There are a lot of similarities between an angry customer and an angry student/parent/colleague. This article answers the question, what is the best way to move forward when anger happens, and breaks down next steps into a pretty simple structure of 3 phases after the initial empathy.
Initial empathy: I am sorry….
Phase 1: Sensing: ask questions to try and understand the issue
Phase 2: Seeking: brainstorm and explore potential solutions
Phase 3: Settling: working with the person to choose the solution that will provide the best outcome.
My two cents, it is important to demonstrate that how the person is feeling matters to you. Be sincere and attentive.