1.

If ABCD is a quadrilateral in which AB is parallel to CD and AD=BC , prove that angle A = angle B

Answer»

AB || DC and AD = BC.


Extend the sides AD and BC till E and F as SHOWN.  


As AB and CD are two parallel lines and AD intersects them both, the angles D and EAB are same.  But angle EAB = 180 - A.  so,

             angle D = 180 - A


Similarly, the line BC intersects parallel lines AB and DC, so

           angle C = angle FBA = 180 - B 


Now, draw perpendiculars from D and C onto AB meeting AB at G and H respectively.  

     Since AB || DC, the sides DG || CH.  

               Also,  DG = CH = distance between the parallel lines.


Looking at the triangles DGA and CHB, we find that

 

            DG = CH,    AD = BC (GIVEN),  angle G = angle H  = 90°.

  

  Δ DGA and Δ CHB are congruent.  Hence angle A = angle B.

 

     So, angle C = 180 - angle A = 180 - angle B = angle

         

HOPE it helps


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