Parallel Lines

Today I worked on some exercises in Analysis with an Introduction to Proof, then read the review of analytic geometry I had queued. Near the end of the review, the author quotes the following theorems, indicating that proofs can be found in his precalculus textbook:

1. Two nonvertical lines are parallel if and only if they have the same slope.
2. Two lines with slopes $m_1$ and $m_2$ are perpendicular if and only if $m_1m_2=-1$; that is, their slopes are negative reciprocals:$$m_2=-\frac{1}{m_1}$$

That got me to thinking about how I would prove those things, and I ended up spending the rest of my study time working on them. I have a complete proof of Theorem 1. It is not that exciting, but it did give me the chance to use some of the logic that I am reviewing in Analysis with an Introduction to Proof.

Proving Theorem 2 seems like the more interesting problem. What does it mean for lines to be perpendicular? If it means that they meet at a right angle, what is that? How do you prove an angle is a right angle? My best idea so far is an argument involving right triangles, but in order for it to work, I need to establish to my satisfaction not just that the Pythagorean theorem holds for all right triangles, but also that only right triangles have sides that relate in the way described in the Pythagorean theorem. It’s pretty easy to convince oneself of the latter, given the former, and I may decide that’s enough. We’ll see.

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Anyway, here’s what I did for Theorem 1:

Proposition: Two nonvertical lines are parallel if and only if they have the same slope.

For two lines to be parallel means that either they do not intersect or they are the same line.

Thus, we need to prove two statements:

1. If two nonvertical lines do not intersect or they are the same line, then they have the same slope.
2. If two nonvertical lines have the same slope, then either they do not intersect or they are the same line.

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To prove statement 2, assume that two nonvertical lines have the same slope, $m$. Their equations can be written as $y=mx+b$ and $y=mx+c$, where $b$ and $c$ are real numbers.

Now assume that the lines intersect, and let $(x_1, y_1)$ be a point that lies on both lines. It follows that $y_1=mx_1+b$ and $y_1=mx_1+c$, and thence that

$$mx_1+b=mx_1+c$$$$b=c$$

When $b=c$, the two lines have the same equation, and are thus the same line.

Thus, when two nonvertical lines have the same slope it follows that if the two lines intersect, then they are the same line.

Hence, if two nonvertical lines have the same slope, then either they do not intersect or they are the same line, which is statement 2.

(The logic in play here is: $[p\implies (q\implies r)]\equiv[p\implies(\lnot q\lor r)]$)

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To prove statement 1 by contraposition, assume that two nonvertical lines have different slopes, $m$ and $n$. Their equations can be written as $y=mx+b$ and $y=nx+c$ where $m\neq n$ and $b$ and $c$ are real numbers.

Since their equations are different, they are not the same line.

Furthermore, consider the point $$\left(\frac{c-b}{m-n},\frac{mc-nb}{m-n}\right)$$ which must exist, since $m\neq n$.

Notice that $$m\left(\frac{c-b}{m-n}\right)+b=\frac{(mc-mb)+(mb-nb)}{m-n}=\frac{mc-nb}{m-n}$$ and $$n\left(\frac{c-b}{m-n}\right)+c=\frac{(nc-nb)+(mc-nc)}{m-n}=\frac{mc-nb}{m-n}\text{.}$$

Thus, the point lies on both lines and the lines intersect.

Therefore, if two nonvertical lines have different slopes, then the two lines intersect and they are not the same line.

By contraposition, it follows that if two nonvertical lines do not intersect or they are the same line, then they have the same slope, which is statement 1.

(The logic in play here is: $[\lnot p\implies (q\land\lnot r)]\equiv[\lnot(q\land\lnot r)\implies p]\equiv[(\lnot q\lor r)\implies p]$)

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Stay tuned for related proofs about perpendicular lines, probably tomorrow.