Math Core

Not every useful quadrilateral is a parallelogram. Trapezoids have only one pair of parallel sides, and kites have no parallel sides at all, yet both have dependable angle, side and diagonal properties. This lesson finishes the family of quadrilaterals.

Trapezoids

Definition

Trapezoid

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases and the other two sides are the legs. A pair of angles that share a base is a pair of base angles. If the legs are congruent, the trapezoid is isosceles.

(Some books define a trapezoid as having at least one pair of parallel sides, which makes every parallelogram a trapezoid. This course uses "exactly one.")

Because the bases are parallel, each leg is a transversal. The two angles along one leg are same-side interior angles, so they are supplementary. In trapezoid $ABCD$ with bases $\overline{AB}$ and $\overline{DC}$ , that means $m\angle A + m\angle D = 180^\circ$ and $m\angle B + m\angle C = 180^\circ$ .

Isosceles trapezoids

An isosceles trapezoid has a line of symmetry through the midpoints of its bases. Reflecting across that line swaps the two legs, the two lower base angles, the two upper base angles and the two diagonals. That symmetry is the idea behind these theorems.

Isosceles trapezoid theorems

If a trapezoid is isosceles, then:

Each pair of base angles is congruent.
Its diagonals are congruent.

The converses are also true: a trapezoid with a pair of congruent base angles, or with congruent diagonals, is isosceles.

Worked example: Angles of an isosceles trapezoid

$ABCD$ is an isosceles trapezoid with bases $\overline{AB}$ and $\overline{DC}$ , and $m\angle A = 64^\circ$ . Find the other three angles.

$\angle A$ and $\angle B$ are base angles of the same base, so $m\angle B = 64^\circ$ . $\angle A$ and $\angle D$ lie along the same leg, so they are supplementary: $m\angle D = 116^\circ$ . $\angle C$ and $\angle D$ are the other pair of base angles, so $m\angle C = 116^\circ$ . Check: $64 + 64 + 116 + 116 = 360$ .

Common mistake

Base angles share a base: in the example, $\angle A \cong \angle B$ and $\angle C \cong \angle D$ . Opposite angles of a trapezoid, like $\angle A$ and $\angle C$ , are not congruent. They are supplementary in an isosceles trapezoid, because $\angle C \cong \angle D$ and $\angle D$ is supplementary to $\angle A$ .

The midsegment of a trapezoid

The midsegment of a trapezoid joins the midpoints of its legs.

Midsegment MN of trapezoid ABCD is parallel to the bases, and its length 6 is the average of 8 and 4.

Trapezoid midsegment theorem

The midsegment of a trapezoid is parallel to both bases, and its length is the average of the base lengths:

MN = \frac{b_1 + b_2}{2}.

You can see why from the triangle midsegment theorem. Draw diagonal $\overline{AC}$ . In $\triangle ABC$ , the segment from $N$ to the midpoint of $\overline{AC}$ is half of $AB$ . In $\triangle ACD$ , the segment from the midpoint of $\overline{AC}$ to $M$ is half of $DC$ . Those two pieces are both parallel to the bases and meet at the midpoint of $\overline{AC}$ , so they line up into $\overline{MN}$ , with length $\dfrac{AB}{2} + \dfrac{DC}{2}$ .

Worked example: Midsegment with algebra

The bases of a trapezoid measure $2x + 3$ and $4x - 1$ , and its midsegment measures $13$ . Find $x$ and both bases.

\begin{aligned} \frac{(2x + 3) + (4x - 1)}{2} &= 13 \\ 6x + 2 &= 26 \\ x &= 4 \end{aligned}

The bases are $11$ and $15$ . Check: $\dfrac{11 + 15}{2} = 13$ .

Kites

Definition

Kite

A kite is a quadrilateral with two pairs of consecutive congruent sides, where opposite sides are not congruent.

In kite $ABCD$ with $AB = AD$ and $CB = CD$ , the vertices $A$ and $C$ where the equal sides meet are the vertex angles. The diagonal $\overline{AC}$ joining them is a line of symmetry.

Kite ABCD with AB = AD and CB = CD. Diagonal AC is the line of symmetry, and it meets BD at a right angle.

Kite theorems

In kite $ABCD$ with $AB = AD$ and $CB = CD$ :

The diagonals are perpendicular.
Exactly one pair of opposite angles is congruent: $\angle B \cong \angle D$ , the angles between the unequal sides.
Diagonal $\overline{AC}$ bisects diagonal $\overline{BD}$ and bisects the vertex angles $\angle A$ and $\angle C$ . The other diagonal is generally not bisected.

These follow from congruent triangles. $\triangle ABC \cong \triangle ADC$ by SSS (they share $\overline{AC}$ ), so $\angle B \cong \angle D$ and $\overline{AC}$ bisects $\angle A$ . Then $\triangle ABE \cong \triangle ADE$ by SAS, so $BE = DE$ and the angles at $E$ are congruent and form a linear pair, so each is $90^\circ$ .

Worked example: Angles of a kite

In kite $ABCD$ , $AB = AD$ and $CB = CD$ . The vertex angles measure $m\angle A = 70^\circ$ and $m\angle C = 50^\circ$ . Find $m\angle B$ .

The angles of a quadrilateral add to $360^\circ$ , and $\angle B \cong \angle D$ :

2 \cdot m\angle B = 360^\circ - 70^\circ - 50^\circ = 240^\circ, \qquad m\angle B = 120^\circ.

Worked example: Sides of a kite

The diagonals of kite $ABCD$ meet at $E$ , with $AE = 5$ , $CE = 9$ and $BE = DE = 12$ . Find the perimeter.

The diagonals are perpendicular, so every side is the hypotenuse of a right triangle:

AB = \sqrt{5^2 + 12^2} = 13, \qquad BC = \sqrt{9^2 + 12^2} = 15.

By symmetry $AD = 13$ and $CD = 15$ , so the perimeter is $2(13) + 2(15) = 56$ .

Tip

To tell the quadrilaterals apart by their diagonals: a kite's diagonals are perpendicular but only one is bisected; a rhombus's diagonals are perpendicular and both are bisected; an isosceles trapezoid's diagonals are congruent but don't bisect each other.