Given: RO = HE, ∠THO = ∠MER, TO ǁ MR

Prove: △THO ≅ △MER

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Given
HO = RO + EO and ER = EO + RO
HO = HE + EO and ER = EO + RO
△THO = △MER
Substitution
HO = ER
Parallel lines form equal corresponding angles
RO = HE, ∠THO = ∠MER, TO ǁ MR
ASA (Angle-Side-Angle)
Substitution
∠TOH = ∠MRE
Segment addition theorem