Question

What makes a triangle a “special” right triangle?

Answer 1

The special right triangles that I remember off the top of my head are ones that have angles 30-60-90 and angles 45-45-90. There are also specials ones with the sides (also multiples of) 3-4-5 and 5-12-13.

Answer 2

these triangles have rules written about them that aid in quickly solving problems.

Answer 3

Some good sites for those rules:
http://en.wikipedia.org/wiki/Special_right_triangles
http://mathworld.wolfram.com/RightTriangle.html
http://en.wikipedia.org/wiki/Right_triangle