I think I get the picture as you've described. If I'm not mistaken the hypotenuse of the lower triangle (which forms the base of the upper one) has length sqrt(3^2+8^2)=sqrt(73) by Pythagoras. So the base of the upper triangle is sqrt(73), and the tangent of the angle between the hypotenuse and base is 3/sqrt(73)=0.3511. The angle in the upper triangle is tan^-1(0.3511)=19.35 deg. The corresponding angle in the lower triangle is tan^-1(3/8)=20.56 deg. The combined angle is 39.90. The other angle in the upper triangle would be 70.65 deg, which ties in with your calculation, and the other angle in the lower triangle is 69.44 deg. So I think we're singing from the same hymn sheet!