Which case reinforced the school's ability to control student speech on school property?

The case that reinforced the school's ability to control student speech on school property is Hazelwood v. Kuhlmeier. In this landmark decision, the Supreme Court ruled that schools have the authority to regulate student expression in school-sponsored activities, such as newspapers, if the actions are reasonably related to legitimate pedagogical concerns. This means that if a school's decision about content serves a valid educational purpose, it can restrict student speech without violating the First Amendment.

The context of Hazelwood is important; the decision specifically addressed instances where school-sponsored publications, like a student newspaper, were found to include material deemed inappropriate or not meeting the educational standards set by the school. This case established a standard whereby schools could exercise editorial control to maintain a certain educational environment, thus emphasizing the balance between a student's right to free expression and the school's role in fostering a productive educational setting.

In contrast, the other cases listed focus on different aspects of constitutional rights, such as libel (New York Times Co. v. Sullivan), desegregation (Brown v. Board of Education), and voluntary school prayer (Engle v. Vitale), which do not relate to the regulation of student speech within the educational context.