The trapezium: what a single pair of parallel sides forces, and what "isosceles" adds

0:00Start with a parallelogram, and then give something up. Both pairs of facing sides run the same way. Take one of those pairs and let it go. The two sides that were parallel swing loose. They can now be any lengths at all, leaning any way they like. One pair is still parallel. That is the only thing left standing. It is the least parallelism a four-sided figure can be asked for and still be given a name of its own.

0:31The name is trapezium. And the whole of this video is one question: what does that single surviving pair still buy you? Here is the definition, and it wants reading slowly. A trapezium is a quadrilateral with at least one pair of parallel sides. At least one. Not exactly one. Those two small words are carrying real weight, and it is very easy to read straight past them. With them, a figure that happens to have two parallel pairs is still a trapezium.

1:08Without them, every parallelogram is thrown out of the family immediately. One word decides whether the family holds all of them or none of them, and that is worth knowing about a definition. So take a mixed collection of 40 figures and put the question to each one. All 40 are trapeziums. Every single one has at least one parallel pair. 24 of them have both pairs parallel, so they are parallelograms as well.

1:4216 of those have four square corners, which makes them rectangles. And 8 of the rectangles have four equal sides on top of that. Squares. Read backwards the chain fails at every step: there are rectangles that are not squares, parallelograms that are not rectangles, trapeziums that are not parallelograms. A kite has no parallel pair at all, so it sits outside the whole picture whichever way you read the definition.

2:15Now for what the one surviving pair actually buys. Here is the figure we will work on: a long side of 20 and a short one of 6, on two lines 12 apart, with the short one slid 5 along. Its two slanting sides come out 13 and 15. Nothing alike, and nothing was assumed about them. Look at one slanting side on its own. It crosses both of the parallel sides.

2:45So it is a transversal, and the two corner angles at its two ends lie between the parallel lines on the same side of it. Those two must total a straight angle. Here, 67.4 and 112.6. The other slanting side says the same thing about its own two ends: 53.1 and 126.9. Two pairs, each totalling 180. That sounds like plenty. But try the other two pairs, the ones at the ends of a parallel side.

3:2367.4 and 53.1. They come to 120.5, and nothing whatever forces that number. Not one trapezium in a sweep of 63, nor in a wider sweep of 171, has its parallel ends totalling a straight angle. Which means the two supplementary pairs never talk to each other. Tell me one angle of a parallelogram and I will hand you the other three. Tell me one angle of a trapezium and I cannot.

3:59There are figures in the sweep sharing that one angle and disagreeing about everything else, so being told it settled nothing. A trapezium therefore has to hand you two angles, one from each pair. Say the two at the ends of the short parallel side are 135 and 105. Each of them sits at one end of a slanting side, so each has a partner at the other end. Follow the first slanting side down. 180 take away 135 is 45.

4:37Follow the second. 180 take away 105 is 75. 45, 75, 135 and 105, and they total 360, which is the check you should always run. Build a figure to those two given angles and it measures exactly those four. The arithmetic and the drawing agree. Now put one thing back, and watch a symmetry come home. Make the two slanting sides the same length as each other. This one has a long side of 18, a short side of 8, the same 12 between the lines, and both slants 13.

5:22A trapezium whose two non-parallel sides are equal is called isosceles. Notice that is read off the finished figure. Nothing about how it was drawn is being used. And its base angles come out equal. 67.4 at both ends of the long side. That is the claim. Now the reason, and it is a nice one. From each end of the short parallel side, drop a perpendicular straight down onto the long one.

5:55Two feet land on the long side, and the figure falls into three pieces: a triangle, then something, then a triangle. Look at the middle piece. Two of its angles are right because that is exactly how the two segments were drawn. And the other two are right as well, because the short side runs parallel to the long one, so a perpendicular to one is a perpendicular to the other.

6:23Four right angles. It is a rectangle, and that was argued rather than assumed. Here it measures 8 by 12. Which hands us something for free: its two upright sides are equal, because facing sides of a rectangle always are. Now the two triangles at the ends. Each has a right angle where its perpendicular meets the long side. Their slanting sides are both 13, because that is what we assumed when we said isosceles.

6:57And their upright sides are both 12, because those are the rectangle's two facing sides. Two triangles, 5, 12 and 13 apiece. The same triangle drawn twice. So the angle at one end of the long side is the angle at the other, and there is the equal pair. 67.4 and 67.4, and every step of that used the equal slants exactly once. It is worth seeing which step needed what, because half of it never needed the assumption at all.

7:35Go back to the first figure, the one whose slants are 13 and 15. Drop the same two perpendiculars. The middle piece is still a rectangle, 6 by 12. Nothing in that step ever mentioned the slanting sides, so it could not have failed. The triangles are where it breaks. One is 5, 12, 13. The other is 9, 12, 15. Different triangles, so different angles at the two ends. 67.4 and 53.1.

8:09Half the argument works on every trapezium. Only the second half was bought, and the equal slanting sides are what bought it. And there is a trap sitting in the wording itself. Take a parallelogram, and look at the two sides that are not the pair you called parallel. They are equal to each other. Word for word, that is the condition the isosceles trapezium asks for. But its base angles are not equal at all. They are 51.3 and 128.7.

8:46They total a straight angle instead, because in a parallelogram the two slanting sides lean the same way rather than opposite ways. So equal slanting sides only mean anything once there is exactly one parallel pair for them to be the other two of. And an isosceles trapezium is never a parallelogram. Not one of 27 in a sweep, and not one of 60 in a wider one. Last, the picture that holds the whole family at once.

9:21One big region, and it is the trapeziums. Every parallelogram sits inside it. Inside those, the rectangles. Inside those, the squares. Each ring is one more condition, and each one is paid for rather than free. The kites reach outside the whole picture, because a kite is promised no parallel sides whatever. And the isosceles trapeziums are a region of their own that overlaps the parallelograms in nothing at all.

9:54Which is the real lesson of the smallest assumption: how much you have to be told is exactly how much you did not assume.