The theorem suggested for congruency between triangles OMN and NMO is likely the Side-Angle-Side (SAS) Postulate, but more context is necessary to determine the exact theorem. Theorems such as ASA or AAS could also be relevant.
The theorem that makes triangle OMN congruent to triangle NMO in triangle measures is likely the Side-Angle-Side (SAS) Postulate, which states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent. However, without a specific diagram or additional context, it is challenging to determine the exact theorem applicable. The fact that the angles in a triangle always add up to 180 degrees might also be relevant in showing that the two mentioned triangles are congruent if they share angles and sides of the same measure.
From the information provided, other possibilities could include the Angle-Side-Angle (ASA) Postulate or the Angle-Angle-Side (AAS) Theorem, both of which also prove triangle congruency. It is important for students to carefully analyze the given information and the specific triangles mentioned in the question to apply the correct congruency theorem.