Trapezoids & Kites

A trapezoid has exactly one pair of parallel sides, called the bases. The other two are the legs.

Its midsegment joins the midpoints of the legs, runs parallel to both bases, and its length is the *average* of them:

midsegment = (base₁ + base₂)/2

An isosceles trapezoid has congruent legs, and it gains two properties: base angles are congruent, and the diagonals are congruent.

A kite has two pairs of adjacent congruent sides — adjacent, not opposite, which is what separates it from a parallelogram. Its diagonals are perpendicular, and one diagonal bisects the other.

For both a kite and a rhombus, the area is half the product of the diagonals, because the diagonals meet at right angles.

A trapezoid has exactly one pair of parallel sides under the exclusive definition used in most US courses; some treatments use the inclusive definition where parallelograms count as trapezoids. Which convention is in force changes the truth of several statements, so it is worth knowing which your course uses.

The midsegment theorem for trapezoids states the midsegment is parallel to both bases with length (b₁ + b₂)/2. It reduces to the triangle midsegment theorem by drawing a diagonal, treating the figure as two triangles.

An isosceles trapezoid has congruent legs, congruent base angles at each base, and congruent diagonals; each of these is also a sufficient condition for a trapezoid to be isosceles. Because its base angles are congruent, it is cyclic — a circle passes through all four vertices.

A kite has two distinct pairs of congruent adjacent sides. Its diagonals are perpendicular, the diagonal between the congruent pairs bisects the other, and exactly one pair of opposite angles is congruent.

The area formula A = ½d₁d₂ applies to any quadrilateral with perpendicular diagonals.