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Q.  Can I claim my sister as a dependent if the custodial parent is receiving child support?

A. Simple answer: Yes. Any close relative, in the household, can claim her.  But, you do have to meet the full rules. In particular, she must be your younger sister (rule .

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test. Only a QC qualifies a taxpayer for the Earned Income Credit and the Child Tax Credit. They are interrelated but the rules are different for each.

The support test is different for each type. The support test, for a QC, is only that the child didn't provide more than half his own support. The support test for a Qualifying Relative is that the taxpayer provided more than half the relative's support.

.A child closely related to a taxpayer can be a “Qualifying Child (QC)” dependent, regardless of the child's income, if:

1. She is under age 19, or under 24 if a full time student for at least 5 months of the year, or  is totally & permanently disabled
2. She did not provide more than 1/2 his own support
3. She lived with the relative (including temporary absences) for more than half the year
4. She is younger than the relative (not applicable for a disabled child)
5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child (this essentially means that you have the parent’s permission to claim the child, if the child also lived with the parent more than half the year)
6. If the parents of a child can claim the child as a qualifying child but no parent so claims the child, no one else can claim the child as a qualifying child unless that person's adjusted gross income (AGI) is higher than the highest AGI of any of the child's parent (only the mom in your case) who can claim the child. This does not apply to the father's income since he does not live in the household. 

See full dependent rules at: https://turbotax.intuit.com/tax-tools/tax-tips/Family/Rules-for-Claiming-a-Dependent-on-Your-Tax-Ret...