Alternative answer:
Interior angle of a regular polygon = 180° × (n - 2)/n, where n is the number of sides/interior angles.
Billy could have already known that a closest possible value to that measurement is the interior angle of a regular 13-gon:
180° × (13 - 2)/13 = 180° × 11/13 = 152 4/13 ° or ≈ 152.31°
The next closest possible value to that measurement is the interior angle of a regular 14-gon :
180° × (14 - 2)/14 = 180° × 12/14 = 154 2/7 ° or ≈ 154.29°
Since 152.5° lies between these two values and it is not possible to have a regular polygon between 13 and 14 sides, the measurement must be wrong.