Object detection has achieved promising success, but requires large-scale fully-annotated data, which is time-consuming and labor-extensive. Therefore, we consider object detection with mixed supervision, which learns novel object categories using weak annotations with the help of full annotations of existing base object categories. Previous works using mixed supervision mainly learn the class-agnostic objectness from fully-annotated categories, which can be transferred to upgrade the weak annotations to pseudo full annotations for novel categories. In this paper, we further transfer mask prior and semantic similarity to bridge the gap between novel categories and base categories. Specifically, the ability of using mask prior to help detect objects is learned from base categories and transferred to novel categories. Moreover, the semantic similarity between objects learned from base categories is transferred to denoise the pseudo full annotations for novel categories. Experimental results on three benchmark datasets demonstrate the effectiveness of our method over existing methods. Codes are available at https://github.com/bcmi/TraMaS-Weak-Shot-Object-Detection.