Question 6543
 Here I you the sketch of the proof:

 Show that quadrilateral ABCD can be inscribed in a circle if and only if angle B and angle D are supplementary.

 --> If ABCD can be inscribed in a circle , then
    angle ABC = 1/2 arc BDC and angle ADC = 1/2 arc ABC.
    But, arc BDC + arc ABC = whole circle = 360 deg
    Hence,  angle B + angle D = 180 deg

 <-- Draw a circle pass through A,B & C.
     If point D is lying inside the circle,then 
    angle D + angle B  > 180 (why?, hint: by extension line AD to
   intersect the circle at point E)
   If point D is lying outside the circle,then 
    angle D + angle B  < 180 (why?)
  Hence, we conclude that D must be lying on the circle.

 Kenny