Diagonal AC of a parallelogram ABCD bisects (See figure). Show that
(i) it bisects also,
(ii) ABCD is a rhombus.
It is given that ABCD is a paralleogram and (1)
and (Opposite sides of parallelogram are parallel)
Therefore, (Alternate Interior Angles) (2)
And, (Alternate Interior Angles) (3)
From (1), (2) and (3) we can say that diagonal AC also bisects .
(Proved above in solution (i))
Therefore, by ASA congruence rule,
(Corresponding parts of congruent triangles are equal) (4)
We already have AB=CD and BC=AD because ABCD is a parallelogram. (5)
From (4) and (5), we can say that ABCD is a parallelogram having all the sides equal. Therefore, ABCD is a rhombus.