What are some good interview techniques that would help to find the angle that you need?

The Laramie Project's creator, Moisés Kaufman, and his theater company, the Tectonic Theater Project, are known for the form of documentary theater. This type of theater's goal is to engage with current events by incorporating the actual words and voices of the people most closely involved in the event. In a way, the documentary playwright partly takes on the role of a documentarian or a journalist.
To "find their angle," it's important for the interviewer to encourage their subjects to talk freely, naturally, and at length. A helpful starting technique to reach that goal would be to favor open-ended questions (e.g., ones involving "why" or "how") that are more likely to garner detailed responses. These are preferable to closed questions, which can be answered with a simple "yes," "no," or short phrase. It would be helpful for the interviewer to come to the interview prepared with a list of questions already generated, but they should also be flexible enough to ask insightful follow-up questions when more information or detail is needed. Even if the interviewer has an initial angle in mind at the start of the research and interview process, they should keep an open mind to chase new discoveries and hidden angles that may present themselves as they learn more about the event and the perspectives of the people most directly affected by it.