Geometry

Trapezoids and Kites

Big Picture

Not all quadrilaterals are parallelograms. Trapezoids and kites are two non-parallelograms with special properties.

Key Terms

Trapezoid: Quadrilateral with exactly one pair of parallel sides.

Isosceles Trapezoid: A trapezoid where the non-parallel sides are congruent.

Midsegment: A line segment that connects the midpoints of the non-parallel sides of a trapezoid.

Kite: Quadrilateral with two sets of adjacent congruent sides.

Trapezoids

The parallel sides of a trapezoid are called the bases, while the nonparallel sides are called the legs. A pair of angles that share the same base are called base angles. A trapezoid has two sets of base angles.

Trapezoids

Isosceles Trapezoids

Isosceles Trapezoids

Think of an isosceles trapezoid as an isosceles triangle with the top cut off. As result, many of the theorems that describe isosceles triangles have corresponding theorems for trapezlids. The properties of isosceles trapezoids are defined by the following theorems:

Theorem: Both pairs of base angles of an isosceles trapezoid are congruent.

  • The converse can also be used: If a trapezoid has congruent base angles, then it is an isosceles trapezoid.

Theorem: The diagonals of an isosceles trapezoid are congruent.

Midsegments

The midsegment is parallel to the bases and is located halfway between them.

Midsegments

Theorem: The length of the midsegment of a trapezoid is the average of the lengths of the bases.

  • If  \( \overline{EF}\) is the midsegment of trapezoid ABCD, then \( \overline{EF} || \overline{AB},\overline{EF} || \overline{DC}, \text{ and } EF = \frac{1}{2}(AB + DC)\)

Geometry

Trapezoids and Kites Cont.

Kites

A kite has two sets of adjacent, distinct congruent sides. Rhombuses and squares are not kites! Kites are also the only quadrilaterals that can be concave. A concave kite (the rightmost kite in the diagram below) is called a dart.

kites

When working with kites, think of the traditional kites that are own in the air.

The angles between the congruent sides are called vertex angles. The other angles are non-vertex angles.

vertex angles

Theorem: The non-vertex angles of a kite are congruent.

Theorem: The diagonal through the vertex angles is the angle bisector for both angles.

Kite Diagonals Theorem: The diagonals of a kite are perpendicular.

  • If KITE is a kite, then \(\overline{KT} \perp \overline{EI}\)
non-vertex angles

Notes