This paper reports on data from investigations on learners' understanding of inclusion relations of quadrilaterals, building on the ideas from our earlier study (Fujita & Jones, 2007). By synthesising past and current theories in the teaching of geometry (van Hiele's model, figural concepts, prototype phenomenon, etc.), we propose a theoretical model and method to describe learners' cognitive development of their understanding of inclusion relations of quadrilaterals, and in order to investigate the topic, data are collected from trainee teachers and lower secondary school students. The findings suggest that in general more than half of above average learners are likely to recognise quadrilaterals primarily by prototypical examples, even though they know the correct definition, and this causes them difficulty in understanding the inclusion relations of quadrilaterals. (Contains 8 tables and 3 figures.)