Label the interior intersection on line FJ as N.
Triangles NMF, MDF, JFI, and FGI are all congruent.
Angles DFM, FMN, FIG, and JFI are all equal.
Angle DFN is a right angle since the grid consists of perpendicular lines.
Angle DFN = angle DFM + angle MFN. Since angle DFM = angle JFI,
angle MFI is also a right angle.
Triangle MFI is a right triangle and sides MF and FI are equal and angles FMI and FIM are also equal.
Triangle MFI is similar to the triangle in the original drawing that included angle C,
so angle FMI equals angle C.
Angle FMI equals angle FMN + angle HMI.
Since angle FMN equals angle B and angle HMI = angle A,
angle A + angle B = angle C.