Final Answer:
The authority to establish and support public schools, establish local governments, and set rules for marriages and divorces falls under the state government's powers.
Step-by-step explanation:
These responsibilities are examples of powers delegated to state governments within a federal system. The establishment and support of public schools fall under the state's jurisdiction, allowing them to shape and regulate the education system within their boundaries. This includes decisions related to curriculum, funding, and policies affecting public schools.
Similarly, the establishment of local governments is a power vested in state authorities. States can create and define the structure, functions, and powers of local governments, such as counties, cities, and municipalities. This decentralized approach allows states to tailor governance to the specific needs of different regions.
Setting rules for marriages and divorces is also within the purview of state governments. While there may be overarching principles at the federal level, states have the authority to enact and enforce laws governing marriage and divorce, including age requirements, licensing procedures, and grounds for divorce.
In summary, these powers are part of the broader framework of federalism, where certain responsibilities are assigned to the state governments to allow for local variations and considerations.