Question

in a quadrilateral if each pair of opposite angles 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

$\huge{hey}$


$\huge { \mathfrak{here \: is \: answer}}$


$\bf{Given}$
ABCD is a quardilateral ..

$\sf{ \angle{1} = \angle{3}}$

$\sf{ \angle{2} = \angle{4}}$


$\bf{to \: prove}$
ABCD is a parallelogram..

$\bf{proof}$
$\sf{ \angle{1} + \angle{2} + \angle{3} + \angle{4} = 360 \degree}$

$\sf{ \angle{1} + \angle{1} + \angle{2} + \angle{2} = 360 \degree}$
(because angle 1 = angle 3 and angle 2 = angle 4)


$\sf{2 \angle{1} + 2 \angle{2} = 360 \degree}$

$\sf{2( \angle{1} + \angle{2} )= 360 \degree}$

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

$\sf{ \angle{1} + \angle{2} = 180 \degree}$
but these are co interior angles

hence

$\bf{ad \parallel \: bc}$
similarly

$\bf{ab \parallel \: cd}$


since

parallelogram have opposite sides are equal


hence .


$\bf{ABCD \: is \: a \: parallelogram}$

$\huge \boxed{hope \: it \: helps}$

$\huge{ \mathbb{THANKS}}$