# ABCD is a quadrilateral in which AB||DC and AD=BC. Prove that angle A = angle B and angle C=angle D

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by shanmu54321
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question is good. nice.

2014-10-16T19:46:41+05:30

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See diagram. 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

I hope that is good enough reasoning and proof.