The converse: equal alternate angles make lines parallel

Turning the last three lessons around — from parallel lines predicting angles, to matching angles guaranteeing parallel lines.

The last three lessons all started from the same place: two lines already agreed to be parallel, and a transversal crossing both. This one starts somewhere different — with an angle that matches — and asks whether that is enough, by itself, to force the lines to be parallel.

A line through A and C, and a transversal through E and F crossing it at G. At F, two candidate second lines: one built to always stay parallel to AC, whichever way A, C, E or F is dragged; the other just an ordinary line aimed at a point Q that tracks nothing at all.

At the start, both candidate lines read the same angle as G, and both look exactly as parallel as each other. The guessed line falls behind because nothing about it was ever built to track the base line — it was only placed to resemble it once, at one moment. Catching it up with Q only holds until the next drag.

Naming the reversed claim

A statement built by swapping a claim’s assumption and conclusion is its converse. The last lesson’s claim was “parallel lines make alternate angles equal”; this lesson’s claim is its converse: “equal alternate angles make lines parallel.” A claim being true never guarantees its converse is too — this lesson is about checking, not assuming.

Telling the two lines apart

The constructed line through F is parallelLine('F', lineAC): built to share line AC’s direction, whatever that direction currently is. Because it shares that direction, it reads exactly the angle the last three lessons already showed a parallel line must read: the same as the angle at G. That is the previous claim, working in its original direction, and it is not what is being tested here.

What is being tested is the guessed line: a plain line through F aimed at a point Q, with nothing connecting Q to A, C, E or F at all. When it happens to read the same angle as G, it looks exactly like the constructed line, because both are lines through F reading one particular number. But only one of them is forced to keep doing that as the figure changes. Dragging the base line is what tells them apart: the constructed line’s reading moves with it, by definition; the guessed line’s reading stays fixed, because a fixed point Q gives a fixed direction, regardless of what the rest of the figure is doing.

This is the difference between a fact that happens to hold once and a fact that is guaranteed to hold. A single matching angle, seen once, is exactly as much evidence for “these lines are parallel” as a single well-placed-looking point was evidence for “B is between A and C”, back when this course’s very first lesson insisted that was arithmetic, not appearance.

A guessed line through F reads the same angle as G. Is it necessarily parallel to line AC?

Not necessarily. It is parallel only if that match is guaranteed to survive the base line moving, which a single reading can never show by itself.

What would make you confident the guessed line actually is parallel, rather than just matching by coincidence?

Dragging A or C to rotate the base line, and watching whether the guessed line’s reading moves to follow it. If it does not, and the two readings stop matching, the guess was never parallel to begin with, only aimed at the right place once.

The constructed line through F is built with parallelLine. Why does it never need this check?

Because “parallel to AC” is what parallelLine('F', lineAC) is defined to mean, not a coincidence noticed after the fact. Its angle with the transversal matching the angle at G is a consequence of that definition, proved in the last lesson, not a claim still waiting to be tested.

Do the exercise

This lesson counts as done once you get its exercise right. Nothing to tick off by hand.