I am taking a math writing course and our first assignment was to create an essay from lecture notes. This is what I came up with.
Also, a nice read on writing mathematics is found at this pdf: http://homepages.math.uic.edu/~kauffman/SuGuidelines.pdf
Elementary Proofs in Geometry Using Features of Pi
Pi is an extraordinarily beautiful number in mathematics. One of pi’s most amusing features is that the measurement of a straight angle is pi radians. Similarly, the sum of the interior angles of a triangle is also equal to pi radians. We seek to prove these facts and use them to determine if parallel lines can ever meet.
First, we will prove that all vertical angles are congruent. After proving this theorem, we will show that parallel lines never meet. To prove this theorem, we will give a definition of what it means for two lines to be parallel and then use the theorem that the sum of the interior angles of a triangle is equal to a straight angle to prove it.
Theorem 1: Vertical angles are congruent.
To prove that vertical angles are congruent we will utilize two axioms.