The line $y = 2x + c$ is tangent to the parabola $y^2 = 8x.$  Find $c.$
Rearranging $y = 2x + c$ gives $2x = y - c.$  Substituting into $y^2 = 8x,$ we get
\[y^2 = 4(y - c) = 4y - 4c,\]or $y^2 - 4y + 4c = 0.$  Since we have a tangent, this quadratic will have a double root.  In other words, its discriminant will be 0.  Hence, $(-4)^2 - 4(4c) = 16 - 16c = 0,$ which means $c = \boxed{1}.$