# Prove that the Diagonals of a Kite are Perpendicular

Here in this lesson we will show how to prove that the diagonals of a kite are perpendicular using the kite shown below.

## A theorem we need to prove that the diagonals of a kite are perpendicular

**Converse of the perpendicular bisector theorem**

If a point is equidistant from the endpoints of a line segment, then the point is on the perpendicular bisector of the segment.

**Given:**
Kite ATBS with

AS ≅

AT and

BS ≅

BT
**Prove:**
AB ⟂

ST
Since

AS ≅

AT, A is equidistant from the endpoints S and T.

By the converse of the perpendicular bisector theorem, A lies on the perpendicular bisector of segment ST or

ST.

Since

BS ≅

BT, B is equidistant from the endpoints S and T.

By the converse of the perpendicular bisector theorem, B lies on the perpendicular bisector of segment ST or

ST.

Therefore, both point A and point B lie on the perpendicular bisector of segment ST or

ST.

Since there is exactly one line through any two points,

AB must be the perpendicular bisector of

ST
We can conclude that

AB ⟂

ST