Mathematical theorems fall into two distinct categories that serve fundamentally different purposes, yet this distinction is almost never explicitly taught. The first kind verifies that a mathematical model is internally consistent and captures the properties we intend — proving things we already know (like vertical angles being equal) to confirm the model isn't broken. The second kind uses the validated model to discover genuinely new, non-obvious results (like the Cauchy-Schwarz inequality or the equal-area parallelogram theorem). This pattern appears throughout mathematics education — Euclidean geometry, Peano arithmetic, Dedekind cuts, Rudin's analysis — but instructors almost never announce the shift between the two modes, leaving students perpetually confused about why they're proving obvious things.
Questions this post answers
Why does Euclid prove theorems that seem completely obvious, like vertical angles being equal?
Euclid's early obvious-seeming theorems serve a model-verification purpose, not a discovery purpose. The goal is not to learn that vertical angles are equal — everyone already knew that — but to confirm that the axiomatic system captures the intended properties correctly and is not broken. Once the model is validated, later theorems (like equal-area parallelograms between the same parallels) reveal genuinely non-obvious facts. Developers and learners exploring formal mathematics track discussions like this on daily.dev.