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Lines A & B are simply the parents name and SSN.  Pub 17 is not always accurate and has not been fully updated to comply with the new 2018 tax law, but the form instructions have been.

 

Also see IRS tax topic https://www.irs.gov/taxtopics/tc553 and pub 929 https://www.irs.gov/pub/irs-pdf/p929.pdf

 

All of the IRS publications are an interpretation of the actual law which is the authority.  The tax law is  26 U.S. Code § 1.   https://www.law.cornell.edu/uscode/text/26/1

 

[quote]

(2) Child to whom subsection appliesThis subsection shall apply to any child for any taxable year if—

(A) such child—
(i)
has not attained age 18 before the close of the taxable year, or
(ii)
(I)
has attained age 18 before the close of the taxable year and meets the age requirements of section 152(c)(3) (determined without regard to subparagraph (B) thereof), and
(II)
whose earned income (as defined in section 911(d)(2)) for such taxable year does not exceed one-half of the amount of the individual’s support (within the meaning of section 152(c)(1)(D) after the application of section 152(f)(5) (without regard to subparagraph (A) thereof)) for such taxable year,
(B)
either parent of such child is alive at the close of the taxable year, and
(C)
such child does not file a joint return for the taxable year.

[end quote]

 

Note that the part of the tax code that taxes the child income at the parents tax rate was changed by the Tax Cuts and Jobs Act (TCJA) passed by congress in 2017 to tax at the qualified dividend rate instead.

**Disclaimer: This post is for discussion purposes only and is NOT tax advice. The author takes no responsibility for the accuracy of any information in this post.**