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Questions & Answers about Odam soğuk.
Why is it odam instead of adding a vowel to make it odaım or odaim?
The root word is oda (room), which already ends in a vowel. The suffix for "my" is generally -(I)m, which means it adds a vowel only if the word ends in a consonant. Since oda ends in a vowel, you skip the extra vowel and just attach the consonant -m to make odam.
Do I need to include the word benim (my) before odam?
No, it isn't necessary. The suffix -m at the end of odam already shows that the room belongs to you. You would only add benim (saying benim odam soğuk) if you wanted to add emphasis, like pointing out that your room is cold while someone else's is warm.