#### theorem

**Theorem:** If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.**Proof:** In quadrilateral ABCD

∠A = ∠D

`=>` ∠A +∠B +∠C + ∠D

Now, ∠A +∠B +∠C + ∠D = 360° (angle sum property of quadrilateral)

`=>` 2(∠A +∠B ) = 360°

`=> ` ∠A +∠B = 180°

`therefore` ∠A +∠B = ∠C + ∠D = 180°

Line AB intersects AD and BC at A and B respectively.

Such that ∠A +∠B = 180°

`therefore` AD || BC (Sum of consecutive interior angle is 180° ) ...(1)

∠A +∠B = 180°

∠A +∠D = 180° (∠B= ∠D)

`therefore` AB || CD ..(2)

From (1) and (2), we get

AB || CD and AD || BC

`therefore` ABCD is a parallelogram.

#### Shaalaa.com | Theorem : If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.

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