Question

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

Answer 1

$\implies$Qυєѕтıσи :

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

$\implies$Aиѕωєя :

Given: A quadrilateral ABCD in which opposite angles are equal i.e., ∠A = ∠C ad ∠B = ∠D

To prove: ABCD is a parallelogram i.e, AB ║ DC and AD ║ BC.

Proof: Since, the sum of the angles of quadrilateral is 3600

⇒ ∠A + ∠B + ∠C + ∠D = 3600

⇒ ∠A + ∠D + ∠A + ∠D = 360

[∠A = ∠C and ∠B = ∠D]

⇒ 2∠A = 2∠D = 3600

⇒ ∠A + ∠D = 1800 [Co-interior angle]

⇒ AB ║ DC

∠A + ∠B + ∠C + ∠D = 3600

⇒ ∠A + ∠B + ∠A + ∠B = 3600 [∠A = ∠C and ∠B = ∠D] ⇒ 2∠A + 2∠B = 3600

⇒ ∠A + ∠B = 1800

[∵This is sum of interior angles on the same side of transversal AB]

∴ AD ║ BC So, AB ║ DC and AD ║ BC

$\implies$ABCD is a parallelogram.

Answer 2

$\huge \sf {\orange{\underline{\purple{\underline{Question :-}}}}}$

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

$\huge \sf {\orange {\underline {\pink{\underline{Answer :-}}}}}$

Given: A quadrilateral ABCD in which opposite angles are equal i.e., ∠A = ∠C ad ∠B = ∠D

To prove: ABCD is a parallelogram i.e, AB ║ DC and AD ║ BC.

Proof: Since, the sum of the angles of quadrilateral is 3600

⇒ ∠A + ∠B + ∠C + ∠D = 3600

⇒ ∠A + ∠D + ∠A + ∠D = 360

[∠A = ∠C and ∠B = ∠D]

⇒ 2∠A = 2∠D = 3600

⇒ ∠A + ∠D = 1800 [Co-interior angle]

⇒ AB ║ DC

∠A + ∠B + ∠C + ∠D = 3600

⇒ ∠A + ∠B + ∠A + ∠B = 3600 [∠A = ∠C and ∠B = ∠D] ⇒ 2∠A + 2∠B = 3600

⇒ ∠A + ∠B = 1800

[∵This is sum of interior angles on the same side of transversal AB]

∴ AD ║ BC So, AB ║ DC and AD ║ BC

ABCD is a parallelogram.

${\huge{\underline{\small{\mathbb{\pink{HOPE\:HELPS\:UH :)}}}}}}$

$\red{\tt{sωєєтcнαям♡~}}$