Final answer:
The Iliolumbar ligament connects the transverse processes of L4 and L5 to the iliac crest. It helps stabilize the lumbar spine to the pelvis.
Step-by-step explanation:
The ligament that passes from the transverse process of L4 and L5 to the medial side of the iliac crest is known as the Iliolumbar ligament. This ligament is not specifically mentioned in the provided materials, but its anatomical position and function match the description given in the student's question. The ligaments discussed such as the posterior sacroiliac ligament, the sacrospinous ligament, and the sacrotuberous ligament serve different connecting functions between the sacrum, ischial spine, ilium, and ischial tuberosity. These ligaments are part of the complex network that supports the stability of the pelvis.