The angles of a triangle

​ ​The triangle is perhaps the most important polygon because you can split up any polygon into triangles. And if you can calculate the area, angles, and lengths of a triangle's sides, then you can calculate almost anything. The angles of a triangle have a special quality. If you measure them carefully and add them together, you will always get the sum of 180 degrees. ​​This is a right triangle. So one of the angles is 90 degrees.

The other angles are 53 and 37 degrees. Ninety plus 53 plus 37 equals 180. The sum of the angles for this triangle is 180 degrees, and this is valid for any triangle as well. ​ ​Draw a triangle with a ruler on a piece of paper; cut it out and tear off the corners. You can place the corners next to each other like this. They form a half circle.

And a half circle is 180 degrees. No matter how you draw your triangle, the corners will always form a half circle if you place them together. ​ ​The triangle on the left is an isosceles triangle. You can tell this because two sides have marks, hence they are equal. Since two sides are equal, these angles will also be equal. We mark them with angle markers to show that they are equal.

Since we know how large the upper angle is, we can calculate how large the two lower equal ones are. All three angles add up to 180 degrees. And the upper angle equals 35. We then have 145 degrees left. The two lower angles must be equal.

Then they have to be 145 degrees divided by two. So each of the two equal angles is 72 and a half degrees. ​ ​This triangle is equilateral. All sides are equal. And if the sides of a triangle are equal, the angles have to be the same too. Try to draw an equilateral triangle where the angles are different, and you will understand why this is impossible.

We could mark the equal angles here as well, but for now, we'll settle for angle markers. Three equal angles add up to 180 degrees, which means that each of them has to be one-third of 180 degrees. 180 divided by three is 60 degrees. In an equilateral triangle, all three angles are always exactly 60 degrees. You never have to measure or calculate that because they can never be anything else in an equilateral triangle. ​ ​The sum of the angles in a triangle is always 180 degrees.

If you know two of the angles in a triangle, you can calculate the third one. A right angle has one angle that is equal to 90 degrees. An isosceles triangle has two equal angles. An equilateral triangle has three angles that are all equal to 60 degrees. All this is true for triangles drawn on a flat surface.

But feel free to draw a triangle on a ball. Then, almost anything is possible. You can draw a triangle with three right angles, for example. But then you are no longer within uclidian geometry anymore.