Trapezoids & Kites

Not every quadrilateral is a parallelogram. A trapezoid has exactly one pair of parallel sides, and a kite has two pairs of consecutive congruent sides. Because they miss the parallelogram definition, they get their own set of properties — and their own style of problem.

The parallel sides of a trapezoid are its bases; the other two sides are its legs. In a kite, the congruent sides are next to each other, not across from each other — picture a real kite, with two short sticks meeting at the top and two long ones meeting at the bottom.

Trapezoids and isosceles trapezoids

In any trapezoid, the two angles along the same leg — one on each base — are supplementary. The bases are parallel and the leg is a transversal, so those co-interior angles add to $180^\circ$ .

An isosceles trapezoid adds three properties: the legs are congruent, each pair of base angles is congruent (the two angles on the same base match), and the diagonals are congruent. So one angle of an isosceles trapezoid determines all four: its base partner matches it, and the two angles on the other base are its supplements.

In the isosceles trapezoid below, the single tick marks show the congruent legs and the matching arcs show the two congruent base angles on the longer base.

Kites

A kite has two pairs of consecutive congruent sides. Its diagonals are perpendicular, and only one of them is bisected: the diagonal connecting the vertices between the congruent pairs (the axis of symmetry) cuts the other diagonal in half, but not the reverse.

For angles, a kite has exactly one pair of congruent opposite angles — the two angles between a long side and a short side. The other two vertex angles, at the ends of the axis of symmetry, are generally different from each other. Combine that with the fact that any quadrilateral's angles total $360^\circ$ and most kite angle problems fall open.

In the kite below, one tick marks the short congruent pair and two ticks mark the long congruent pair. The dashed diagonals meet at a right angle, and the arcs at the left and right vertices flag the one pair of congruent opposite angles.

Telling the figures apart

Count parallel pairs first. Two pairs parallel — parallelogram family. Exactly one pair — trapezoid; congruent legs or congruent diagonals upgrade it to isosceles. No parallel pairs, but two pairs of consecutive congruent sides — kite.

The diagonal behavior is a fast fingerprint too: bisecting each other means parallelogram, congruent without bisecting suggests isosceles trapezoid, and perpendicular with only one bisected means kite.

Worked examples

Example 1: angles of an isosceles trapezoid

In isosceles trapezoid $ABCD$ , $\overline{AB} \parallel \overline{DC}$ and $m\angle D = 70^\circ$ . Find $m\angle C$ and $m\angle A$ .

\angle C

is the other base angle on

\overline{DC}

, and base angles of an isosceles trapezoid are congruent

m\angle C = 70^\circ

\angle A

and

\angle D

sit on the same leg between the parallel bases, so they are supplementary

m\angle A + 70 = 180

Solve

m\angle A = 110^\circ

Answer: $m\angle C = 70^\circ$ and $m\angle A = 110^\circ$

Example 2: perimeter of a kite

A kite has two sides that measure $7$ ft each and two sides that measure $12$ ft each. Find its perimeter.

A kite's four sides come in two congruent pairs, so add both pairs

P = 2(7) + 2(12)

Multiply

P = 14 + 24

Add

P = 38

Answer: $38$ ft

Example 3: the congruent angles of a kite

In a kite, the angles at the ends of the axis of symmetry measure $50^\circ$ and $110^\circ$ . Find the measure of each of the other two angles.

The angles of any quadrilateral total

360^\circ

50 + 110 + x + x = 360

Subtract the known angles

2x = 200

The remaining two angles are the kite's congruent pair, so divide

x = 100

Answer: Each measures $100^\circ$ .

Try one yourself

Common questions

Can a trapezoid have two pairs of parallel sides?

Not under the definition we use — a trapezoid has exactly one pair of parallel sides. A figure with two parallel pairs is a parallelogram, and it gets classified with the parallelogram family instead.

Are the base angles of every trapezoid congruent?

No, only in an isosceles trapezoid. In a general trapezoid the legs can have different lengths and different tilts, so the two angles on a base can differ. What is always true is that the two angles along the same leg are supplementary.

Are the diagonals of a kite congruent?

No — they are perpendicular, which is different. The axis of symmetry bisects the cross diagonal, but the two diagonals usually have different lengths. Congruent diagonals belong to rectangles and isosceles trapezoids.

Is a rhombus a kite?

Most textbooks say no. A rhombus does have two pairs of consecutive congruent sides, but the usual kite definition requires the two pairs to be distinct — opposite sides of a kite are not congruent. Treat kites and rhombi as separate categories unless your course says otherwise.

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Trapezoids & Kites

Trapezoids and isosceles trapezoids

Kites

Telling the figures apart

Worked examples

Example 1: angles of an isosceles trapezoid

Example 2: perimeter of a kite

Example 3: the congruent angles of a kite

Try one yourself

Common questions

Keep learning

Properties of Parallelograms

Rectangles, Rhombi & Squares

Proving a Quadrilateral Is a Parallelogram

Polygon Interior & Exterior Angle Sums