Parallelograms


   
 
Theorem 5
 
Statement
 
The diagonals of a rhombus are perpendicular to each other.
 
 
Given:
 
ABCD is a rhombus. Diagonal AC and BD intersect at O.
 
To prove:
 
AC and BD bisect each other at right angles.
 
Proof:
 
 
A rhombus is a parallelogram such that
 
AB=DC=AD=BC ---(i)
 
Also the diagonals of a parallelogram bisect each other.
 
Hence BO=DO and AO=OC ---(ii)
 
Now compare triangles AOB and AOD,
 
AB=AD (from (i) above)
 
BO=DO (from (ii) above)
 
AO=AO (common side)
 
(SSS congruency condition)
 
(corresponding parts of corresponding parts)
 
BD is a straight line segment.
 
 
 
 
i.e., the diagonals bisect at right angles.
 
Hence the theorem is proved.
 
 
     
   
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