There are, apparently, two convex bicyclic pentagons with rational-degree angles. One is the regular pentagon. Thrle other is given here, with angles measuring 140°, 120°, 80°, 80°, 20° in rotational order. If the long side, opposite the 140°angpe, is the unit length, then the other sides are the positive roots of

$3x^3-3x+1=0,$

and the two longest diagonals (connecting each 80°angle with the opposing 120° one) measures the negative root except for sign.