A straightedge is a ruler-like device that has no measurements marked on it; it is used to construct lines through any two given points. We investigate the famous question of which geometric figures can be constructed using only a straightedge and compass. It is easily shown that angles such as those of 60 degrees, 45 degrees, and 30 degrees are all constructible. On the other hand, it is quite difficult to prove that an angle of 20 degrees is not constructible and that, therefore, an angle of 60 degrees cannot be trisected (that is, divided into three equal parts) using only a straightedge and compass. We present a complete proof of this famous and remarkable fact. This is accomplished by studying the properties of “constructible numbers,” which also produces corollaries concerning constructibility of other geometric figures.