By the law of sines, m∠EFG is such that
sin(m∠EFG) / (8 in.) = sin(m∠G) / (7.5 in)
so you need to find m∠G.
The interior angles to any triangle sum to 180°, so
m∠DEG = m∠D + m∠G + 43°
m∠DEG + m∠D + m∠G = 2 (m∠D + m∠G) + 43°
180° = 2 (m∠D + m∠G) + 43°
137° = 2 (m∠D + m∠G)
68.5° = m∠D + m∠G
But ∆DEG is isosceles, so m∠D = m∠G, which means
68.5° = 2 m∠G
34.25° = m∠G
Then
sin(m∠EFG) = (8 in.) sin(34.25°) / (7.5 in)
m∠EFG ≈ sin⁻¹(0.600325) ≈ 36.8932°
