# Properties of a Parallelogram - Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.

#### theorem

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

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### Shaalaa.com

Theorem : If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram. [00:13:07]
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