The postulates, stated plainly, including the parallel postulate
What a postulate is, why some things must be agreed rather than proved, and Playfair's form of the parallel postulate.
Every lesson so far has rested on the one before it. This one is different: it states the handful of things that nothing rests on, because they cannot be proved from anything more basic.
Here is a line through A and C, and a point P that starts off it. A second line runs through P, drawn alongside. Drag P to a new spot, still off the line, or drag A or C to swing the first line to a whole new direction.
A line through A and C, and a point P off it, with the one line through P parallel to it drawn alongside. All three points can be dragged. This needs JavaScript switched on.
Try to make a third line through P that also never meets AC, running some other way — there is not one to find: any other direction through P will eventually cross AC somewhere, however far off the frame that somewhere is.
A statement agreed to be true, rather than proved from something earlier, is a postulate. The one the widget above is showing is the parallel postulate: through a point not on a line, there is exactly one line parallel to it. Two lines that run the same way forever, never meeting, are parallel.
“Exactly one” is doing two jobs in that sentence, and both matter. At least one exists: however P is dragged, a line through it running the same way as AC is there to be drawn, and the widget draws it every time, for every position tried. No more than one exists: any other line through P, in any other direction, is not parallel to AC, because two different directions through the same point are exactly the two things that cross. Put together, there is one, and there is only one — that is what “exactly one” means, and it is what the widget lets you check by dragging rather than take on trust.
This is not proved because there is nothing more basic to prove it from, and unlike an unstated assumption smuggled into a diagram, it is named here, out loud, so it can be checked against later. It is not the only such agreement: later lessons will also agree, rather than prove, that two triangles with matching sides are the same shape, and that the area of a rectangle is its base times its height. Each will be named as an agreement at the point it is needed, the same way this one is named now.
Dropping this one agreement is not a small change: whole other geometries exist where it is false, and triangles in them do not add up to 180° the way they do here. A lesson far ahead in this course (the angle sum of a triangle) is impossible without agreeing to this one first.
A line through A and C. A point P not on it. How many lines through P never meet AC?
Is that a proved fact or an agreed one?
Agreed: the parallel postulate. Nothing earlier in the course derives it.
P is dragged so that it lands exactly on the line through A and C. Is there still a unique parallel line through P to talk about?
No. The postulate is about a point not on the line; once P sits on it, “the line through P parallel to AC” is just AC itself, and the question the postulate answers has not been asked yet.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.