Among all the different right triangles, there are two particular shapes that show up again and again in mathematics. They are called special right triangles, and they are often referred to by their angles: 30-60-90 and 45-45-90.
If we take an equilateral triangle and slice it in half through one of the vertices, each half is a 30-60-90 triangle. Using geometry, we can figure out the relationships between the three sides. These relationships are always the same in 30-60-90 triangles, and give us a rule that we can use help find the lengths of the side (if we know one side, we can find the others). This video explains the rule and shows how to use it:
The other special right triangle is formed by cutting a square in half diagonally. Each half is 45-45-90 triangle. These triangles are even easier to work with, since two of the sides are equal.
These rules may seem arbitrary — why do we use these particular numbers, why are there radicals involved in certain sides but not others, and so on? In fact, these rules come from geometry. The following videos show how these rules were created (the first covers 30-60-90 triangles, and the second covers 45-45-90 triangles).
Note: In our examples of 30-60-90 triangles above, the shorter leg is referred to as x — but in the video below, x represents the hypotenuse. This makes the resulting rule look slightly different, but the relationships between the sides are the same.