Use 1.2 and 1.3 above to prove that \( \triangle \mathrm{POQ} \equiv \triangle \mathrm{BOA} \)

To prove that \( \triangle \mathrm{POQ} \equiv \triangle \mathrm{BOA} \), we start by establishing two key elements: the relevant properties of the triangles and what conditions lead to triangle congruence. First, we note that the triangles must have corresponding sides equal in length and corresponding angles equal in measurement. From the information given in parts 1.2 and 1.3, let’s assume that we have established that \( OP = BO \), \( OQ = OA \), and the angles at point \( O \) are equal, i.e., \( \angle POQ = \angle BOA \). These established equalities can be viewed through the lens of congruence criteria, such as Side-Angle-Side (SAS). Secondly, having confirmed these equalities, we can invoke the SAS congruence criterion, which states that if two sides and the included angle of one triangle are respectively equal to two sides and the included angle of another triangle, then those triangles are congruent. Therefore, \( \triangle POQ \equiv \triangle BOA \) follows directly from these conclusions, showcasing the beauty of geometry in establishing relationships.