The Kite | Moosa Academy

The Kite — Moosa Academy

A rhombus has all four sides equal. Loosen that to two pairs of equal sides — but neighbouring ones rather than opposite ones — and you get a kite. It keeps the perpendicular diagonals, but loses almost everything else. Comparing the two shows exactly which property comes from where.

Concept Adjacent pairs, not opposite ones

The two short sides meet at

A

and the two long sides meet at

C

. So the equal pairs are

AB = AD

and

CB = CD

— sides that touch each other, unlike a parallelogram where the equal sides are opposite.

Theorem The properties of a kite

Two pairs of equal adjacent sides:

AB = AD

and

CB = CD

One pair of equal angles:

\angle B = \angle D

— the two between unequal sides

Perpendicular diagonals:

AC \perp BD

One diagonal bisects the other:

BM = DM

, but

AM \neq MC

One line of symmetry: the diagonal

AC

That fourth line is the one students most often get wrong. In a kite only one diagonal is cut in half — the other is not.

Concept Why the diagonals are perpendicular

The argument is the same one used for a rhombus, and it needs only the equal pairs:

AB = AD

, so

A

is equidistant from

B

and

D

CB = CD

, so

C

is equidistant from

B

and

D

Two points equidistant from $B$ and $D$ both lie on the perpendicular bisector of $BD$ , and the line through them is $AC$ . Therefore:

AC \perp BD \quad \text{and} \quad BM = DM

Notice the argument only ever mentions $B$ and $D$ . Nothing forces $AM$ to equal $MC$ , which is precisely why the other diagonal is not bisected.

Example Finding the remaining angles

In kite $ABCD$ , $\angle A = 108^\circ$ and $\angle C = 62^\circ$ . Find $\angle B$ and $\angle D$ .

All four angles add to

360^\circ

\angle B + \angle D = 360^\circ - 108^\circ - 62^\circ = 190^\circ

\angle B = \angle D

, so each is

190^\circ \div 2 = 95^\circ

The equal pair is always $\angle B$ and $\angle D$ — the angles where a short side meets a long one. The angles at $A$ and $C$ are generally different from each other.

Note Kite against rhombus

Property	Kite	Rhombus
Sides	Two adjacent pairs	All four equal
Equal angles	One pair (B and D)	Both pairs
Diagonals	Perpendicular; one bisected	Perpendicular; both bisected
Lines of symmetry	1	2

Make the two pairs equal to each other and the kite becomes a rhombus. So every rhombus is a kite, but a kite is a rhombus only in that special case.

Summary

A kite has two pairs of equal sides that are adjacent, not opposite.
Its diagonals are perpendicular, for the same reason as in a rhombus.
Only one diagonal is bisected — the one joining the unequal-angle corners.
Exactly one pair of opposite angles is equal.
It has one line of symmetry, along the diagonal that bisects the other.
When all four sides become equal, the kite is a rhombus.