The votes Wednesday send the bills to the House, while more bills on the subject are being considered by each chamber.
A task force issued detailed recommendations late last year, providing a blueprint for the bills.
The bills approved by the Senate would set broader, statewide rules for who is responsible for a child's welfare, who can be considered a perpetrator of child abuse and who must report a case of suspected child abuse to authorities.
Another bill would require medical professionals to report a case of suspected child abuse immediately to the county child welfare agency.
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