Question 568387
I don't think that statement is true. If one pair of opposite angles in a cyclic quadrilateral is congruent, then both angles must be 90 (since opposite angles in a cyclic quadrilateral add up to 180, 180/2 = 90). It is definitely possible to construct a cyclic quadrilateral satisfying that property without any parallel sides.


{{{drawing(300,300,0,10,0,10,


line(2,2,6,2),
line(2,2,2,8),
line(6,2,7,5),
line(7,5,2,8),
locate(2,2,A),
locate(6,2,B),
locate(7,5,C),
locate(2,8,D)
)
}}}


Not completely drawn well, but you get the picture. If angles A and C are congruent in cyclic quadrilateral ABCD, then A and C are right angles. There are no parallel sides.