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Theorem 3 - In a Parallelogram Opposite Angles are Equal

Given : A quadrilateral ABCD is a parallelogram.

To Prove :
A = C and B = D

Proof :
AB
ïï DC and AD is a transversal
...  
A + D = 180° ...... (Consecutive Interior angles)...(i)
Again, AD
ïï BC and DC is a transversal
...  
D + C = 180° ...... (Consecutive Interior angles) ...(ii)
Comparing eqns (i) and (ii), we get
                 
A + D = D + C
...                      
A = C
Similarly               
B = D
Hence A = C and B = D
 

Example :

ABCD is a parallelogram and line segments DX and BY bisect the angles D and B respectively (see Fig.). Show that DX ïï BY.

Solution

Since opposite angles are equal in a parallelogram,
...            
D = B
...            D =B
...             1 = 2 ......(i)
Now AB
ïï DC and BY is the transversal,
...            
2 = 3 ...... (Alternative Interior angles)...(ii)
Comparing equations (i) and (ii), we get
               
1 = 3
But
1 and 3 are the corresponding angles of lines DX and BY.
...             DX
ïï BY





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