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Converse of Theorem 3


"If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram." 


Given: A quadrilateral ABCD in which A = C and D = B.
To Prove: ABCD is a parallelogram.


Proof :
In quadrilateral ABCD, we have

A = C......(i)
B = D......(ii)
Adding (i) and (ii), we get

A + B = C + D.....(iii)
We know that, sum of four angles of a quadrilateral is 360°
A + B + C + D = 360°.....(iv)
...         A + B + A + B = 360°
...                        A + B = 180°
ïï BC......(Consecutive Interior angles)
Similarly from (i), (ii), (iii) and (iv) we can prove that
A + D = 180°
...                                AB
ïï DC
Quadrilateral ABCD is a parallelogram (opposite sides are parallel).

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