Thread: Finding the Angle in this Figure (see the figure)

I am trying to figure out the measure of the angle labeled "?" in the figure. All the labeled variables besides "?" are known. I have been staring at this for the past 2 hours and haven't found anything leading to a solution. As this figure is derived from a separate problem (approximation of the geometric apparent radius of Earth's penumbra at a given distance -- ignore if this makes things confusing), I may have forgotten a known value somewhere though it seems like there is only one unique figure for a given set of variables. Any ideas?

a, b and "?" are angles
r and d are distances

Thanks.

I actually just figured it out. The right triangle containing angle a was not needed. That left a trapezoid remaining. Creating a right triangle from the top of the trapezoid and using the tangent function allowed me to find the longer height of the trapezoid. Then, it just took a simple tangent inverse to find the result.

There, ? = atan(tan(90 - b) + r/d) where tan() uses degrees.