Answer to Question #99798 in Geometry for Claudia Mae Baron

Question

Explain further about Theorems on Kite "In a kite the perpendicular bisector of at least one diagonal is the other diagonal

Answer 1

Given:

The kite has the shape of a quadrangle, the two sides of which are equal in pairs:

i.e., AB = BC, AD = CD.

Bisector - divides an angle into 2 equal angles.

Diagonal - connects 2 opposite corners.

The bisector is perpendicular to the diagonal.

To prove:

That the bisector is ​​the diagonal.

Proof:

The fulfillment of the conditions of perpendicularity of the bisector gives the equality of two angles of adjacent triangles.

1 if AC is the diagonal

angle ABO = angle CBO

angle AOB = angle BOC

BO is a common side;

that is, based on the congruence of the triangles, triangle ABO = triangle BCO by ASA theorem.

From the congruence of triangles it follows that AO = СО

Continuing the bisector further than point O, any triangles constructed will be equal by SAS theorem:

AO = CO

OD is the common side;

angle AOD = angle COD= 90⁰, since the bisector is perpendicular.

Therefore, the bisector is a diagonal.

2 if BD is the diagonal

The reasoning will be the same.

angle BAO = angle DAO

angle AOB = angle AOD

AO is a common party;

that is, based on the congruence of the triangles, triangle ABO = triangle ADO by ASA theorem.

From the equality of triangles it follows that BO = DO

Continuing the bisector further than point O, any triangles constructed will be equal by SAS theorem:

BO = DO

OC is a common side;

angle BOC = angle COD 90⁰, since the bisector is perpendicular.

Therefore, the bisector is a diagonal.

From proofs 1, 2 and the congruence of all four triangles it follows that AB = BC = AD = CD, the kite has the shape of a rhombus.