Scoliosis is a congenital disease in which the spine is deformed from its normal shape. Measurement of scoliosis requires labeling and identification of vertebrae in the spine. Spine radiographs are the most cost-effective and accessible modality for imaging the spine. Reliable and accurate vertebrae segmentation in spine radiographs is crucial in image-guided spinal assessment, disease diagnosis, and treatment planning. Conventional assessments rely on tedious and time-consuming manual measurement, which is subject to inter-observer variability. A fully automatic method that can accurately identify and segment the associated vertebrae is unavailable in the literature. Leveraging a carefully-adjusted U-Net model with progressive side outputs, we propose an end-to-end segmentation model that provides a fully automatic and reliable segmentation of the vertebrae associated with scoliosis measurement. Our experimental results from a set of anterior-posterior spine X-Ray images indicate that our model, which achieves an average Dice score of 0.993, promises to be an effective tool in the identification and labeling of spinal vertebrae, eventually helping doctors in the reliable estimation of scoliosis. Moreover, estimation of Cobb angles from the segmented vertebrae further demonstrates the effectiveness of our model.