Stanbon is right that the triangle is the basis for so many structures.
Go where they are building a building.  How many triangles do you see?

Maybe the answer to your question is that if we understand the
right triangle, then we understand all triangles because every 
triangle can be split into two right triangles, by drawing a line 
segment from one of the vertices to the opposite side.  For instance, 
take this triangle, which is not a right triangle at all.



But look what happens if we draw a perpendicular from
the top vertex to the bottom side:



We have split it into two right triangles:



Edwin