Modern Geometry. Here's the question: "Say that ABC and DBE so that angle ABD and angle CBE are vertical angles. Show that if r is a ray emanating from B such that it bisects one of angle ABD or angle CBE, then the ray opposite to r bisects the other angle."
Here's where I'm at with my proof so far: "Let A, B, C, D, and E be points and let ABC and DBE so that angle ABD and angle CBE are vertical angles. Let r be a ray emanating from B such that it bisects angle ABD. Suppose there is a point X on r such that r = ray BX. By Proposition 3.15, angle ABD is congruent to angle CBE since they are vertical angles. By definition of angle bisector and Proposition 4.4.a., we now have angle ABX and angle XBD such that angle ABX is congruent to angle XBD."
I'm confused on where to go next from here. The drawing I made seems apparent enough that the opposite ray bisects angle CBE, but how do I prove that? I must use the propositions and theorems that have been presented so far in this class.