Trapezoids and Kites
Define and classify trapezoids (one pair of parallel sides) and isosceles trapezoids. Prove the Trapezoid Midsegment Theorem. Identify properties of kites (two pairs of consecutive congruent sides) and apply them.
A lesson of Geometry, in Quadrilaterals and Polygons, about 10 minutes.
By the end of this lesson, you will be able to apply properties of trapezoids (including isosceles trapezoids) and kites, find the midsegment of a trapezoid, and solve algebraic problems using these properties.
The Big Idea
Trapezoids and kites are not parallelograms (they don't have two pairs of parallel sides), but they still have special properties worth studying. The trapezoid's midsegment theorem mirrors the triangle midsegment theorem; the isosceles trapezoid has diagonals that mirror rectangles. Kites bring symmetry without parallelism.
Trapezoids
Definition: a quadrilateral with exactly one pair of parallel sides.
- Bases: the parallel sides ($\overline{BC} \parallel \overline{AD}$)
- Legs: the non-parallel sides ($\overline{AB}$ and $\overline{CD}$)
- Base angles: pairs of angles that share a base
Co-interior angles: each leg creates two supplementary co-interior angles with the bases.
$$\angle A + \angle B = 180° \quad \text{and} \quad \angle C + \angle D = 180°$$
Isosceles Trapezoid
Definition: a trapezoid with congruent legs ($\overline{AB} \cong \overline{CD}$).
Properties of isosceles trapezoids:
- Base angles are congruent: $\angle A \cong \angle D$ and $\angle B \cong \angle C$
- Diagonals are congruent: $AC = BD$
Theorem converse: If the base angles of a trapezoid are congruent, it is isosceles.
Trapezoid Midsegment Theorem
The midsegment (median) of a trapezoid connects the midpoints of the legs.
$$m = \frac{b_1 + b_2}{2}$$
where $b_1$ and $b_2$ are the two bases.
The midsegment is parallel to both bases and its length is the average of the two base lengths.
Example 1 — Isosceles Trapezoid Angles
Isosceles trapezoid $ABCD$ with $\angle A = 75°$. Find all angles.
$\angle D = \angle A = 75°$ (base angles)
$\angle B = 180° - 75° = 105°$ (co-interior with $\angle A$)
$\angle C = \angle B = 105°$ (base angles)
Example 2 — Trapezoid Midsegment
Trapezoid with bases 9 and 15. Find the midsegment.
$$m = \frac{9 + 15}{2} = 12$$
Example 3 — Algebra
Midsegment $= 3x + 2$, bases $= 2x + 5$ and $5x - 1$. Find $x$ and the midsegment length.