A Thurston map whose postcritical set consists of exactly four points and for which the local degree at each of its critical points is 2 is called textit{nearly Euclidean}. These maps were specified to parse Thurston's combinatorial characterization of rational functions. We determine an extension of the half-space theorem which provides an open hyperbolic half-space such that the negative reciprocal of any fixed slope value is excluded from the boundary of the half-space. / Master of Science / Thurston proved necessary and sufficient conditions under which a certain class of mappings defined topologically are equivalent, in a precise sense which can be considered less strict than topological conjugacy, to a rational map. The conditions presented in the proof of this theorem are not ones for which computational algorithms are easily admitted in all settings. Nearly Euclidean Thurston maps are a sub-class of the maps to which this theorem is applicable and for which an abundance of information is algorithmically attainable. We extend a theorem in this setting. One main example which speaks to the utility of this extension is in determining when certain rational maps arise as matings of polynomials.