I was reading Euclid’s propositions and proofs of the Linear Pair Theorem and converse, and the Vertical Angles Theorem. It finally occurred to me how I should prove the vertical Angle theorem through superposition. In the past I always proved it in the way Euclid did, while informally pointing out that vertical angles are reflections of each other. In the summer I came up with all of the proofs listed in the geometry section of the common core standards. I typed all of my proofs. Some I thought could improve when I got around to going through it and making a second draft. However, I didn't completely make the move to talking about rigid transformations in my proofs. So, I put it on the shelf.
By definition, vertical angles are formed by the intersection of two lines.
So, angle AEB is a Straight Angle Measuring 180 degrees.
Similarly, angle CED is a straight angle measuring 180 degrees.
Therefore, if angle AEC is rotated 180 degrees, then ray EA would be superimposed on top of ray EB since AEB is a straight angle. Ray EC would be superimposed upon ray ED since angle CED is a straight angle.
Thus, by definition angle AEC is congruent to angle DEB since they can be superimposed to coincide through a rigid tranformation.
