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Chapter 1 · Geometric Twins

Naming congruent figures so the correspondence is unambiguous

यह वीडियो हिंदी में भी · Watch in Hindi

Congruence criteria for triangles9 min

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9 min.

Also recorded in Hindi.Englishहिन्दी

The congruence sign is not the equals sign. Equals does not care which number you write first; this one relates two ordered lists.

The idea

≅ is not =. Writing ΔABC ≅ ΔXYZ does not merely report that two triangles match; it commits to a particular pairing of the six vertices, read off position by position, and shuffling the letters on one side turns a true statement into a false one without changing the picture at all. The chapter proves this the hard way: it takes two triangles it has already shown to be congruent, writes down a plausible-looking pairing, and demonstrates that superimposing according to that pairing lays a side onto a side of a different length. So the notation is not bookkeeping — it is the claim.

What you should be able to do

  • Read a congruence statement and say which vertex is claimed to match which
  • List, from a congruence statement alone and without the figure, which vertices correspond, which sides correspond and which angles correspond
  • Rewrite a given congruence in every other correct way, and say how many there are
  • Judge whether a proposed statement expresses the correspondence correctly
  • Explain what goes wrong when the letters are shuffled on only one side
  • Use tick marks on a diagram to work out which vertex should pair with which
  • Find the correspondence in a figure where two triangles overlap or share a side

Words to know

TermDefinition in one lineFirst introduced
corresponding verticesthe pairs of corners that coincide once one triangle is laid over the otherprinted in §1.2, Part II, pp.6–7
corresponding sidesthe pairs of sides that land on each otherprinted in §1.2, Part II, pp.7, 13
corresponding anglesthe pairs of angles that land on each otherprinted in §1.2, Part II, p.7
vertexa corner of a triangleprinted in §1.2, Part II, pp.6–7, 13
≅the sign read "is congruent to", written between two triangle names in a fixed orderprinted in §1.2, Part II, p.7 onwards
conventionan agreed way of writing something, so that everyone reads it the sameprinted as the subheading "Conventions to Express Congruence", §1.2, Part II, p.6
incorrectthe book's own verdict on a congruence written with the letters shuffledprinted in bold in §1.2, Part II, p.7
common sidethe side that belongs to both triangles being comparedprinted in §1.2, Part II, p.8 and in §1.3, p.18
rectanglea four-sided figure with opposite sides equal, used here as the worked exampleprinted in §1.2, Part II, pp.7–8
correspondencethe pairing itself, considered as one objectprinted in §1.2, Part II, p.8
diagonalthe segment joining opposite corners of the rectanglean added term, not printed in this chapter — the book names the segment BD by its endpoints only
order-sensitivethe explanation's shorthand for a notation whose meaning depends on the order of its lettersan added coinage, not printed in this chapter

Where people slip up

  • "Once I know they are congruent, any way of writing it will do." The rectangle on pp.7–8 is the counterexample the chapter builds for exactly this. The triangles genuinely are congruent and the proposed statement is still wrong.
  • "ΔABC ≅ ΔXYZ is the same kind of sentence as AB = XY." The equals sign relates two numbers and does not care which is written first. ≅ relates two ordered triples of vertices. Say this out loud; it is the reason for everything else in the topic.
  • "You can reorder the letters as long as you keep the same triangle." ΔACB does name the same triangle as ΔABC. That is precisely why the trap works: the picture is unchanged and the claim is not. Only a matched reordering of both names survives.
  • "The letters are chosen to match, so just read them left to right." In the printed pair on p.6 the second triangle is drawn at a different tilt, and reading the two pictures left to right does not give the right pairing. The tick marks do. Train the eye on the marks.
  • "A pairing that lists each letter once must be valid." The failed table on p.8 lists each letter exactly once and is still wrong, because it sends a side of one triangle onto a side of the other that has no reason to match it.
  • "There is one right way to write a congruence." There are six ways to write the same pairing, and the chapter's own exercise on p.8 asks for the other five. Uniqueness is not the point; consistency between the two names is.

Worked answers to this chapter’s exercises · this video explains Figure it Out · 2 Q1, Figure it Out · 2 Q4, Figure it Out · 1.3 Q1, Figure it Out · 1.3 Q4

Transcript1,425 words

Here are two triangles, and they are congruent. The same shape, and the same size. One of them is sitting upright. The other has been turned and put down at a different angle. The corners of the first are labelled A, B and C. The corners of the second are X, Y and Z. Now here is the question. Which corner of the first one matches which corner of the second?

That has a definite answer, and it matters far more than it looks. Because when we write the congruence down, the writing has to carry that answer inside it. Not as a footnote underneath. As the order of the letters themselves. Get the order wrong and you have written something false, with the picture completely unchanged. So how do you find the matching? Not by reading the two pictures from left to right.

The second triangle has been turned. Left and right on the page tell you nothing at all. What tells you is the little marks on the sides. One side of each triangle carries a single stroke. Those two sides are the equal pair. One side of each carries a double stroke. One side of each carries a triple. Equal side has to land on equal side, so the strokes do the pairing for you.

Follow them and you get A with X, B with Y, and C with Z. Every corner is now spoken for, and not one step of that was a guess. That is one decision, and three lists fall out of it. The corresponding vertices first. A goes with X, B goes with Y, C goes with Z. Then the corresponding sides, which you read straight off the vertices without looking up.

The side from A to B goes with the side from X to Y. B to C goes with Y to Z. And C back round to A goes with Z back round to X. Then the corresponding angles, from that same pairing a third time. The angle at A goes with the angle at X, and so on down the list. Three lists, and not one of them needed a second look at the drawing.

Now the notation. There is a sign for this, and it is read as: is congruent to. Write triangle A B C, then the sign, then triangle X Y Z. And look carefully at what the order of those letters is doing. The first letter on the left is paired with the first letter on the right. The second with the second. The third with the third. Three little arrows running across the sign, and each one says: corresponds to.

So the statement is not merely reporting that the two triangles match. It is naming, in the order it is written, exactly which corner goes with which. Which means you cannot shuffle the letters on one side and leave the other side alone. Take the true statement and swap the last two letters of the first name only. Triangle A C B is congruent to triangle X Y Z. Now, A C B names the same triangle as A B C. Nothing whatsoever has moved on the board.

But the claim has changed. It now pairs C with Y, and B with Z. And that is simply false, because those sides are not equal. Swap the last two on both sides at once, though, and it survives. A C B with X Z Y is true. Permute both names the same way, or do not permute at all. That gives a small and rather useful fact. One pairing can be written in more than one way.

Suppose you are handed this. Triangle H E N is congruent to triangle B I G. H goes with B, E goes with I, N goes with G. Three pairs, and that is the entire content. And you can list three pairs in any order you please. There are six orders, so there are six ways of writing that same statement. Lead with the E pair and you get E H N is congruent to I B G.

Lead with the N pair and you get N E H is congruent to G I B. Six statements, one single claim, and five of them besides the one you started with. Now a figure where all of this really bites. Here is a rectangle. Its corners are A at the top left, B at the top right, C at the bottom right, D at the bottom left. Draw one segment, from B straight across to D.

That cuts the rectangle into two triangles, and we want to know whether they are congruent. Opposite sides of a rectangle are equal, so A B equals C D. That is one pair. And A D equals C B. That is a second pair. The third pair is the segment itself, which belongs to both triangles at once. Three pairs of equal sides. So the two triangles are congruent, and that much is settled.

So all that is left is to write it down. And this is exactly where it goes wrong. Suppose somebody proposes this pairing. A with C, B with B, and D with D. It looks tidy. Every letter appears exactly once, on each side. Nothing has been left out, and nothing has been used twice. It even leaves the two shared corners paired with themselves, which feels sensible enough. And it is wrong.

Being a neat list is not the test. It was never the test. The test is whether the pairing puts equal sides onto equal sides. So follow it through. Under that pairing A goes to C, and B goes to B. Which means the side from A to B is being claimed to land on the side from C to B. Cut the two triangles out of paper and slide them together the way the pairing says.

A B is a long side of the rectangle. C B is a short one. On the board here they are five and three. So a side of five is being laid on a side of three, and it hangs off the end. The two triangles really are congruent. The statement about them is still false. And that is the whole point. Being congruent is one thing. Writing it correctly is another.

So repair it, and repair it the only way that ever works. Follow the equal sides. A B has to land on something equal to it, and the only candidate is C D. That forces A to go with C, and B to go with D. Then A D has to land on C B, which sends D to B. And the shared segment goes to itself, which those two decisions have already arranged.

So triangle A B D is congruent to triangle C D B. I tried all six orderings of that second name, and exactly one of them comes out true. There is one repair. Not several, and not none. One warning now, because two different sixes live in this topic and they are easy to merge. The six we just counted was six ways of writing one single pairing. There is another six, and it counts something else entirely.

It asks how many different pairings between two triangles are true at the same time. For a triangle with three different sides the answer is one. The lengths leave no choice. For a triangle with exactly two equal sides it is two, because either of them can play either part. Only when all three sides are equal do all six pairings work at once. Cut a square along the line joining two opposite corners, and each half has exactly two equal sides. So: two.

One last figure, and it is the one that catches people out. Here is a dart. A point at the top, two points at the bottom, and a notch pushed up between them. Call the top corner D, the lower left F, the lower right G, and the notch E. The two upper sides are equal. The two lower sides are equal. And the line running from D down to E belongs to both halves of the shape.

So the halves are congruent. Three pairs of equal sides, exactly as before. But now write it down, and be careful, because only one ordering survives. D F E goes with D G E. The letters follow the equal sides. What the picture looks like is never the reason.

Where this fits

Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.

Builds on

Comes up again in

The book

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