Giant cell arteritis (GCA) and Takayasu's arteritis (TAK) are diseases that cause swelling of the arteries in the head, neck, upper body, and arms. TAK specifically affects the aorta, the largest blood vessel in the body, and its branches. Therapies are available to improve the symptoms of GCA and TAK, but relapse often occurs, and better treatments are needed. Abatacept is a drug that interacts with certain cells in the body that are involved with GCA and TAK. This study will evaluate the effectiveness of abatacept in treating GCA and TAK and preventing disease relapse.