DISCOVER

# What Quadrilaterals Must Have Diagonals of Equal Length?

Updated March 23, 2017

A polygon is any two-dimensional shape with three or more sides. The term "quadrilateral" refers to a polygon with four sides. The corner of a quadrilateral is also known as a vertex; the plural of vertex is vertices. A diagonal is a line draw between opposite vertices of a quadrilateral.

All quadrilaterals have four sides and are two-dimensional; the sum of any quadrilateral's interior angles is 360 degrees. Some examples of quadrilaterals include parallelograms, kites, trapezoids, rhombuses, isosceles trapezoids, rectangles and squares. The only quadrilaterals that have diagonals of equal length are rectangles, squares and isosceles trapezoids.

## Rectangles

All rectangles have specific geometric properties; a rectangle has four 90 degree angles. The opposite sides of a rectangle are parallel; this means that a rectangle is also a parallelogram. The diagonals of a rectangle bisect each other; this means that the point where the diagonals intersect divides each of them exactly in half. The diagonals of a rectangle are also congruent, or of equal length.

## Squares

Squares are rectangles with four congruent sides. Therefore, squares also have the properties of a rectangle: all four interior angles are 90 degrees, the opposite sides are parallel, the diagonals are bisecting and the diagonals are of equal length. The diagonals of a square are also perpendicular, meaning that they form 90 degree angles at their intersection. The diagonals of a square also bisect the opposite angles. In the case of a square, the diagonals split the 90 degree vertices into two 45 degree angles.

## Isosceles Trapezoid

An isosceles trapezoid is a quadrilateral with only two opposing sides that are parallel, and the other two sides not parallel but are equal in length. An isosceles trapezoid's diagonals are congruent or equal in length. The base angles, those at the ends of the parallel lines, are congruent. This means that the opposite angles of the figure are supplementary, adding up to 180 degrees.