We present a general solution to the long-standing problem of determining the masses of pair-produced, semi-invisibly decaying particles at hadron colliders. We define two new transverse kinematic variables M(CT)(⊥) and M(CT)(∥), which are suitable one-dimensional projections of the contransverse mass M(CT). We derive analytical formulas for the boundaries of the kinematically allowed regions in the (M(CT)(⊥),M(CT)(∥)) and (M(CT)(⊥),M(CT)) parameter planes and introduce suitable variables D(CT)(∥) and D(CT) to measure the distance to those boundaries on an event per event basis. We show that the masses can be reliably extracted from the end-point measurements of M(CT)(⊥)(max) and D(CT)(min) (or D(CT)(∥)(min)). We illustrate our method with dilepton tt events at the LHC.