Why do diagonals of a kite cross at right angles?

Answer 1

The short answer is: because that is what a kite is.

Long answer (although some minor steps have been skipped):

Consider a kite, ABCD, where AB = DA and BC = CD

Draw the diagonal AC.

Then triangles ABC and ACD are congruent (SSS).

So angle BAC = angle DAC.

Now draw diagonal BD which intersects AC at E.

Then triangles ABE and ADE are congruent (SAS)

So angle AEB = angle AED.

But these angles are supplementary. Therefore they must be right angles.