Question

ABCD is a quadrilateral in which AB parallel to CD and AD=BC. Prove that angle A = angle B and angle C =
angle D
​

Answer 1

Answer:

You can see a general quadrilateral with 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 D .

Step-by-step explanation:

Answer 2

Answer:

Angle A= Angle B

Angle C=Angle D

Step-by-step explanation:

In triangle ABC and triangle ABD

AC=BD

AB=AB

BC=AD

Therefore, triangle ABC is congruent to triangle ABD

Angle A=Angle B(CPCT)

Similarly, Angle C=Angle D

Hence proved//.