The Angle Sum of a Triangle

Here is a fact that holds for every triangle ever drawn: its three angles add to exactly one hundred and eighty degrees. You can prove it to yourself with paper. Draw any triangle at all, tear off the three corners and slide them together so their points touch. They always fit into a straight line, and a straight angle is one hundred and eighty degrees. The triangle here measures sixty-five, seventy-five and forty, and those three add to one hundred and eighty. Try it with a long thin triangle or a wide flat one and you still get the same total. That single fact is what makes almost every triangle question solvable.

Once you know the total is one hundred and eighty, two angles are always enough to find the third. Add the two you have been given, then take that total away from one hundred and eighty. In this triangle you know fifty-five and seventy-five. Fifty-five plus seventy-five is one hundred and thirty, and one hundred and eighty minus one hundred and thirty leaves fifty, so the missing angle is fifty degrees. Then check by adding all three back up: fifty-five plus seventy-five plus fifty really is one hundred and eighty. Adding the two known angles first and subtracting once is safer than subtracting them one at a time, because it gives you only one place to slip.

A right-angled triangle gives you a shortcut. One of its angles is already ninety degrees, which uses up half of the whole total, so the other two angles must share the remaining ninety between them. That means in any right-angled triangle the two other angles add to ninety. Here one of them is twenty-five, so the other is ninety minus twenty-five, which is sixty-five. You can always fall back on the full method, since one hundred and eighty minus ninety minus twenty-five gives the same sixty-five, but the shortcut saves a step. It also tells you both other angles must be acute, and that a triangle can never have two right angles.

An isosceles triangle hands you extra information, because its two equal sides sit opposite two equal angles. If you are told the odd angle at the top, subtract it from one hundred and eighty and then halve what is left, since the two remaining angles are equal to each other. An apex of forty leaves one hundred and forty to share, so each base angle is seventy. Going the other way, if you are told a base angle, double it first and then subtract from one hundred and eighty. Base angles of fifty-five use up one hundred and ten, leaving seventy for the apex. Always check which angle you were actually given, because the two routes are different.

You can also use the angle sum the other way round, to test whether a set of angles is even possible. Add all three. If the total is exactly one hundred and eighty then the triangle can exist, and if the total is anything else it cannot. Sixty, seventy and fifty add to one hundred and eighty, so that one is fine. Ninety, sixty and forty add to one hundred and ninety, so no such triangle exists, and the correct answer is to say so rather than bend a number to make it fit. There is a second condition as well: every angle must be bigger than zero, because a zero angle folds the triangle flat.