Question

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

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 360 

⇒ ∠A + ∠B + ∠C + ∠D = 360  

⇒ ∠A + ∠D + ∠A + ∠D = 360.. [∠A = ∠C and ∠B = ∠D]  

⇒ 2∠A = 2∠D = 360  

⇒ ∠A + ∠D = 180 [Co-interior angle]  

⇒ AB ║ DC  

Similarly,  

∠A + ∠B + ∠C + ∠D = 360  

⇒ ∠A + ∠B + ∠A + ∠B = 360 [∠A = ∠C and ∠B = ∠D]  

⇒ 2∠A + 2∠B = 360  

⇒ ∠A + ∠B = 180 [∵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.