Parallelograms


   
 
Converse of Theorem 6
 
Statement
 
If in a parallelogram, the diagonals are equal and perpendicular, then it is a square.
 
 
Given:
 
ABCD is a parallelogram.
 
AC=BD and AC ^ BD
 
To prove:
 
ABCD is a square.
 
Proof:
 
 
Since the diagonals AC and BD are equal,
 
ABCD is a rectangle - - -(i)
 
(Diagonal property of rectangle)
 
Since the diagonals are perpendicular to each other.
 
ABCD is a rhombus.
 
AB=AD - - -(ii)
 
ABCD is a rectangle. (from i)
 
With consecutive sides equal. (from ii)
 
ABCD is a square. (by definition of a square)
 
 
Hence the theorem is proved.
 
 
     
   
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