As seen from the carpenter's example at the beginning of this lesson, diagonals play an important role in identifying special quadrilaterals.  The theorems about the diagonals of special quadrilaterals are easily proved by identifying congruent triangles formed.  Rather than go through all these theorems and their proofs, we will list the main results in a table:

The X's in this table specify that the diagonals of the given quadrilateral have the indicated properties, and conversely if the diagonals have the indicated properties then the quadrilateral is of the specified type.  For example, if the diagonals of a quadrilateral are congruent and bisect each other, then that quadrilateral is a rectangle.  The "special kite" is a kite in which the diagonals are congruent: