Parallelograms


   
 
Converse of Theorem 4
 
Now consider the converse of this theorem.
 
Statement
 
If two diagonals of a parallelogram are equal, it is a rectangle.
 
 
Given:
 
ABCD is a parallelogram in which AC=BD.
 
To prove:
 
Parallelogram ABCD is a rectangle.
 
Proof:
 
 
In triangles ABC and DBC,
 
 
AB=DC (opposite sides of parallelogram)
 
BC=BC (common side)
 
AC=BD (given)
 
 
(corresponding parts of
 
corresponding triangles)
 
But these angles are consecutive interior angles on the same side of transversal BC and AB||DC.
 
 
 
 
By definition of rectangle, parallelogram ABCD is a rectangle.
 
 
Hence the theorem is proved.
 
 
     
   
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