An isosceles trapezoid (isosceles trapezium in British English) is a quadrilateral with a line of symmetry bisecting one pair of opposite sides, making it automatically a trapezoid. Two opposite sides (bases) are parallel, the two other sides (legs) are of equal length. The diagonals are of equal length. An isosceles trapezoid's base angles are equal in measure. Any quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite.
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Diagonals
The diagonals of an isosceles trapezoid have the same length and divide each other into segments of the same length. In the picture below, the diagonals AC and BD have the same length, that is AC = BD; and they divide each other in segments of the same length, that is, AE = DE and BE = CE.
The ratio in which each diagonal is divided is equal to the ratio of the lengths of the parallel sides that they intersect, that is,
The length of each diagonal is given by:
(where c is the length of each leg AB and CD).
Angles
In an isosceles trapezoid the base angles have the same measure. In the picture on the right, angles 
and 
are obtuse angles of the same measure, while angles 
and 
are acute angles, also of the same measure.
Since the lines AD and BC are parallel, angles adjacent to opposite bases are supplementary, that is, angles 
.
Area
The area of an isosceles (or any) trapezoid is equal to the average of the lengths of the base and top (the parallel sides) times the height. In the diagram to the right, if we write AD = a, and BC = b, and the height h is the length of a line segment between AD and BC that is perpendicular to them, then the area is given as follows
If instead of the height of the trapezoid, the length of the leg AB = c is known, then the area can be computed using the formula
,
where
is the semi-perimeter of the trapezoid. This formula is analogous to Heron's formula to compute the area of a triangle.The previous formula for area can also be written as
.
Characterization of isosceles trapezoids
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.
- The diagonals divide each other into segments with lengths that are pairwise equal; in terms of the picture above, AE = DE, BE = CE (and
if one wishes to exclude rectangles).
An isosceles trapezoid can also be defined as "a cyclic quadrilateral with equal diagonals". [1]
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