(Reuters) - The South Dakota House of Representatives failed on Thursday to override a veto by Governor Dennis Daugaard of a bill that would have made the state the first in the United States to dictate what bathrooms transgender students can use in public schools.

By a vote of 36-29, supporters of the bill did not muster the two-thirds vote required in both chambers to override the veto. The Republican-controlled House had approved the bill beyond that threshold in January.

The proposed law would have required transgender pupils to use restrooms and locker rooms in public schools that correspond to their gender at birth.

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Supporters, including conservative Christian groups such as Family Heritage Alliance Action, said the bill would enhance the privacy of all students.

But civil rights groups said the measure would expose schools to legal challenges over access to restrooms and that it violated Title IX, a federal rule regarding discrimination in public schools.

"Fairness and equality have prevailed over this unconscionable legislative assault on transgender children," said Chad Griffin, president of the Washington-based Human Rights Campaign, a gay, lesbian, bisexual and transgender civil rights organization, reacting to the vote on Thursday.

The bill would have required schools to provide "reasonable" accommodations for transgender students. Those accommodations include a single-occupancy restroom, a unisex restroom or the supervised use of a restroom, locker room or shower room designated for use by faculty.

In vetoing the bill on Tuesday, Daugaard, a Republican, said it would invite conflict and litigation, diverting resources from education.

In December, a suburban Chicago school district reached an accord with the U.S. government over locker room access for a transgender student after the U.S. Department of Education's Office of Civil Rights found the district discriminated against the student in violation of Title IX.

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Daugaard's veto came about a month after a U.S. appeals court heard arguments over whether a high school in Virginia should be ordered to allow a transgender male student to use the boys' bathroom.

Last week, local lawmakers in Charlotte, North Carolina, voted to allow transgender people to use public bathrooms matching their gender identity. Republican State House of Representatives Speaker Tim Moore said he would consider legislation to block the measure.