Question

in a quadrilateral if each pair of opposite angle is equal then it is a parallelogram

Answer 1

Hey there !!

➡ Given:-

→ ABCD is a quadrilateral.

→ $\angle 1 = \angle 3$ and $\angle 2 = \angle 4 .$

➡ To Prove :-

→ ABCD is a parallelogram.

➡ Proof :-

→ ABCD is a quadrilateral.

Then,

=> $\angle 1 + \angle 2 + \angle 3 + \angle 4 = 360 \degree .$

=> $\angle 1 + \angle 1 + \angle 2 + \angle 2 = 360 \degree .$

[ → $\angle 1 = \angle 3$ and $\angle 2 = \angle 4 .$

=> $2 \angle 1 + 2 \angle 2 = 360 \degree .$

=> $2 ( \angle 1 + \angle 2 ) = 360 \degree .$

=> $\angle 1 + \angle 2 = \frac{ 360 \degree }{2} .$

=> $\angle 1 + \angle 2 = 180 \degree .$

▶ But, these are the pairs of co-interior angles.

→ So, AD || BC.

▶Now, Again

=> $\angle 1 + \angle 2 + \angle 3 + \angle 4 = 360 \degree .$

=> $\angle 1 + \angle 1 + \angle 4 + \angle 4 = 360 \degree .$

[ → $\angle 1 = \angle 3$ and $\angle 2 = \angle 4 .$

=> $2 \angle 1 + 2 \angle 4 = 360 \degree .$

=> $2 ( \angle 1 + \angle 4 ) = 360 \degree .$

=> $\angle 1 + \angle 4 = \frac{ 360 \degree }{2} .$

=> $\angle 1 + \angle 4 = 180 \degree .$

▶ But, these are the pairs of co-interior angles.

→ So, AB || CD.

▶ Now, we have AB || CD and AD || BC.......(1).

And, in parallelogram opposite pairs of sides are parallel..........(2).

From equation (1) and (2), we get

=> ABCD is a parallelogram.

✔✔ Hence, it is proved ✅✅.

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THANKS

#BeBrainly.

Answer 2

Given : ABCD is a quadrilateral
Angle A = Angle C
angle B = angle D

To prove : ABCD is a parallelogram

Proof :

In quadrilateral ABCD ,

angle A + Angle B + Angle C + Angle D = 360°

angle A + Angle C + Angle B + Angle D = 360°

Angle A + Angle A + Angle D + Angle D = 360°

2(Angle A + Angle D ) = 360°

Dividing both sides by 2 ,

Angle A + Angle D = 180° .

And it is a pair of interior angles on same side of a transversal also .

So AB//DC .

Similarly we can prove Angle C + Angle B = 180° .

And the result will be AD//BC .

Both pairs of opposite sides are equal. So it is a parallelogram.

#Hence proved .