Assume that X is a point between A and C, that X is also between B and D, and that these points are not all collinear. Then the angles AXB and CXD are called vertical angles. Prove that vertical angles are congruent.
I'm not sure how to do this without using angle measure, but since I am in Euclidean Geometry we can only use the Axioms we have so far and previous problems. We only have SSS and SAS and from these axioms we have proven how to construct right angles, perpendicular angles, midpoints, and angle bisectors.
I'm really confused on where to start with this so any help would be appreciated.
That's Euclid book 1 prop 15.
You have $\angle AXC + \angle CXB =\angle CXB + \angle BXD $ since both sum to a straight angle.
Therefore $\angle AXC =\angle BXD $.