Now that we have the guidelines set up we can label some angles. Let's call the angle that the seven ball could go φ, and the angle you can hit the cue ball θ. There are two of each because depending on which way you hit the ball depends on the direction the ball will go.
Because so many of the angles are parallels, we can use geometry to find the measurements of most of the angles. Like on the seven ball; we know all vertical angles must be congruent, so therefore we can place φ into the vertical angles on the seven ball.
Because interior angles are congruent, we can move θ into it's corresponding place.
