\ D BOA @ D BOC
\ Ð BOA @ Ð BOC.
Corresponding angles of congruent triangles are congruent.
But Ð BOA and Ð BOC form a linear pair. i.e. they are supplementary. Supplementary angles are congruent if and only if they are right angles. Therefore, AC is perpendicular to BD. The converse of this theorem is used as a test for rhombus.
Theorem: If the diagonals of a parallelogram are perpendicular, the parallelogram is a rhombus.
Square
A quadrilateral is called a square if all its sides are congruent and all its angles are congruent. Thus a square is a parallelogram with the properties of a rectangle as well as those of a rhombus.
Properties of a square
1) The diagonal of a square divides it into two congruent triangles.
2) The opposite sides of a square are equal.
3) The opposite angles of a square are equal.
4) The consecutive angles of a square are supplementary.
5) The diagonals of a square are equal and bisect each other at right angles.
6) The diagonals of a square bisect the opposite angles.
Example 1
If ABCD is a parallelogram and m Ð A = 60^{0}, find m Ð B, m Ð L and m Ð D.
Solution:
m ÐB = 120^{0}  The consecutive angles are supplementary.
m ÐL = 60^{0}  The opposite angles are equal.
m ÐD = 120^{0 }  The opposite angles are equal.
