Final answer:
The structure at risk when placing a retractor inferior to the transverse acetabular ligament is the ligament of the head of the femur, which contains an important artery for blood supply to the femoral head.
Step-by-step explanation:
The placement of a retractor inferior to the transverse acetabular ligament during a surgical procedure risks damaging the ligament of the head of the femur, also known as the ligamentum teres. This intracapsular ligament is crucial as it contains an artery that supplies blood to the head of the femur. Caution must be taken during surgery to avoid compromising the blood supply to this critical area of the hip joint.