They never show them getting married but in the last episode of the Final Act Kagome calls up to Sesshomaruo and calls him big brother so THAT is how viewers know that they got married.
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they don't get married in any of the episodes but in the last chapter
of the manga they do after kagome graduates out of high school.
episode 100
Episode 18 of InuYasha the final Act
inuyasha and kagome don't get married yet in anime but they are married in manga version in chapter 558. kagome is trapped in her time for 3 years and is later reuided with inuyasha and lives in the past with her now husband inuyasha .
Episode 79: Jaken's Plan to Steal Tetsusaiga
There are only 6 seasons provided to the US and none has him proposing to her, but the manga shows in the last book it shows them married but it never showed InuYasha proposing to lady Kagome.