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In this figure, ABED is a cyclic quad. therefore  \angle A =\angle BED ( as opposite angles of a cyclic quad. are supplementary) ...
BCFE is also a cyclic Quad. so\angle C = \angle BEF for the same reason!
when we add these angles ...
\angle A + \angle BED +\angle C + \angle BEF=  360 degree ..1
angle bed + angle bed = 190 degree as they form a linear pair..
so equation 1 becomes
\angle A + \angle C=180 degree

 but they are co interior angles and if co interior angles sum up to 180 degrees they are parallel... so ab is parallel to cf!
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