Measuring the angle between bone structures is a routine task in medical image analysis and provides a key quantitative parameter for diagnosis and treatment planning. Automated methods can reduce time and cost while improving reproducibility. In this work, we address automatic bone pose estimation using a learning-based point candidate proposal followed by a line model to extract axis parameters. Since conventional line models such as least squares are sensitive to outliers, we incorporate false-positive reduction strategies and robust fitting techniques, such as RANSAC and Hough transforms, to improve robustness. We evaluate our method on three clinically relevant paediatric angle estimation tasks: fracture fragment assessment in radiographs and ultrasound and developmental dysplasia of the hip evaluation in ultrasound using the Graf method. Our approach achieves mean errors of $4.1^\circ$, $5.4^\circ$, and $5.51^\circ$, respectively, not only remaining within the expected clinical observer variability, but also significantly outperforming landmark-based methods. Our code and annotations for fracture angle assessment in radiographs are publicly available on GitHub.