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.
Axiom 1. Two points determine a line.
Axiom 2. All straight angles are equal and the sum of straight angles is π /2 radians.
Giving a situation where two lines intersect at a particular point P, we will show that vertical angle α is equal to vertical angle β.
Using the axioms, we see that the straight angle created by α and ρ is equal to π radians, while, similarly, the straight angle created by β and ρ also equals π radians. Setting these two equations equal to each other and solving we come to the conclusion that vertical angles are congruent.
α + ρ = π and β + ρ = π,
α + ρ = β + ρ,
α = β.
Theorem 2: Parallel lines never meet.
We define line L and L’ to be parallel if, giving a line T transverse to both L and L’, then the corresponding angles between T and L and between T and L’ are equal. Transverse line T intersects line L at exactly one point and intersects line L’ at exactly one point. This means that, in 2-dimensions, line L is a shifted version of line L’. We will prove that parallel lines never meet by introducing the parallel line axiom and proving an additional theorem.
Axiom 3. Giving a point P not on a line L, then there is exactly one line through point P parallel to line L.
Theorem 3: The sum of the interior angles of a triangle is equal to a straight angle.
Giving a set of parallel lines labeled L and L’, we can form a triangle by connecting a point on L’ with two additional points on L. Let us label the interior angles as α, β, and ϕ.
Since we have two points, axiom 1 allows us to expand the sides of the triangle out into lines. In doing so, we have created three new angles, α’, β’, and ϕ’.
From figure three we see that α’, β’, and ϕ’ make up a straight angle. We note that, by theorem 1, ϕ and ϕ’ are vertical angles and thus, are congruent.
α’ + β’ + ϕ’= π,
ϕ’ = ϕ,
Since we have two parallel lines, by axiom 3 we conclude that angles α and α’ are equal and angles β and β’ are equal.
β = β’,
α = α’,
Since angles α, β, and ϕ are equal to angles α’, β’, and ϕ’ respectively, we conclude that α, β, and ϕ sum to a straight angle. Since angles α, β, and ϕ are the interior angles of the triangle, we have shown that the interior angles of a triangle sum up to a straight angle.
Now, using theorem 3, we seek to prove theorem 2, that parallel lines never meet. We will prove this theorem by contradiction. We assume that two parallel lines, L and L’, meet at a point.
We now add a transverse line to figure 4 which connects through a point on line L and L’. We have now formed a triangle between the two parallel lines which angles α, β, and ϕ.
EDIT: Not really sure what happened to the labels here on figure 5. They are not working on my MS Word. The lines L and L’ are the same, there is now a transverse line creating the necessary angles.
Since we has assumed that the two lines are parallel, the angle created by the transverse line represented by α’ and ϕ is a straight angle.
α’ + ϕ = π,
By definition of parallel lines, we note that angle α and α’ are equal.
α’ = α,
α + ϕ = π,
By theorem 3, we conclude that angles α, β, and ϕ must equal a straight angle. However, since angles α and ϕ sum up to a straight angle.
α + ϕ + β ≠ π.
Therefore, we have a contradiction and have shown that parallel lines can never meet.
In our analysis here, we have proven three theorems and have shown that parallel lines can never meet. In proving theorem 1, we used the fact that all straight angles are equal to show that vertical angles are congruent. We proved theorem 3 by using the parallel line axiom to show that the interior angles of a triangle sum up to a straight angle. Using theorem 3, we were able to come to a contradiction and show that parallel lines never meet.