Isosceles Trapezoid - Characterizations

Characterizations

If a quadrilateral is known to be a trapezoid, it is not necessary to check that the legs have the same length in order to know that it is an isosceles trapezoid; any of the following properties also distinguishes an isosceles trapezoid from other trapezoids:

• The diagonals have the same length.
• The base angles have the same measure.
• An isosceles triangle is formed by the base and the extensions of the legs.
• The segment that joins the midpoints of the parallel sides is perpendicular to them.
• Opposite angles are supplementary, which in turn implies that isosceles trapezoids are cyclic quadrilaterals.
• The diagonals divide each other into segments with lengths that are pairwise equal; in terms of the picture below, AE = DE, BE = CE (and AE ≠ CE if one wishes to exclude rectangles).

If rectangles are included in the class of trapezoids then one may concisely define an isosceles trapezoid as "a cyclic quadrilateral with equal diagonals" or as "a cyclic quadrilateral with a pair of parallel sides."