Reasserting the Epistemic Role of Non-Parallel Line Contexts in Introducing Angle Relationships

Abstract

Angle relationships are an important topic in Euclidean plane geometry and form a conceptual basis for understanding line parallelism. In scholarly geometry, these relationships were introduced through configurations of two non-parallel lines intersected by a transversal. However, in several instructional contexts, angle relationships were presented first through the context of parallel lines. This practice risked obscuring the epistemic status of angle relationships within the structure of Euclidean geometry. This study aimed to reaffirm the epistemic role of non-parallel line contexts as the conceptual foundation for introducing angle relationships. A qualitative study was adopted. Data were collected through both of the analysis of mathematical documents such as geometry textbooks and Focus Group Discussion (FGD) involving scholars in geometry, pure mathematics, and advanced mathematics education. The analysis focused on logical coherence and structural relationships among concepts. The results showed that the configuration of non-parallel lines provided the most conceptually neutral framework for introducing angular relationships. This can be seen, among other things, from the use of the term special pair of angles (in this case same-side interior angles) in Euclid's parallel postulate. Consequently, angle congruences or supplementary were interpreted as consequences of Euclid’s fifth postulate rather than as their conceptual origin. These findings highlight the epistemic necessity of introducing angle relationships through non-parallel contexts to preserve coherence and conceptual hierarchy in Euclidean geometry.