Final answer:
The county had a legal basis for denying the right of African Americans to attend white schools in the case of Plessy v. Ferguson. However, this precedent was later overturned in the landmark case of Brown v. Board of Education.
Step-by-step explanation:
The Supreme Court case that gave the county a legal basis for denying the right of African Americans to attend white schools is Plessy v. Ferguson. This case, decided in 1896, established the 'separate but equal' precedent, allowing for racial segregation in public facilities. It was not until the landmark Supreme Court case of Brown v. Board of Education in 1954 that the principle of 'separate but equal' was overturned and the right of African Americans to attend white schools was recognized.