Triangular Numbers and Complete Graphs

I had some extra time during my mathematical reasoning class this semester, so I gave my classes a brief lecture on the relationship between triangular numbers and properties of complete graphs. Students had seen both of these concepts, and some related ones, throughout the semester but not in this more integrated fashion. I also felt it would be helpful review for the final exam since I could touch on a bunch of things we had covered throughout the semester. With that in mind, I decided to turn the exposition into a blogpost.