These are a pain to typeset. Bear with me on the notation; I'll write A for the measure of angle A and small letters de for the measure of arc DE.
Let's write the givens
A=20
de=116 (drawing apparently not to scale)
We're asked for bc
Note the measure of an arc is the measure of the angle at the center of the circle to that arc. There's a theorem that says the angle subtended by the arc will be half that. For us that means
DBE = DCE = de/2 = 116/2 = 58
Also
D = E = bc/2
ABE = ACD = 180 - DBE = 122
D = E = 180 - ABE - A = 180 - 122 - 20 = 38
bc = 2D = 2(38) = 76
We didn't seem to need the intersecting chord theorem, which tells us the average of the arcs gives the measure of the angles at F.
BFC = (bc + de)/2
