Angles ∠MOK and ∠NOL are identical by opposite angle theorem.
∠M = ∠N through given information
∠MKO and ∠NLO are congruent. Since the other two angles of each triangle are equivalent, the last angle must be equivalent to the last angle of the other triangle. (180 - ∠1 -∠2 = ∠3); you could also prove ∠MKO = ∠NLO by stating that the opposite sides of the two angles are congruent.
ΔMKO and ΔNLO by Angle-Angle-Side theorem
since ΔMKO and ΔNLO, all corresponding parts of both triangles are congruent, so therefore MO ≅ NO