Trigonometry/Angles of a triangle sum to 180 Degrees
In any triangle the angles
always sum to
Contents
The Sum of Angles[edit]
In any triangle the angles always sum to
This is a perhaps surprising fact.
Because 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.
The examples suggest it is true, but they don't prove it.[edit]
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[edit]
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[edit]
Given any triangle with angles 123° and 60°. Evaluate the third angle. Is it possible?

A triangle has angles 15° and 65°, what is the third angle? 
A triangle has angles 100° and 79.5°, what is the third angle?

What is the measure of each angle of an equilateral triangle? 
Roadsign Exercise

The measure of one angle of an isosceles triangle is . What are the measures of the other two angles? 