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Chapter 7 · A Tale of Three Intersecting Lines

Why the three angles of any triangle add to 180°

यह वीडियो हिंदी में भी · Watch in Hindi

Constructing from sides and angles9 min

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9 min.

Also recorded in Hindi.Englishहिन्दी

Measuring a hundred triangles gives a hundred totals near 180° and never tells you why. One line does.

The idea

Measuring a hundred triangles would give you a hundred totals near 180° and never tell you why. The chapter gets the why from a single line that was not there before: through the apex, draw a line running parallel to the side opposite it, and the triangle's three angles reappear along that line, where their total is already known. The extra line adds no information — it is not part of the triangle, and no measurement is taken on it — yet it converts a fact about a closed shape into a fact about a straight one. That move, an auxiliary construction, is the lesson; the 180° is what falls out of it.

What you should be able to do

  • Given two angles of a triangle, find the third by adding a parallel line and arguing, not by measuring
  • Identify the two pairs of alternate angles created when a line through the apex runs parallel to the base
  • Reproduce the general argument for the angle sum property using letters rather than chosen values
  • State the angle sum property and say what makes it a property of every triangle
  • Explain what an auxiliary line contributes when it carries no new measurement
  • Verify the property by folding a paper triangle, and say what the folding demonstrates and what it does not
  • Define an exterior angle of a triangle and locate one in a labelled figure
  • Compute an exterior angle from two given interior angles, in two steps
  • Notice the relation between an exterior angle and the two interior angles at the other vertices, and say how it follows

Words to know

TermDefinition in one lineFirst introduced
angle sum propertythe fact that the three angles of any triangle total 180°printed in bold in §7.3, p.166; the SUMMARY on p.171 states the fact but does not use the term
exterior angleat one vertex, the angle lying between the neighbouring side and the extension of the other oneprinted in bold in §7.3, p.167
straight anglethe 180° angle that a straight line makes at a point on itprinted on p.166 and again on p.167
alternate anglesthe equal pair on opposite sides of a transversal cutting two parallel linesprinted on p.164
transversala line cutting across two othersprinted on p.164 and earlier on p.163
parallelsaid of two lines in a plane that never meetprinted on pp.164–166
vertexa corner point of the triangleprinted on p.146 and again on p.166
trianglethe closed figure whose three angles this topic totalsprinted throughout, from p.146
The Elementsthe book by Euclid, around 300 BCE, that this argument is attributed toprinted on p.166
auxiliary linea line added to a figure purely so that an argument can be made about itan added term; the phrase is not printed in this chapter, though the line itself is drawn as XY in Fig. 7.6 and Fig. 7.7
exterior angle theoremthe name usually given to the relation set up at the end of p.167not printed in this chapter — the chapter builds the relation and leaves the student to state it

One caution: The chapter uses the word proved for the lettered argument on p.166. That is deliberate — it is the same standard set in Part I, Chapter 5 — so do not downgrade it to "we can see that". The paper fold on the same page is presented as a way of verifying, which is a different and weaker claim.

Where people slip up

  • "It adds to 180° because I measured lots of triangles and it did." Every measurement is approximate, and a hundred approximations do not make a certainty. The chapter's own reflection on p.166 is about precisely this — the relationship was invisible until a line was added, and then it was forced.
  • "The parallel line is part of the triangle." It is not. Nothing in the triangle changes when XY is drawn, and no length on XY is ever used. It exists so that three angles that were scattered can be seen sitting on one line.
  • "A very large triangle would have a bigger angle sum." Size is irrelevant; the argument never mentions a length. This is the same lesson as the previous topic's "the base does not matter", arriving one step further on.
  • "The paper fold is the proof." It is a check. Folding shows the three angles fitting along a straight edge for that triangle, cut with scissors and folded by hand. The chapter separates the two on the same page — it uses the language of proof for the lettered argument and the language of verifying for the fold.
  • "A triangle can have two right angles if it is big enough." Two right angles already use the whole 180°, leaving nothing for the third. Note that this is the same fact the previous topic reached from the parallel-line side, now available as a one-line consequence.
  • "The exterior angle is outside, so it is the angle between the two extended sides." It is measured between one extended side and the side next to it at the same vertex, and it shares an arm with the interior angle there. Reading it off the wrong pair of arms is the standard error, and the figure on p.167 is what fixes it.
  • "Euclid proved it, so it is true." The chapter's point is the opposite way round: the argument is what makes it true, and it is credited to The Elements because that is where this particular idea is found.

Worked answers to this chapter’s exercises · this video explains Figure it Out · 8 Q1, Figure it Out · 8 Q2, Figure it Out · 8 Q3

Transcript1,348 words

Here is a loose end worth pulling on. Two angles, sixty and seventy, with a five centimetre side between them. Build it, then measure the third angle. Fifty degrees. Now the same two angles on a seven centimetre side. A bigger triangle, the same two corners. Measure the third angle again. Fifty degrees. It did not move. Try a base of half a centimetre, or forty, and it still will not.

Which is suspicious. If the third angle is completely decided by the other two, there ought to be a rule. There is. But finding it by measuring is exactly what will not work. So here is the target. A triangle. Fifty degrees at B, seventy at C. What is the angle at A? You could draw it carefully and lay a protractor on it. Let us not. A protractor reads to the nearest degree, and a hand-drawn line is a hand-drawn line.

I measured a thousand triangles that way, rounding every angle as a protractor would. Seven hundred and forty-eight of them came to a hundred and eighty. The other two hundred and fifty-two read a hundred and seventy-nine, or a hundred and eighty-one. And every single one of those thousand triangles really does total a hundred and eighty. It was the measuring that wobbled. So no amount of measuring will settle this. Something has to be argued instead.

And the argument opens with something odd. Draw a line. Not a side of the triangle — a new line, through A, running parallel to B C. Call its two ends X and Y. Now look at what that did to the triangle. Nothing. Not one corner moved. Not one side changed length. The line is not part of the shape, and no length on it will ever be measured.

It is there for exactly one reason. So that something can be said. A line added purely so an argument can be made about it. Watch what it buys. X Y runs parallel to B C. And A B cuts straight across both of them. A line cutting across two parallels like that is a transversal. Which means the angle up at A on the X side, between X A and A B, is an alternate angle to the one at B.

Alternate angles across a transversal are equal. So that angle at A is fifty degrees — the same as the angle at B. Now take A C. That cuts across both parallels too. So the angle at A on the Y side, between A C and A Y, matches the angle at C. Seventy degrees. And here is the thing. Neither of those was measured. Both were argued. The angle from B has turned up at A. So has the angle from C.

So count what is sitting at A now. Three angles, side by side. Fifty on the left. The triangle's own angle in the middle. Seventy on the right. And all three of them lie along X Y, which is a straight line. Angles filling a straight line come to a hundred and eighty degrees. That much you already knew. So fifty, plus the angle at A, plus seventy, makes a hundred and eighty.

Which puts the angle at A at a hundred and eighty, minus fifty, minus seventy. Sixty degrees. Go and draw it if you like. It will be sixty. Now the step that turns one answer into every answer. Rub out the numbers. Call the three angles simply A, B and C. Draw the same line through the apex, parallel to the opposite side. A B is still a transversal, so the angle on the X side still equals the angle at B.

A C is still a transversal, so the angle on the Y side still equals the angle at C. And the three angles at the apex still lie along one straight line. So angle B, plus angle A, plus angle C, is a hundred and eighty degrees. No numbers went in and nothing was measured. That is a proof, and it holds for every triangle there is. That is the angle sum property. The three angles of any triangle total a hundred and eighty degrees.

Any triangle. Enormous ones, needle-thin ones, ones nobody has drawn yet. And it is worth stopping to notice how strange that is. Nothing about a triangle looks as though it has a straight line hidden inside it. The three corners point in three different directions. The relationship was simply invisible. Then one extra line, carrying no information whatsoever, and it was forced. It settles some things instantly, too. Two right angles use up the whole hundred and eighty, leaving nothing at all for the third corner.

Two obtuse angles are worse. Ninety-one and ninety-one is already a hundred and eighty-two. So a triangle gets at most one of each. This argument is old. It is credited to Euclid, who was working around three hundred years before the common era. He gathered it, along with a great deal else, into a set of books called The Elements. That is roughly twenty-three centuries ago. The line through the apex has been doing this same job ever since.

Though it is worth being careful about why any of us believe it. Not because Euclid said so. The argument is what makes it true. The name attached to it is history, not authority. There is also something you can do with your hands. Cut a triangle out of paper. Any triangle at all. Fold the top corner straight down until its point touches the bottom edge. You are left with a four-sided shape, and one corner of the triangle now lying flat along the bottom.

Now fold the two bottom corners inwards to meet it. All three corners are lying side by side along a single straight edge. Which is lovely. And it is not the proof. It shows three angles fitting along a line for that one triangle, cut with scissors and folded by hand. The argument covers all of them at once. One more idea, and this one comes free. Take the triangle again, and extend one of its sides.

B to C, carried on past C to a new point, D. There is a new angle at C now. Between C A and C D. That is called an exterior angle. And it is worth being careful about which angle it actually is. It is not the angle between the two extended sides. That one sits somewhere else, and it comes out equal to the inside angle instead. The exterior angle shares an arm with the interior one, and together the two of them lie along the straight line B C D.

So work one out. The angle at A is fifty. The angle at B is sixty. First, the angle inside the triangle at C. The three of them total a hundred and eighty, so C is a hundred and eighty, minus fifty, minus sixty. Seventy degrees. Now the exterior angle. It lies on the straight line with the interior one. A hundred and eighty, minus seventy. A hundred and ten degrees.

Two steps, and neither of them was a measurement. Now put those two facts side by side, because there is something hiding in them. The three angles inside the triangle total a hundred and eighty. The interior and the exterior angle at C also total a hundred and eighty. So the angle at C turns up in both, and it is the only thing they do not have in common.

Take it out of each, and whatever is left has to match. The exterior angle at C equals the angle at A plus the angle at B. Check it against the one we just did. Fifty and sixty make a hundred and ten, which is the exterior angle exactly. I put it through every whole triangle there is — fifteen thousand nine hundred and thirty-one of them — and the exterior angle was the other two added together, every single time.

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

Either side of this one

The book

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