Test 3 Section 5 - #15 (page 530)

This one's gonna get a little messy.  Here's our starting point:

Note that I've just gone ahead and put 40° in for x and 30° in for y.  If we just work in the order they tell us things, we're going to scribble all over our diagram before we can fill in any actual NUMBERS for our angle measures.  Instead, let's start with the last thing they tell us, since that'll allow us to start putting numbers in right away.  If OB bisects angle AOD, then:

Next, we can use the first thing they told us: OD bisects angle AOF.  That means angle DOF is going to be 80 as well:

And now we're almost there. Since y was 30°, we know angle DOE is 50°, so fill that in.  At this point, we kinda have everything we need.  The question asked us for angle BOE, which has to be the sum of DOE (which we just figured out is 50°) and BOD (which we already know is 40°):

In this last diagram, I've also gone ahead and filled in all the other angles just for practice (based on what they tell us about OC bisecting AOE, which we actually don't need).  Angle BOE is 90°.