Trigonometry/Angles of a triangle sum to 180 Degrees

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

The Sum of Angles
In any triangle the angles always sum to $$180^\circ$$

This is a perhaps surprising fact.

Because $$90^\circ$$ is a right angle, it means that the sum of the angles of any triangle is the same as two right angles. If we 'tore the corners off' and placed them together at the same point, we could arrange them so that they exactly formed a straight line. There doesn't need to be anything special about the triangle. It works for any triangle.

Some examples that we had before of triangles are shown below


 * The first example shows an equilateral triangle. All of the sides are equal. All of the angles are equal. Each angle is 60 degrees. The sum of the angles is $$60^\circ+60^\circ+60^\circ$$ which is $$180^\circ$$.
 * The second triangle shows a right angle triangle. One of the angles is a right angle. This right angle triangle has two sides the same length. It is symmetric. It fulfills our criteria for being an isosceles triangle. This is a particularly special isosceles triangle because it is isosceles and it is a right triangle.  There is one angle of 90° and each of the two remaining angles is 45°. The sum of the angles is $$45^\circ+45^\circ+90^\circ$$ which is $$180^\circ$$.
 * The third triangle is sometimes called the 30°-60°-90° triangle, because of its angles. It is actually half an equilateral triangle. The sum of the angles is $$30^\circ+60^\circ+90^\circ$$ which is $$180^\circ$$.

The pattern is pretty clear. -->
 * Next we have a more arbitrary triangle. All the sides are different. The angles are 50°, 60° and 70°. The sum of the angles is $$50^\circ+60^\circ+70^\circ$$ which is $$180^\circ$$.
 * Finally we have a triangle with an obtuse angle, that is one of the angles is larger than 90°. The angles happen to be 20°, 40° and 120°, and the sum of the angles is $$20^\circ+40^\circ+120^\circ$$ which is $$180^\circ$$.

The examples suggest it is true, but they don't prove it.
We could keep on doing this for other triangles, and keep finding the same answer, unless we make a mistake. This might convince us that our statement that the angles sum to 180 is true for all triangles, but it does not prove that it is so. To prove it we need some kind of general argument that could convince a mathematician that it is true. How do we know it is always true?

How could it go wrong? Well, if we hadn't tried with a triangle with an obtuse angle, it might be the case that the formula only works for triangles which don't have obtuse angles. Even having tried the triangle with an obtuse angle we could have not been trying hard enough to find an example that doesn't work. For all we know the formula only works if the angles are multiples of 5°.

Proof will show it works for all triangles
The formula does in fact work for all triangles. We can for example make a triangle with angles of 33° and 66° and the third angle will have to be 81°. Making more and more examples unfortunately doesn't get us anywhere closer to proving it is true of all triangles. We need a different approach. We'll show a proof later. The point of having a proof is to show that it is true for all triangles, not just the ones we've chosen to look at.

Exercises

 * It is not possible because the sum of all angles of a triangle cannot exceed 180°.


 * 100


 * 0.5
 * Do you think all the sides of this triangle will be about the same length?


 * 60

The following road signs from Tanland show how steep the road ahead is. Put the road signs in order, least steep to steepest.
 * In these signs a sign that shows, for example, 5:8 means that the road is 5m higher when you've travelled 8m horizontally.


 * 36 each