Equal slopes mean parallel; slopes multiplying to minus one mean perpendicular

0:00Two lines are parallel exactly when their slopes are equal. That looks like one fact. It is two. Forwards it says: if the lines are parallel, the slopes come out equal. Backwards it says: if the slopes come out equal, the lines are parallel. Those are different claims, they need different arguments, and one of them is much harder than the other. The same goes for the second criterion, which says two lines meet at a right angle exactly when their slopes multiply to minus one.

0:36Four claims altogether. The hard halves are where the content of this topic actually lives, so this video does all four. Start with the picture. Two parallel lines, both falling as you read them left to right, each crossing the horizontal axis somewhere of its own. At each crossing there is an angle, drawn the same way both times: start along the positive horizontal direction and sweep anticlockwise to the line.

1:10For a line that falls, that sweep goes past the sharp angle and lands on the blunt one. So both arcs here are blunt. It would be easy to draw the sharp angles instead. They look tidier, and they are the wrong angles. Call the two of them alpha and beta. Everything that follows is about whether those two are the same. Take the forward direction first, because it goes through without complaint.

1:41If the lines are parallel, the horizontal axis cuts them both as a transversal, so the corresponding angles at the two crossings are equal. That gives alpha equals beta. Not their tangents. The angles themselves. Now hand both angles to the tangent. Equal things given to the same machine come out equal, so the tangent of alpha is the tangent of beta. And those two tangents are the two slopes. Parallel lines have equal slopes.

2:16Notice what that never needed: nothing about the tangent except that it is a machine. Give it the same angle twice and it gives the same number twice. Now run it backwards, and watch where the difficulty is. You are given that the slopes are equal, so the tangent of alpha is the tangent of beta. You want to conclude that alpha IS beta, because from equal angles the transversal argument hands you parallel lines back.

2:49But equal answers do not in general mean equal questions. So this step needs a licence, and the licence is a property of the tangent: over the directions a line can point in, it never returns a value it has already returned. Each value at most once. That is what turns equal tangents back into equal angles, and it is the sentence that usually gets skipped. That is easy to nod at. Here is what it is worth, measured.

3:24Take eighty-eight directions: every way a line can point, built out of small whole-number steps. Ask the tangent to tell them apart. Pairs of different directions it cannot separate: none. Now swap it for something a student might reach for. Read both legs as lengths, so the slope loses its sign, and what you have is the SIZE of the tangent. Pairs of different directions that reading cannot separate: forty-three.

3:56Try the sine of the inclination instead: the same number, forty-three. And the pairs are not scattered. Of the forty-three each of them confuses, the number that are anything other than a direction and its supplement is zero. Both mistake a line that rises for the line that falls at the same steepness, which is exactly the mistake the converse rules out. There is one honest wrinkle in each value at most once, worth facing rather than stepping over.

4:32The inclination is given a range running from nothing round to a straight angle, with both ends allowed in. On that closed range the tangent is not quite one to one, because it gives nothing at the start and nothing again at the finish. Counted: eighty-nine angles across the range, and exactly one pair of different angles sharing a tangent. The two ends, sharing the value nothing. But look at what those two angles are. An angle of nothing and a straight angle both describe a line lying flat along the horizontal.

5:11So the eighty-nine angles name eighty-eight lines, one fewer, and the two that shared a tangent were never two directions to begin with. Every other value is taken once, and the right angle is taken by nothing at all: the one place the tangent refuses. So the criterion holds both ways. Here it is, run against lines rather than argued about. Sixty-four lines, four thousand and thirty-two ordered pairs, each pair looked at both ways round so nothing depends on which line you call the first.

5:51Whether two lines are parallel is settled here without any slope: they have a single meeting point or they do not. Pairs whose slopes agree but which meet in a single point: none. Pairs that never meet in a single point but whose slopes differ: none either. Now the thing it cannot do. Of the one hundred and eighty pairs the criterion calls parallel, ten are one line wearing two different names.

6:20Equal slopes buys equal direction and nothing else. Whether you have two lines is settled by a point, not by a slope. Second criterion. Two lines crossing, with a small square at the crossing to say the angle there is a right angle. You need that square, because you cannot see a right angle. Stretch the picture sideways and a right angle stops looking like one. Each line has its own inclination, measured the same way, at its own crossing with the horizontal.

6:57One line rises and its inclination is sharp. Call that one alpha. The other falls, and its inclination is blunt. Here is the relation the whole criterion comes out of: the blunt one is exactly a right angle further round the sweep than the sharp one. Beta is alpha plus a right angle. That is not read off the drawing. Turn any direction through a quarter turn and it stands at a right angle to where it started: over all eighty-eight, the number that fails is zero.

7:34Now turn that into arithmetic, and watch the minus sign arrive. The tangent of an angle increased by a right angle is minus the cotangent of that angle, and the cotangent is one over the tangent. So the tangent of beta is minus one over the tangent of alpha. Those two tangents are the two slopes. The second slope is minus the reciprocal of the first, and their product is minus one.

8:05Measured: of the eighty-six directions that have a tangent and whose quarter turn has one too, the number failing to give minus the reciprocal is zero. Now the labelling, because it quietly confuses people. Of those eighty-six directions, forty-three have their quarter turn later in the sweep and forty-three have it earlier, and the split is exactly sharp against blunt. Which is why you name the sharp one alpha. The equation is the same; you are choosing which end to write first.

8:44That was the forward half. The backward half starts from a product of minus one and has to reach a right angle. Take it as a measurement first. Seven thousand six hundred and fifty-six ordered pairs of different directions. Whether two of them stand at a right angle is decided by the directions alone, and eighty-eight pairs do: one partner apiece for the eighty-eight directions. Of the eighty-six where both slopes exist, the number whose slopes fail to multiply to minus one is zero.

9:21And the other way round: pairs NOT at a right angle whose slopes nevertheless multiply to minus one, also zero. The argument behind that second zero is the first one played backwards: a product of minus one forces the tangent of beta to be minus one over the tangent of alpha, which forces beta a quarter turn from alpha. There is a perpendicular pair the criterion cannot say anything about, and it is not an exotic one.

9:57Look hard at the equation. Two numbers multiply to minus one. Neither of them can be nothing, because nothing times anything is nothing. So the criterion has quietly told you that neither line is flat. And a flat line's partner at a right angle is an upright line, which has no slope to offer. Take the two axes. They meet at a right angle. One has slope nothing and the other has no slope at all, and there is no product to form.

10:31Measured: of the eighty-eight perpendicular pairs, two have no product available, and the number of those that are anything other than a flat line with an upright one is zero. That is not a counterexample. It is what was excluded when the criterion said, before either half, that both lines must have slopes. Time to use it, on a question you can answer without drawing anything. One line runs through minus two and six, and through four and eight. Another runs through eight and twelve, and through a point whose second coordinate is twenty-four and whose first you have to find.

11:15The two lines meet at a right angle. Find the missing coordinate. The first line's slope is one third. So the second line's slope has to be minus three, the negative reciprocal. Searching whole numbers for one that does it turns up exactly one: four. And the line through eight and twelve and through four and twenty-four does have slope minus three. Notice what was never done: the two lines were never made to cross. A slope is a property of direction, and the meeting point was never needed.

11:54Two more, and neither measures a single length. Three corners: four and four, three and five, and minus one and minus one. Show it is a right triangle. From the first corner to the second the slope is minus one. From the first corner to the third it is one. Their product is minus one. So the right angle sits at four and four, and no distance was worked out.

12:21Now four corners in order: minus two and minus one, four and nothing, three and three, minus three and two. Show it is a parallelogram. Going round, the four side slopes come out one sixth, minus three, one sixth, minus three. Pairs of opposite sides that fail to be parallel: none. Adjacent sides that are parallel, which would flatten the shape: none either. One last consequence, and it is the parallel criterion doing an unexpected job.

12:58Take three points. Measure the slope from the first to the second, and the slope from the second to the third. If those agree, the two segments are parallel, and they already share a point. Parallel plus a shared point is not two lines. It is one. So three points lie on a single line exactly when those two slopes agree. Try three and nothing, minus two and minus two, and eight and two: both slopes come to two fifths.

13:30That is the same test as asking whether the triangle on the three points has no area, and the two agree across thirteen thousand eight hundred triples with zero disagreements. One caveat. Stated as a comparison of two slopes, the test walks straight into the case where a slope is not there. It gets three hundred of those triples wrong, and the number of the three hundred that are anything other than points standing one above another is zero.

14:04So what was bought. Not two rules of thumb. Four claims, each with its own argument, and the two easy ones needed almost nothing. Both hard halves lean on one fact: over the directions a line can point in, the tangent never returns a value twice. Take that away, by reading the slope as a size or reaching for the sine, and forty-three pairs of genuinely different directions collapse into each other.

14:36The minus sign in the second criterion is not decoration and not a convention. It is what a quarter turn does to a tangent. And the restriction that both lines must have slopes is not small print: it keeps a flat line and an upright one, perpendicular and untestable, out of a rule they would break. The two criteria are one shape written twice: equal inclinations the first time, inclinations a quarter turn apart the second.

15:10Which is why they were never two separate things to remember.