Question

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

Answer 1

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   Similarly,  ∠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.