Trigonometry/Proof: Angles sum to 180

In any triangle the angles always sum to $$180^{\circ}\,$$

Two Proofs
We gave two proofs of the Pythagorean theorem. The first one was short and, we hope, convinced you that the Pythagorean Theorem is true. The second followed Euclid and was more technical.

We're going to do exactly the same in proving that the sum of the angles in a triangle is 180 degrees.

Do you need to learn the proofs?

 * If you are learning trigonometry you need to know that the sum of angles in a triangle is $$180^{\circ}\,$$.


 * If you are learning trigonometry in preparation for an exam, check with your teacher whether you also need to learn a proof that the angles sum to $$180^{\circ}\,$$ at all. If you do need to learn a proof, you may need to learn the second proof. It all depends what your syllabus is.  The proof given first is not the one you will usually want for an exam, but it does give a clearer picture of why the angles sum to $$180^{\circ}\,$$.   Reading and understanding it will help you to understand more about the idea of mathematical proof.  The idea of proof gets more and more important as you progress in mathematics, so it is a good idea to get a headstart in understanding proof.  If you want to skip both proofs at this stage, that's fine too.  You can always come back to them later.  However, do not skip over all the proofs in this book.  Some of them are essential to understanding trigonometry.  You'll find it easier to learn the formulae of trigonometry if you also know how to prove them.  Doing 'more' is actually less work and more fun.

More About Proof
We saw earlier examples of triangles in which the angles add up to $$180^{\circ}\,$$. Just as with the Pythagorean Theorem, mathematicians want to know why it is so, and show that it is always so. It is not enough to show that it is true in lots of examples.

On this page we'll give two proofs.


 * The first proof shows something of why it is true. Unfortunately that first proof depends on some other facts about triangles which seem reasonable but for which we don't have a proof here that they are true.

Actually, when you try to prove something you always end up depending on other 'facts' which may be very reasonable but be 'facts' that you haven't proven. In the Pythagoras proof we relied on the idea that if you move shapes around then they keep the same area. That's very reasonable, but we haven't proved that.

If you haven't proven the facts that you depend on, have you proven the theorem? How far do you have to go to prove something? What 'facts' is it acceptable to choose as ones that you can rely on? These are not easy questions to answer. There is a way forward though. It is to have some kind of agreement about what facts one is allowed to use.

On the introduction page we mentioned the mathematician Euclid who lived around 2,300 years ago. He made some choices about what facts were reasonable to assume in geometry and trigonometry. Everything in geometry and trigonometry in his system should follow from those allowed principles.


 * The second proof on this page either directly or indirectly only uses the principles that Euclid allowed.

First Proof that angles sum to 180 degrees
The more traditional proof is later.



That's what we were trying to prove. We're done!
 * 1) First we draw some triangle.  Whilst the diagram on the right shows a particular example we can argue that our proof will work whatever triangle we drew.
 * 2) Next find the mid point of each edge, divide each edge in half.  Join up these midpoints with lines as shown in the next diagram to get four smaller triangles.
 * 3) The four smaller triangles are all congruent to each other, and each one is a quarter the size of the large triangle.  Each of the four smaller triangles is similar to the large triangle.  The angles are the same but the lengths of the sides are halved.
 * 4) Now look at the mid point of any one of the sides.  Three 'corners' meet there, and the three corners have one of each of the three sizes of angles.
 * 5) The sum of those three angles make a straight line, i.e. they sum to $$180^o\,$$


 * Whilst step 3 is very reasonable, we would actually need to do a bit more work to fully prove that step. A mathematician might say:
 * "I'm happy that in the middle triangle the sides are exactly one half of the sides of the original triangle, but you still haven't proved that this triangle is similar to the big triangle."
 * This just shows how careful we have to be. For us, this is OK.  We wanted this proof to show you why the theorem is true.  If we had to, and if the mathematician told us more precisely what assumptions we were allowed to make, we could fully prove that the middle triangle is congruent with the other small triangles.

<!-- The diagram on the right shows one rather nice approach to proving that in all triangles the angles sum to 180 degrees. We have to assume some things in a proof, and it may be that a mathematician would question even some of the very reasonable assumptions we are going to make. Let's see how it goes.

Firstly we say that we can divide any triangle into four smaller triangles as we've done for this one. That's pretty hard for anyone to disagree with. We just mark the mid point of each side and then join the three midpoints.

Next we say that the three outermost small triangles are just smaller versions of the large triangle. We can see that if we shrink the large triangle down towards the top point, until it is half the size it currently is, it will fit exactly over the top small triangle. Try it out with an example if you are not sure. When we shrink a shape the angles stay the same and all the sides decrease by the same proportion. In this case all the sides end up half the original length.

This is true for all of the three outer small triangles. They are all the 'congruent' to the large triangle, they are half its size, and they are congruent to each other.

''You may want to make a copy of the picture and mark on it which lengths and which angles we have proved are the same. Be careful! The triangle is not an equilateral triangle. Some of the angles look very similar, but they are not the same. Each of the three sides of the original triangle is a different length and that's true of the small triangles too. There is an answer here.''

Now we look at the small triangle in the centre. Each of its sides is half the length of one of the lengths in the large triangle. Check that we have proved this. This central triangle is also congruent to the large triangle.


 * This is where we might get into an argument with a mathematician. They might say "Have you really proved that the inner triangle is congruent to the outer triangle?  All you have proved is that its sides are in proportion, exactly half the size.  You haven't proved that the angles are the same as the large triangle's."


 * Well, at that point you have to question the mathematician and ask him or her to tell you what you are allowed to assume. Once they give you a list of things you are allowed to assume you can start looking for a different proof.


 * We've chosen to give this prettier proof first because it shows better what is going on. It is a more directly convincing demonstration that the angles sum to 180 degrees than other proofs.  We've also chosen to discuss proof early on in your mathematical education.  If no one takes the time to talk about proof it is very easy to wonder "What is the point of proof? Why not just show it is true for lots of examples".  A good discussion of proof will also explain that there are conventions about what you can assume and what you can't.  There are conventions about how big steps you are allowed to take in a proof and they depend on what kind of statement you are proving.

Anyway, we haven't quite finished this proof.

If you now look at any one of the midpoints of the large triangles sides you'll see that we've shown the three angles of the large triangle are present there. But we also know that we got this diagram by marking mid points on straight lines. So those three angles sum to a straight line, or in other words 180 degrees. So the three angles of our original large triangle sum to 180 degrees. -->

Variation of Euclid's Proof that the angles sum to 180

 * Triangle angle sum.svg

The two lines marked with $$\displaystyle >>$$ are parallel.


 * $$\angle 1 = \angle A$$. Euclid has a proposition about lines crossing parallel lines - that they cross at the same angle, and this is a consequence of that.
 * $$\angle 2 = \angle C$$. This is also true for the same reason.

On the top line we have


 * $$\angle 1 + \angle B + \angle 2 = 180^\circ$$

But since $$\angle 1 = \angle A$$ and because $$\angle 2 = \angle C$$ this is the same as the statement:


 * $$\angle A + \angle B + \angle C = 180^\circ$$

Which is what we wanted to show.