# Converse of Theorem 3

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

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**Given:**A quadrilateral ABCD in which âˆ A = âˆ CÂ and âˆ D = âˆ B.

**To Prove:**ABCD is a parallelogram.

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**Proof** **:**

In quadrilateral ABCD, we have

âˆ A = âˆ C......(i)

âˆ B = âˆ D......(ii)

Adding (i) and (ii), we get

âˆ A + âˆ B = âˆ C + âˆ D.....(iii)

We know that, sum of four angles of a quadrilateral is 360Â°

.^{.}.Â Â Â Â Â Â Â âˆ A + âˆ B + âˆ C + âˆ D = 360Â°.....(iv)

.^{.}.Â Â Â Â Â Â Â Â âˆ A + âˆ B +Â âˆ A + âˆ B = 360Â°

.^{.}.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â âˆ A + âˆ B = 180Â°

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â AD Ã¯Ã¯ BC......(Consecutive Interior angles)

Similarly from (i), (ii), (iii) and (iv) we can prove that

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â âˆ A + âˆ D = 180Â°

.^{.}.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â AB Ã¯Ã¯ DC

Quadrilateral ABCD is a parallelogram (opposite sides are parallel).