ . Find all four angles.

Solution

In a parallelogram, opposite angles are equal and adjacent angles are supplementary.

Two angles are $65°$
Two angles are $180° - 65° = 115°$

The four angles are $65°$ , $115°$ , $65°$ , $115°$ .

Related concepts

Geometry· Triangles

TrianglesThree-sided polygons with rich properties connecting angles, sides, area, and similarity.

Geometry· Polygons

PolygonsClosed figures formed by straight line segments — classified by number of sides, regularity, and interior angle sums.

Geometry· Measurement

Perimeter & AreaMeasuring the boundary length and the enclosed space of two-dimensional figures.